Oscillatory Motion and Waves
The Simple Pendulum
Learning Objectives
- Measure acceleration due to gravity.
Pendulums are in common usage. Some have crucial uses, such as in clocks; some are for fun, such as a child’s 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 simple pendulum 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 (Figure). 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.
We begin by defining the displacement to be the arc length
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
The displacement
so that
For small angles, then, the expression for the restoring force is:
This expression is of the form:
where the force constant is given by
Using this equation, we can find the period of a pendulum for amplitudes less than about
Thus,
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
Note the dependence of
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?
Strategy
We are asked to find
Solution
- Square
and solve for :
Substitute known values into the new equation:
Calculate to find
Discussion
This method for determining
Knowing
Use a simple pendulum to determine the acceleration due to gravity
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
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’s length) and by the acceleration due to gravity.
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’s 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
Section Summary
- A mass
suspended by a wire of length is a simple pendulum and undergoes simple harmonic motion for amplitudes less than aboutThe period of a simple pendulum is
where
is the length of the string and is the acceleration due to gravity.
Conceptual Questions
Pendulum clocks are made to run at the correct rate by adjusting the pendulum’s 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.
Problems & Exercises
As usual, the acceleration due to gravity in these problems is taken to be
What is the length of a pendulum that has a period of 0.500 s?
6.21 cm
Some people think a pendulum with a period of 1.00 s can be driven with “mental energy” or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?
What is the period of a 1.00-m-long pendulum?
2.01 s
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?
The pendulum on a cuckoo clock is 5.00 cm long. What is its frequency?
2.23 Hz
Two parakeets sit on a swing with their combined center of mass 10.0 cm below the pivot. At what frequency do they swing?
(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is
(a) 2.99541 s
(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
A pendulum with a period of 2.00000 s in one location
(a) What is the effect on the period of a pendulum if you double its length?
(b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?
(a) Period increases by a factor of 1.41 (
(b) Period decreases to 97.5% of old period
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
At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is
Slow by a factor of 2.45
Suppose the length of a clock’s 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.
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?
length must increase by 0.0116%.
Glossary
- simple pendulum
- an object with a small mass suspended from a light wire or string