{"id":850,"date":"2017-10-27T16:30:56","date_gmt":"2017-10-27T16:30:56","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/the-simple-pendulum\/"},"modified":"2017-11-08T03:25:39","modified_gmt":"2017-11-08T03:25:39","slug":"the-simple-pendulum","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/the-simple-pendulum\/","title":{"raw":"The Simple Pendulum","rendered":"The Simple Pendulum"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Measure acceleration due to gravity.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id3178394\">\n<div class=\"bc-figcaption figcaption\">A simple pendulum has a small-diameter bob and a string that has a very small mass but is strong enough not to stretch appreciably. The linear displacement from equilibrium is [latex]s[\/latex], the length of the arc. Also shown are the forces on the bob, which result in a net force of [latex]-\\text{mg}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\theta [\/latex] toward the equilibrium position\u2014that is, a restoring force.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2075212\" data-alt=\"In the figure, a horizontal bar is drawn. A perpendicular dotted line from the middle of the bar, depicting the equilibrium of pendulum, is drawn downward. A string of length L is tied to the bar at the equilibrium point. A circular bob of mass m is tied to the end of the string which is at a distance s from the equilibrium. The string is at an angle of theta with the equilibrium at the bar. A red arrow showing the time T of the oscillation of the mob is shown along the string line toward the bar. An arrow from the bob toward the equilibrium shows its restoring force asm g sine theta. A perpendicular arrow from the bob toward the ground depicts its mass as W equals to mg, and this arrow is at an angle theta with downward direction of string.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_04_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"In the figure, a horizontal bar is drawn. A perpendicular dotted line from the middle of the bar, depicting the equilibrium of pendulum, is drawn downward. A string of length L is tied to the bar at the equilibrium point. A circular bob of mass m is tied to the end of the string which is at a distance s from the equilibrium. The string is at an angle of theta with the equilibrium at the bar. A red arrow showing the time T of the oscillation of the mob is shown along the string line toward the bar. An arrow from the bob toward the equilibrium shows its restoring force asm g sine theta. A perpendicular arrow from the bob toward the ground depicts its mass as W equals to mg, and this arrow is at an angle theta with downward direction of string.\" width=\"225\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2589696\">Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child\u2019s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A <span data-type=\"term\">simple pendulum<\/span> is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in <a href=\"#import-auto-id3178394\" class=\"autogenerated-content\">(Figure)<\/a>. Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period. <\/p>\n<p id=\"import-auto-id1401960\">We begin by defining the displacement to be the arc length <em data-effect=\"italics\">[latex]s[\/latex]<\/em>. We see from <a href=\"#import-auto-id3178394\" class=\"autogenerated-content\">(Figure)<\/a> that the net force on the bob is tangent to the arc and equals [latex]-\\text{mg}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex]. (The weight [latex]\\text{mg}[\/latex] has components [latex]\\text{mg}\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] along the string and [latex]\\text{mg}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex]<em data-effect=\"italics\"> tangent to the arc.) Tension in the string exactly cancels the component [latex]\\phantom{\\rule{0.25em}{0ex}}\\text{mg}\\phantom{\\rule{0.25em}{0ex}}\\text{cos}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] parallel to the string. This leaves a <em data-effect=\"italics\">net<\/em> restoring force back toward the equilibrium position at [latex]\\theta =0[\/latex].<\/em><\/p>\n<p id=\"import-auto-id1999298\">Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about [latex]\\text{15\u00ba}[\/latex]), [latex]\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta \\approx \\phantom{\\rule{0.25em}{0ex}}\\theta \\phantom{\\rule{0.25em}{0ex}}[\/latex]([latex]\\text{sin}\\phantom{\\rule{0.25em}{0ex}}\\theta [\/latex] and [latex]\\theta [\/latex] differ by about 1% or less at smaller angles). Thus, for angles less than about [latex]\\text{15\u00ba}[\/latex], the restoring force <em data-effect=\"italics\">[latex]F[\/latex]<\/em> is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-64\">[latex]F\\approx -\\text{mg}\\mathrm{\\theta .}[\/latex]<\/div>\n<p id=\"import-auto-id2639190\">The displacement <em data-effect=\"italics\">[latex]s[\/latex]<\/em> is directly proportional to [latex]\\theta [\/latex]. When [latex]\\theta [\/latex] is expressed in radians, the arc length in a circle is related to its radius (<em data-effect=\"italics\">[latex]L[\/latex]<\/em> in this instance) by:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]s=\\mathrm{L\\theta },[\/latex]<\/div>\n<p id=\"import-auto-id1120184\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\theta =\\frac{s}{L}.[\/latex]<\/div>\n<p id=\"import-auto-id3116588\">For small angles, then, the expression for the restoring force is:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]F\\approx -\\frac{\\text{mg}}{L}s[\/latex]<\/div>\n<p id=\"import-auto-id1917186\">This expression is of the form:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]F=-\\text{kx},[\/latex]<\/div>\n<p id=\"import-auto-id2672250\">where the force constant is given by [latex]k=\\text{mg}\/L[\/latex] and the displacement is given by [latex]x=s[\/latex]. For angles less than about [latex]\\text{15\u00ba}[\/latex], the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator.<\/p>\n<p id=\"import-auto-id1588131\">Using this equation, we can find the period of a pendulum for amplitudes less than about [latex]\\text{15\u00ba}[\/latex]. For the simple pendulum:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-424\">[latex]T=2\\pi \\sqrt{\\frac{m}{k}}=2\\pi \\sqrt{\\frac{m}{\\text{mg}\/L}}.[\/latex]<\/div>\n<p id=\"import-auto-id3080441\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-152\">[latex]T=2\\pi \\sqrt{\\frac{L}{g}}[\/latex]<\/div>\n<p id=\"import-auto-id1917752\">for the period of a simple pendulum. This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. As with simple harmonic oscillators, the period [latex]T[\/latex] for a pendulum is nearly independent of amplitude, especially if <em data-effect=\"italics\">[latex]\\theta [\/latex]<\/em> is less than about [latex]\\text{15\u00ba}[\/latex]. Even simple pendulum clocks can be finely adjusted and accurate.<\/p>\n<p id=\"import-auto-id1080305\">Note the dependence of [latex]T[\/latex] on <em data-effect=\"italics\">[latex]g[\/latex]<\/em>. If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity. Consider the following example.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1538976\">\n<div data-type=\"title\" class=\"title\">Measuring Acceleration due to Gravity: The Period of a Pendulum<\/div>\n<p id=\"import-auto-id1418508\">What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s?<\/p>\n<p id=\"import-auto-id2382213\"><strong>Strategy<\/strong><\/p>\n<p>We are asked to find <em data-effect=\"italics\">[latex]g[\/latex]<\/em> given the period [latex]T[\/latex] and the length <em data-effect=\"italics\">[latex]L[\/latex]<\/em> of a pendulum. We can solve [latex]T=2\\pi \\sqrt{\\frac{L}{g}}[\/latex] for <em data-effect=\"italics\">[latex]g[\/latex]<\/em>, assuming only that the angle of deflection is less than [latex]\\text{15\u00ba}[\/latex].<\/p>\n<p id=\"import-auto-id3079401\"><strong>Solution<\/strong><\/p>\n<ol id=\"fs-id1381777\" data-number-style=\"arabic\">\n<li id=\"import-auto-id1413046\">Square <em data-effect=\"italics\">[latex]T=2\\pi \\sqrt{\\frac{L}{g}}[\/latex] and solve for [latex]g[\/latex]:<br>\n    [latex]g={4\\pi }^{2}\\frac{L}{{T}^{2}}.[\/latex]<\/em><\/li>\n<\/ol>\n<\/div>\n<p>Substitute known values into the new equation:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]g={4\\pi }^{2}\\frac{0\\text{.}\\text{75000}\\phantom{\\rule{0.25em}{0ex}}\\text{m}}{{\\left(1\\text{.}\\text{7357 s}\\right)}^{2}}.[\/latex]<\/div>\n<p>Calculate to find [latex]g[\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]g=9\\text{.}\\text{8281}\\phantom{\\rule{0.25em}{0ex}}\\text{m}\/{\\text{s}}^{2}.[\/latex]<\/div>\n<p id=\"import-auto-id959726\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2423472\">This method for determining [latex]g[\/latex] can be very accurate. This is why length and period are given to five digits in this example. For the precision of the approximation [latex]\\text{sin \u03b8}\\approx \\theta [\/latex] to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about [latex]\\text{0.5\u00ba}[\/latex].<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2008833\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Career Connections<\/div>\n<p>Knowing [latex]g[\/latex] can be important in geological exploration; for example, a map of <em data-effect=\"italics\">[latex]g[\/latex]<\/em> over large geographical regions aids the study of plate tectonics and helps in the search for oil fields and large mineral deposits.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Take Home Experiment: Determining [latex]g[\/latex]<\/div>\n<p id=\"import-auto-id2931439\">Use a simple pendulum to determine the acceleration due to gravity [latex]g[\/latex] in your own locale. Cut a piece of a string or dental floss so that it is about 1 m long. Attach a small object of high density to the end of the string (for example, a metal nut or a car key). Starting at an angle of less than [latex]\\text{10\u00ba}[\/latex], allow the pendulum to swing and measure the pendulum\u2019s period for 10 oscillations using a stopwatch. Calculate [latex]g[\/latex]. How accurate is this measurement? How might it be improved?<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2990952\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2588598\">\n<p id=\"import-auto-id1927228\">An engineer builds two simple pendula. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of [latex]\\text{10}\\phantom{\\rule{0.25em}{0ex}}\\text{kg}[\/latex]. Pendulum 2 has a bob with a mass of [latex]\\text{100 kg}[\/latex]. Describe how the motion of the pendula will differ if the bobs are both displaced by [latex]\\text{12\u00ba}[\/latex].<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1431810\" data-print-placement=\"here\">\n<p id=\"import-auto-id2032212\">The movement of the pendula will not differ at all because the mass of the bob has no effect on the motion of a simple pendulum. The pendula are only affected by the period (which is related to the pendulum\u2019s length) and by the acceleration due to gravity.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Pendulum Lab<\/div>\n<p id=\"eip-id1720216\">Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It\u2019s easy to measure the period using the photogate timer. You can vary friction and the strength of gravity. Use the pendulum to find the value of [latex]g[\/latex] on planet X. Notice the anharmonic behavior at large amplitude.<\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/41f9bd00e8c525aeae6d78fdd171c9228fb8e5f3\/pendulum-lab_en.jar\">Pendulum Lab<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_17.4\" data-alt=\"\"><a href=\"\/resources\/41f9bd00e8c525aeae6d78fdd171c9228fb8e5f3\/pendulum-lab_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3189523\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1909628\">\n<li id=\"import-auto-id3008898\">A mass <em data-effect=\"italics\">[latex]m[\/latex]<\/em> suspended by a wire of length [latex]L[\/latex] is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about [latex]\\text{15\u00ba}.[\/latex]\n<p id=\"import-auto-id1516413\">The period of a simple pendulum is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]T=2\\pi \\sqrt{\\frac{L}{g}},[\/latex]<\/div>\n<p id=\"import-auto-id2459109\">where [latex]L[\/latex]<em data-effect=\"italics\"> is the length of the string and [latex]g[\/latex] is the acceleration due to gravity.<\/em><\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1920537\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2588490\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2591478\">\n<p id=\"import-auto-id3422253\">Pendulum clocks are made to run at the correct rate by adjusting the pendulum\u2019s length. Suppose you move from one city to another where the acceleration due to gravity is slightly greater, taking your pendulum clock with you, will you have to lengthen or shorten the pendulum to keep the correct time, other factors remaining constant? Explain your answer.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2216361\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<p id=\"import-auto-id2206674\"><strong data-effect=\"bold\">As usual, the acceleration due to gravity in these problems is taken to be<\/strong>[latex]g=9.80\\phantom{\\rule{0.25em}{0ex}}\\text{m}\/{\\text{s}}^{2}[\/latex], <strong data-effect=\"bold\">unless otherwise specified.<\/strong><\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2593990\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2057678\">\n<p id=\"import-auto-id2446121\">What is the length of a pendulum that has a period of 0.500 s?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1987861\">\n<p id=\"import-auto-id1990575\">6.21 cm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1997560\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2409357\">\n<p>Some people think a pendulum with a period of 1.00 s can be driven with \u201cmental energy\u201d or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1986261\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2398036\">\n<p id=\"import-auto-id3013730\">What is the period of a 1.00-m-long pendulum?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1385763\">\n<p id=\"import-auto-id2684901\">2.01 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2449227\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2382598\">\n<p id=\"import-auto-id3069879\">How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2382885\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2404428\">\n<p id=\"import-auto-id1486338\">The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3246373\">\n<p id=\"import-auto-id3035100\">2.23 Hz<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1945602\">\n<p id=\"import-auto-id1930247\">Two parakeets sit on a swing with their combined center of mass 10.0 cm below the pivot. At what frequency do they swing?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2454166\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2679099\">\n<p id=\"import-auto-id3009247\">(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is [latex]9\\text{.}\\text{79}\\phantom{\\rule{0.25em}{0ex}}{\\text{m\/s}}^{2}[\/latex] is moved to a location where it the acceleration due to gravity is [latex]9\\text{.}\\text{82}\\phantom{\\rule{0.25em}{0ex}}{\\text{m\/s}}^{2}[\/latex]. What is its new period? (b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2669182\">\n<p id=\"import-auto-id1598830\">(a) 2.99541 s<\/p>\n<p id=\"import-auto-id3007596\">(b) Since the period is related to the square root of the acceleration of gravity, when the acceleration changes by 1% the period changes by [latex]\\left(0\\text{.}\\text{01}{\\right)}^{2}=0\\text{.}\\text{01%}\\text{}[\/latex] so it is necessary to have at least 4 digits after the decimal to see the changes.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1980938\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3032227\">\n<p id=\"import-auto-id3046066\">A pendulum with a period of 2.00000 s in one location [latex]\\left(g=9\\text{.}\\text{80}\\phantom{\\rule{0.25em}{0ex}}{\\text{m\/s}}^{2}\\right)[\/latex] is moved to a new location where the period is now 1.99796 s. What is the acceleration due to gravity at its new location?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1324396\">\n<p id=\"import-auto-id1908149\">(a) What is the effect on the period of a pendulum if you double its length?<\/p>\n<p id=\"import-auto-id3008429\">(b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2971904\">\n<p id=\"import-auto-id1561895\">(a) Period increases by a factor of 1.41 ([latex]\\sqrt{2}[\/latex])<\/p>\n<p id=\"import-auto-id3080834\">(b) Period decreases to 97.5% of old period<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1931004\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id2682373\">Find the ratio of the new\/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is [latex]1\\text{.}\\text{63}\\phantom{\\rule{0.25em}{0ex}}{\\text{m\/s}}^{2}[\/latex]. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1864432\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1816344\">\n<p id=\"import-auto-id1587045\">At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is [latex]1\\text{.}\\text{63}\\phantom{\\rule{0.25em}{0ex}}{\\text{m\/s}}^{2}[\/latex], if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock\u2019s hour hand to make one revolution on the Moon.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2621292\">\n<p id=\"import-auto-id3062813\">Slow by a factor of 2.45<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3229314\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2442364\">\n<p id=\"import-auto-id3047589\">Suppose the length of a clock\u2019s pendulum is changed by 1.000%, exactly at noon one day. What time will it read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2688094\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1348806\">\n<p id=\"import-auto-id3027530\">If a pendulum-driven clock gains 5.00 s\/day, what fractional change in pendulum length must be made for it to keep perfect time?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"import-auto-id3259736\">length must increase by 0.0116%.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2979651\">\n<dt>simple pendulum<\/dt>\n<dd>an object with a small mass suspended from a light wire or string<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Measure acceleration due to gravity.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id3178394\">\n<div class=\"bc-figcaption figcaption\">A simple pendulum has a small-diameter bob and a string that has a very small mass but is strong enough not to stretch appreciably. The linear displacement from equilibrium is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/>, the length of the arc. Also shown are the forces on the bob, which result in a net force of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3af1cac7d5a1fd545ff94fe19521905b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"72\" style=\"vertical-align: -3px;\" \/> toward the equilibrium position\u2014that is, a restoring force.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2075212\" data-alt=\"In the figure, a horizontal bar is drawn. A perpendicular dotted line from the middle of the bar, depicting the equilibrium of pendulum, is drawn downward. A string of length L is tied to the bar at the equilibrium point. A circular bob of mass m is tied to the end of the string which is at a distance s from the equilibrium. The string is at an angle of theta with the equilibrium at the bar. A red arrow showing the time T of the oscillation of the mob is shown along the string line toward the bar. An arrow from the bob toward the equilibrium shows its restoring force asm g sine theta. A perpendicular arrow from the bob toward the ground depicts its mass as W equals to mg, and this arrow is at an angle theta with downward direction of string.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_17_04_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"In the figure, a horizontal bar is drawn. A perpendicular dotted line from the middle of the bar, depicting the equilibrium of pendulum, is drawn downward. A string of length L is tied to the bar at the equilibrium point. A circular bob of mass m is tied to the end of the string which is at a distance s from the equilibrium. The string is at an angle of theta with the equilibrium at the bar. A red arrow showing the time T of the oscillation of the mob is shown along the string line toward the bar. An arrow from the bob toward the equilibrium shows its restoring force asm g sine theta. A perpendicular arrow from the bob toward the ground depicts its mass as W equals to mg, and this arrow is at an angle theta with downward direction of string.\" width=\"225\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2589696\">Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child\u2019s swing; and some are just there, such as the sinker on a fishing line. For small displacements, a pendulum is a simple harmonic oscillator. A <span data-type=\"term\">simple pendulum<\/span> is defined to have an object that has a small mass, also known as the pendulum bob, which is suspended from a light wire or string, such as shown in <a href=\"#import-auto-id3178394\" class=\"autogenerated-content\">(Figure)<\/a>. Exploring the simple pendulum a bit further, we can discover the conditions under which it performs simple harmonic motion, and we can derive an interesting expression for its period. <\/p>\n<p id=\"import-auto-id1401960\">We begin by defining the displacement to be the arc length <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/em>. We see from <a href=\"#import-auto-id3178394\" class=\"autogenerated-content\">(Figure)<\/a> that the net force on the bob is tangent to the arc and equals <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c4341455a0190acf9073b147265b801b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"76\" style=\"vertical-align: -3px;\" \/>. (The weight <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-649ee3316590a2ad3b7d61c9dfffad9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"24\" style=\"vertical-align: -3px;\" \/> has components <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b41f0438ee8d3d31ae0b130b68571cb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"65\" style=\"vertical-align: -3px;\" \/> along the string and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-227f707bf2a3b9eacbe4c0184aabaa92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"63\" style=\"vertical-align: -3px;\" \/><em data-effect=\"italics\"> tangent to the arc.) Tension in the string exactly cancels the component <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-86760a7fa10959397e346d5e8b0fb381_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#111;&#115;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" style=\"vertical-align: -3px;\" \/> parallel to the string. This leaves a <em data-effect=\"italics\">net<\/em> restoring force back toward the equilibrium position at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f40e5ee17fcf76df6a4841818adbee5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/>.<\/em><\/p>\n<p id=\"import-auto-id1999298\">Now, if we can show that the restoring force is directly proportional to the displacement, then we have a simple harmonic oscillator. In trying to determine if we have a simple harmonic oscillator, we should note that for small angles (less than about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d0913ff5fe480f07f7fecfc6f0108ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/>), <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-55e01550888dac7cbde09fe7d811295d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"72\" style=\"vertical-align: 0px;\" \/>(<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b691404dafbb2033aee29a4307ec732d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"35\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> differ by about 1% or less at smaller angles). Thus, for angles less than about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d0913ff5fe480f07f7fecfc6f0108ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/>, the restoring force <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2510519bbe1660dfdffb4195c7287343_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/><\/em> is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-64\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-46c19ee9ba5bc1a2e967e4f5750644b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"88\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id2639190\">The displacement <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/><\/em> is directly proportional to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>. When <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is expressed in radians, the arc length in a circle is related to its radius (<em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/em> in this instance) by:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c6e877df478b8cbc22a15a6ad5539f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#115;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#76;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1120184\">so that<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5aaba7caf0b239832e13558dd5283230_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#115;&#125;&#123;&#76;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id3116588\">For small angles, then, the expression for the restoring force is:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7721394a67512f5c1166436491cf7655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;&#125;&#123;&#76;&#125;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1917186\">This expression is of the form:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-05fef78392a0ad61ff4566e678a4ec13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#61;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#120;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"74\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2672250\">where the force constant is given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39049bc9c6fc10c355366aba7a91688d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;&#47;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -5px;\" \/> and the displacement is given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-292b1d808b4183c2a33c55bbecfb9460_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"42\" style=\"vertical-align: 0px;\" \/>. For angles less than about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d0913ff5fe480f07f7fecfc6f0108ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/>, the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator.<\/p>\n<p id=\"import-auto-id1588131\">Using this equation, we can find the period of a pendulum for amplitudes less than about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d0913ff5fe480f07f7fecfc6f0108ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/>. For the simple pendulum:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-424\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-de0a9026ccb26d140fecf6b6b2c4bc58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#50;&#92;&#112;&#105;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#107;&#125;&#125;&#61;&#50;&#92;&#112;&#105;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#109;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#103;&#125;&#47;&#76;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"193\" style=\"vertical-align: -14px;\" \/><\/div>\n<p id=\"import-auto-id3080441\">Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-152\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b483c652fb750e3f70b37efe2f4d64a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#50;&#92;&#112;&#105;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#76;&#125;&#123;&#103;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"88\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id1917752\">for the period of a simple pendulum. This result is interesting because of its simplicity. The only things that affect the period of a simple pendulum are its length and the acceleration due to gravity. The period is completely independent of other factors, such as mass. As with simple harmonic oscillators, the period <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> for a pendulum is nearly independent of amplitude, especially if <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-761998727948942ceb1b5763e45f01e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/><\/em> is less than about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d0913ff5fe480f07f7fecfc6f0108ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/>. Even simple pendulum clocks can be finely adjusted and accurate.<\/p>\n<p id=\"import-auto-id1080305\">Note the dependence of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> on <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><\/em>. If the length of a pendulum is precisely known, it can actually be used to measure the acceleration due to gravity. Consider the following example.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1538976\">\n<div data-type=\"title\" class=\"title\">Measuring Acceleration due to Gravity: The Period of a Pendulum<\/div>\n<p id=\"import-auto-id1418508\">What is the acceleration due to gravity in a region where a simple pendulum having a length 75.000 cm has a period of 1.7357 s?<\/p>\n<p id=\"import-auto-id2382213\"><strong>Strategy<\/strong><\/p>\n<p>We are asked to find <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><\/em> given the period <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> and the length <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/em> of a pendulum. We can solve <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b483c652fb750e3f70b37efe2f4d64a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#50;&#92;&#112;&#105;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#76;&#125;&#123;&#103;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"88\" style=\"vertical-align: -12px;\" \/> for <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><\/em>, assuming only that the angle of deflection is less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2d0913ff5fe480f07f7fecfc6f0108ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<p id=\"import-auto-id3079401\"><strong>Solution<\/strong><\/p>\n<ol id=\"fs-id1381777\" data-number-style=\"arabic\">\n<li id=\"import-auto-id1413046\">Square <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b483c652fb750e3f70b37efe2f4d64a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#50;&#92;&#112;&#105;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#76;&#125;&#123;&#103;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"88\" style=\"vertical-align: -12px;\" \/> and solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>:<br \/>\n    <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-838887657aaba030f953a67b6cc7dd22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#123;&#52;&#92;&#112;&#105;&#32;&#125;&#94;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#76;&#125;&#123;&#123;&#84;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"85\" style=\"vertical-align: -7px;\" \/><\/em><\/li>\n<\/ol>\n<\/div>\n<p>Substitute known values into the new equation:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4034ebf19897cf8bb48045ac2ab82ed8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#123;&#52;&#92;&#112;&#105;&#32;&#125;&#94;&#123;&#50;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&#48;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#51;&#53;&#55;&#32;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"134\" style=\"vertical-align: -12px;\" \/><\/div>\n<p>Calculate to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b5c763ad9823b81bd69e7681a0103069_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#50;&#56;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#47;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"129\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"import-auto-id959726\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2423472\">This method for determining <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> can be very accurate. This is why length and period are given to five digits in this example. For the precision of the approximation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5375af41f8eac73ad9bf83bdffeb23bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#32;&theta;&#125;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"60\" style=\"vertical-align: 0px;\" \/> to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c72c9cb1c0ce6429a66049001d5b960b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2008833\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Career Connections<\/div>\n<p>Knowing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> can be important in geological exploration; for example, a map of <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><\/em> over large geographical regions aids the study of plate tectonics and helps in the search for oil fields and large mineral deposits.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Take Home Experiment: Determining <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2931439\">Use a simple pendulum to determine the acceleration due to gravity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> in your own locale. Cut a piece of a string or dental floss so that it is about 1 m long. Attach a small object of high density to the end of the string (for example, a metal nut or a car key). Starting at an angle of less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-79ab0ab38e640bbcc4b344d94e2250d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"17\" style=\"vertical-align: -1px;\" \/>, allow the pendulum to swing and measure the pendulum\u2019s period for 10 oscillations using a stopwatch. Calculate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>. How accurate is this measurement? How might it be improved?<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2990952\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2588598\">\n<p id=\"import-auto-id1927228\">An engineer builds two simple pendula. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c19ea54aac68b41e3e64302c51b0494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"39\" style=\"vertical-align: -3px;\" \/>. Pendulum 2 has a bob with a mass of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9517de36e3afb617bd7d3c518066ed42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#32;&#107;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" style=\"vertical-align: -3px;\" \/>. Describe how the motion of the pendula will differ if the bobs are both displaced by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d4ecb52206f4214e49071271c2b0ff3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"16\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1431810\" data-print-placement=\"here\">\n<p id=\"import-auto-id2032212\">The movement of the pendula will not differ at all because the mass of the bob has no effect on the motion of a simple pendulum. The pendula are only affected by the period (which is related to the pendulum\u2019s length) and by the acceleration due to gravity.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Pendulum Lab<\/div>\n<p id=\"eip-id1720216\">Play with one or two pendulums and discover how the period of a simple pendulum depends on the length of the string, the mass of the pendulum bob, and the amplitude of the swing. It\u2019s easy to measure the period using the photogate timer. You can vary friction and the strength of gravity. Use the pendulum to find the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> on planet X. Notice the anharmonic behavior at large amplitude.<\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/41f9bd00e8c525aeae6d78fdd171c9228fb8e5f3\/pendulum-lab_en.jar\">Pendulum Lab<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_17.4\" data-alt=\"\"><a href=\"\/resources\/41f9bd00e8c525aeae6d78fdd171c9228fb8e5f3\/pendulum-lab_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3189523\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1909628\">\n<li id=\"import-auto-id3008898\">A mass <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6b41df788161942c6f98604d37de8098_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\" \/><\/em> suspended by a wire of length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> is a simple pendulum and undergoes simple harmonic motion for amplitudes less than about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1e3afdbefa4e1f8b8c91f1e9ca5a8208_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&ordm;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"21\" style=\"vertical-align: -1px;\" \/>\n<p id=\"import-auto-id1516413\">The period of a simple pendulum is<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-04244947f2286ba0e3630a1147258653_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#50;&#92;&#112;&#105;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#76;&#125;&#123;&#103;&#125;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"91\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id2459109\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\"> is the length of the string and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d208fd391fa57c168dc0f151de829fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> is the acceleration due to gravity.<\/em><\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1920537\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2588490\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2591478\">\n<p id=\"import-auto-id3422253\">Pendulum clocks are made to run at the correct rate by adjusting the pendulum\u2019s length. Suppose you move from one city to another where the acceleration due to gravity is slightly greater, taking your pendulum clock with you, will you have to lengthen or shorten the pendulum to keep the correct time, other factors remaining constant? Explain your answer.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2216361\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<p id=\"import-auto-id2206674\"><strong data-effect=\"bold\">As usual, the acceleration due to gravity in these problems is taken to be<\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2bd2a8245b9f7a6ee94d4ec4972d394b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#103;&#61;&#57;&#46;&#56;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#47;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" style=\"vertical-align: -5px;\" \/>, <strong data-effect=\"bold\">unless otherwise specified.<\/strong><\/p>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2593990\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2057678\">\n<p id=\"import-auto-id2446121\">What is the length of a pendulum that has a period of 0.500 s?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1987861\">\n<p id=\"import-auto-id1990575\">6.21 cm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1997560\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2409357\">\n<p>Some people think a pendulum with a period of 1.00 s can be driven with \u201cmental energy\u201d or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1986261\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2398036\">\n<p id=\"import-auto-id3013730\">What is the period of a 1.00-m-long pendulum?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1385763\">\n<p id=\"import-auto-id2684901\">2.01 s<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2449227\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2382598\">\n<p id=\"import-auto-id3069879\">How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2382885\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2404428\">\n<p id=\"import-auto-id1486338\">The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3246373\">\n<p id=\"import-auto-id3035100\">2.23 Hz<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1945602\">\n<p id=\"import-auto-id1930247\">Two parakeets sit on a swing with their combined center of mass 10.0 cm below the pivot. At what frequency do they swing?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2454166\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2679099\">\n<p id=\"import-auto-id3009247\">(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8921ce1213b37eb450b7b15022d4f780_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"74\" style=\"vertical-align: -4px;\" \/> is moved to a location where it the acceleration due to gravity is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-381ebc08c76ba45b16fb854868a8f8f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"74\" style=\"vertical-align: -4px;\" \/>. What is its new period? (b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2669182\">\n<p id=\"import-auto-id1598830\">(a) 2.99541 s<\/p>\n<p id=\"import-auto-id3007596\">(b) Since the period is related to the square root of the acceleration of gravity, when the acceleration changes by 1% the period changes by <\/p>\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#49;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#49;&#37;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#125;\n\n*** Error message:\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#49;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#49;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;\r\n&#70;&#105;&#108;&#101;&#32;&#101;&#110;&#100;&#101;&#100;&#32;&#119;&#104;&#105;&#108;&#101;&#32;&#115;&#99;&#97;&#110;&#110;&#105;&#110;&#103;&#32;&#117;&#115;&#101;&#32;&#111;&#102;&#32;&#92;&#116;&#101;&#120;&#116;&#64;&#46;\r\n&#69;&#109;&#101;&#114;&#103;&#101;&#110;&#99;&#121;&#32;&#115;&#116;&#111;&#112;&#46;\r\n\n<\/pre>\n<p> so it is necessary to have at least 4 digits after the decimal to see the changes.\n<\/p><\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1980938\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3032227\">\n<p id=\"import-auto-id3046066\">A pendulum with a period of 2.00000 s in one location <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d50fe4644768ee66bfa7511464cd587c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#103;&#61;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"123\" style=\"vertical-align: -12px;\" \/> is moved to a new location where the period is now 1.99796 s. What is the acceleration due to gravity at its new location?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1324396\">\n<p id=\"import-auto-id1908149\">(a) What is the effect on the period of a pendulum if you double its length?<\/p>\n<p id=\"import-auto-id3008429\">(b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2971904\">\n<p id=\"import-auto-id1561895\">(a) Period increases by a factor of 1.41 (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0f51c9c2ccb4ec4dcca8926b645d14c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#113;&#114;&#116;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"23\" style=\"vertical-align: -2px;\" \/>)<\/p>\n<p id=\"import-auto-id3080834\">(b) Period decreases to 97.5% of old period<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1931004\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id2682373\">Find the ratio of the new\/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-65d73ebb7da11dc1882e40464f2681a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"73\" style=\"vertical-align: -4px;\" \/>. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1864432\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1816344\">\n<p id=\"import-auto-id1587045\">At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-65d73ebb7da11dc1882e40464f2681a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#47;&#115;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"73\" style=\"vertical-align: -4px;\" \/>, if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock\u2019s hour hand to make one revolution on the Moon.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2621292\">\n<p id=\"import-auto-id3062813\">Slow by a factor of 2.45<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3229314\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2442364\">\n<p id=\"import-auto-id3047589\">Suppose the length of a clock\u2019s pendulum is changed by 1.000%, exactly at noon one day. What time will it read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2688094\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1348806\">\n<p id=\"import-auto-id3027530\">If a pendulum-driven clock gains 5.00 s\/day, what fractional change in pendulum length must be made for it to keep perfect time?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"import-auto-id3259736\">length must increase by 0.0116%.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2979651\">\n<dt>simple pendulum<\/dt>\n<dd>an object with a small mass suspended from a light wire or string<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-850","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":826,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/850","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/850\/revisions"}],"predecessor-version":[{"id":851,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/850\/revisions\/851"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/826"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/850\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=850"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=850"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=850"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=850"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}