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Physics LibreTexts

16.4: The Simple Pendulum

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Learning Objectives

By the end of this section, you will be able to:

  • Measure acceleration due to gravity.
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.
Figure 16.4.1: 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 s, the length of the arc. Also shown are the forces on the bob, which result in a net force of - mgsinθ toward the equilibrium position—that is, a restoring force.

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 16.4.1. 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 s. We see from Figure 16.4.1 that the net force on the bob is tangent to the arc and equals mgsinθ. (The weight mg has components mgcosθ along the string and mgsinθ tangent to the arc.) Tension in the string exactly cancels the component mgcosθ parallel to the string. This leaves a net restoring force back toward the equilibrium position at θ=0.

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 15o), sinθθ(sinθ and θ differ by about 1% or less at smaller angles). Thus, for angles less than about 15o, the restoring force F is Fmgθ. The displacement s is directly proportional to θ. When θ is expressed in radians, the arc length in a circle is related to its radius (L in this instance) by:

s=Lθ,

so that

θ=sL.

For small angles, then, the expression for the restoring force is:

FmgLs.

This expression is of the form:

F=kx,

where the force constant is given by k=mg/L and the displacement is given by x=s. For angles less than about 15o the restoring force is directly proportional to the displacement, and the simple pendulum is a simple harmonic oscillator.

Using this equation, we can find the period of a pendulum for amplitudes less than about 15o. For the simple pendulum:

T=2πmk=2πmmg/L.

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 T for a pendulum is nearly independent of amplitude, especially if θ is less than about 15o. Even simple pendulum clocks can be finely adjusted and accurate.

Note the dependence of T on g. 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.

Example 16.4.1: Measuring Acceleration due to Gravity: The Period of a Pendulum

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 g given the period T and the length L of a pendulum. We can solve T=2πLg for g, assuming only that the angle of deflection is less than 15o.

Solution

  1. Square T=2πLg and solve for g: g=4π2LT2.
  2. Substitute known values into the new equation: g=4π20.75000m(1.7357s)2.
  3. Calculate to find g: g=9.8281m/s2.

Discussion

This method for determining g can be very accurate. This is why length and period are given to five digits in this example. For the precision of the approximation sinθθ to be better than the precision of the pendulum length and period, the maximum displacement angle should be kept below about 0.5o.

MAKING CAREER CONNECTIONS

Knowing g can be important in geological exploration; for example, a map of g over large geographical regions aids the study of plate tectonics and helps in the search for oil fields and large mineral deposits.

TAKE-HOME EXPERIMENT: DETERMINING g

Use a simple pendulum to determine the acceleration due to gravity g 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 10o, allow the pendulum to swing and measure the pendulum’s period for 10 oscillations using a stopwatch. Calculate g. How accurate is this measurement? How might it be improved?

Exercise 16.4.1

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 10kg. Pendulum 2 has a bob with a mass of 100kg. Describe how the motion of the pendula will differ if the bobs are both displaced by 12o.

Answer

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.

PHET EXPLORATIONS: PENDELUM LAB

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 g on planet X. Notice the anharmonic behavior at large amplitude.

PhET_Icon.png
Figure 16.4.2: Pendulum Lab

Glossary

simple pendulum
an object with a small mass suspended from a light wire or string
 

This page titled 16.4: The Simple Pendulum is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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