Simple Harmonic Motion

The Ideal Spring and Simple Harmonic Motion

xkF Appliedx

spring constant

Units: N/m

Sample Problem

Hooke’s Law

If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?

k mg

x

(0.55 kg)(9.81 m/s2 )

–0.020 m

k 270 N/m

Simple Harmonic Motion

Simple harmonic motion describes any periodic motion that is the result of a restoring force that is proportional to displacement. The motion of a vibrating mass-spring system is an

example of simple harmonic motion.

Because simple harmonic motion involves a restoring force, every simple harmonic motion is a back-and-forth motion over the same path.

The Simple Pendulum

The forces acting on the bob at any point are the force exerted by the string and the gravitational force.

A simple pendulum consists of a mass called a bob, which is attached to a fixed string.

Amplitude, Period, and Frequency

Amplitude of the vibration is the maximum displacement from equilibrium.A pendulum’s

measured by the angle between the pendulum’s equilibrium position and its maximum displacement.

For a mass-spring system, the maximum amount the spring is stretched or

compressed from its equilibrium position.

SI units are radian (rad) and meter (m).

The period (T) is the time that it takes a complete cycle to occur. The SI unit of period is seconds (s).

The frequency (f) is the number of cycles or vibrations per unit of time. The SI unit of frequency is hertz (Hz). Hz = s–1

f 1

T or T

1

f

Period of a Simple Pendulum

The period of a simple pendulum depends on the length and on the free-fall acceleration.

T 2L

ag

The period does not depend on the mass of the bob or on the amplitude (for small angles).

period 2length

free-fall acceleration

Period of a Mass-Spring System

The period of an ideal mass-spring system depends on the mass and on the spring constant.

T 2m

kThe period does not depend on the amplitude.This equation applies only for systems in

which the spring obeys Hooke’s law.

period 2mass

spring constant

Simple Harmonic Motion and the Reference Circle

tAAx coscos

DISPLACEMENT

AmplitudeAmplitude

Constant angular speed Constant angular speed (rad/s)(rad/s)

Simple Harmonic Motion and the Reference Circle

tAAx coscos

AmplitudeAmplitude

Constant angular speed Constant angular speed (rad/s)(rad/s)

period T: the time required to complete one cycle

frequency f: the number of cycles per second (measured in Hz)

Tf

1

Tf

22

amplitude A: the maximum displacement

The frequency of motion is 1.0 KHz and the amplitude is 0.20 mm.

(a)What is the maximum speed of the diaphragm?(b)Where in the motion does this maximum speed

occur?

The Maximum Speed of a Loudspeaker Diaphragm

tAvvv

Tx sinsinmax

(a) sm3.1

Hz100.12m1020.02 33max

fAAv

(b)The maximum speedoccurs midway betweenthe ends of its motion.