Peoples Physics Concepts

http://www.ck12.org Chapter 10. Simple Harmonic Motion

10.3 Pendulum

• Describe the harmonic motion of a pendulum and calculate its period.

Students will learn about the harmonic motion of a pendulum and how to calculate its period.

Key Equations

Tpendulum= 2 π

√

L

g

; Period of a pendulum of length L

Guidance

• In harmonic motion there is always arestorative force, which acts in the opposite direction of the velocity.
  The restorative force changes during oscillation and depends on the position of the object. In a pendulum it is
  the component of gravity along the path of motion. The force on the oscillating object is directly opposite that
  of the direction of velocity.


• For pendulums, the period gets larger as the length of the pendulum increases.

Example 1

You have a mass swinging on the end of 1 m pendulum. If the maximum linear velocity of the mass is 2 m/s, (a)

calculate the period of the pendulum and (b) calculate the amplitude of the pendulum.

Solution

To calculate the period of the pendulum, we can just plug in the given length into the equation above.

T= 2 π

√

l

g

T= 2 π

√

1 m

9 .8 m/s^2

T=2 s

To find the amplitude, we’ll use the equation given in the Period and Frequency lesson that gives us the velocity as
a function of time. Since the problem says that the given velocity is the maximum velocity, we know that the
pendulum is at the bottom of it’s arc and 1/4th (or 3/4th’s) of it’s way through one period. Based on this knowledge,
we can plug in 1/4 of the period for the change in time. We also know the frequency because we just found the
period, so all we have to do is solve for the amplitude.