Introduction to Oscillations and Simple Harmonic Motion

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Springs and Waves behave very similar to objects that move in circles.

The radius of the circle is symbolic of the displacement, x, of a spring or the amplitude, A, of a wave.

Slide 18

The Pendulum

A simple pendulum consists of a particle of mass m, attached to a frictionless pivot P by a cable of length L and negligible mass.

Slide 19

Pendulums

Consider the free body diagram (FBD) for a pendulum. Here we have the weight and tension. Even though the weight isn’t at an angle let’s draw an axis along the tension.

q

q

mgcosq

mgsinq

Slide 20

Pendulums

What is x? It is the amplitude! In the picture to the left, it represents the chord from where it was released to the bottom of the swing (equilibrium position).

Slide 21

The Reference Circle

The reference circle compares the circular motion of an object with its horizontal projection.

x = Horizontal displacement.

A = Amplitude (xmax).

q = Reference angle.

Slide 22

Velocity in SHM

The velocity (v) of an oscillating body at any instant is the horizontal component of its tangential velocity (vT).

vT = wR = wA; w = 2f

v = -vT sin  ;  = wt

v = -w A sin w t

v = -2f A sin 2f t

Slide 23

Acceleration Reference Circle

The acceleration (a) of an oscillating body at any instant is the horizontal component of its centripetal acceleration (ac).

a = -ac cos q = -ac cos(wt)

R = A

a = -w2A cos(wt)

Slide 24

The Period and Frequency as a Function of a and x.

For any body undergoing simple harmonic motion:

Since a = -4p2f2x and T = 1/f

The frequency and the period can be found if the displacement and acceleration are known. Note that the signs of a and x will always be opposite.

Slide 25

Period and Frequency as a Function of Mass and Spring Constant.

For a vibrating body with an elastic restoring force:

Recall that F = ma = -kx:

The frequency f and the period T can be found if the spring constant k and mass m of the vibrating body are known. Use consistent SI units.

Slide 26

Example 3:

A visitor to a lighthouse wishes to determine the height of the tower. She ties a spool of thread to a small rock to make a simple pendulum, which she hangs down the center of a spiral staircase of the tower. The period of oscillation is 9.40 s. What is the height of the tower?