Harmonic MotionHarmonic Motion

Vector ComponentsVector Components Circular motion can be Circular motion can be

described by components.described by components.• xx = = rr cos cos • yy = = rr sin sin

For uniform circular motion For uniform circular motion the angle is related to the the angle is related to the angular velocity.angular velocity.• = = tt

The motion can be described The motion can be described as a function of time.as a function of time.• xx = = rr cos cos tt• yy = = rr sin sin tt

r

r sin

r cos

Velocity ComponentsVelocity Components The velocity vector can also be The velocity vector can also be

described by components.described by components.• vvxx = - = -vv sin sin

• vvyy = = vv cos cos

This velocity is related to the This velocity is related to the angular frequency.angular frequency.

v

-v sin

v cos

trvtrv

y

x

cossin

Acceleration ComponentsAcceleration Components For uniform circular motion the For uniform circular motion the

acceleration vector points inward.acceleration vector points inward.• aaxx = - = -aa cos cos

• aayy = = -a-a sin sin

The acceleration is also related to The acceleration is also related to the angular frequency.the angular frequency.

a -a sin

-a cos

tra

tra

y

x

sin

cos2

2

Changing Angle to PositionChanging Angle to Position If only one component is viewed the motion is If only one component is viewed the motion is

sinusoidal in time.sinusoidal in time.

This is called harmonic motion.This is called harmonic motion. Springs and pendulums also have harmonic motion.Springs and pendulums also have harmonic motion.

xx = = AA cos cos tt

1 period

Acceleration and PositionAcceleration and Position In uniform circular motion acceleration is opposite to In uniform circular motion acceleration is opposite to

the position from the center .the position from the center .

In harmonic motion the acceleration is also opposite In harmonic motion the acceleration is also opposite to the position.to the position.

xtrax22 cos

This is true for all small oscillations

Spring OscilationsSpring Oscilations From the law of action the From the law of action the

force is proportional to the force is proportional to the acceleration.acceleration.

Harmonic motion has a Harmonic motion has a position-dependent force.position-dependent force.• Force is negativeForce is negative• Restoring forceRestoring force

xmmaF x2

mk

xmkxF

/

2

Spring Constant CurveSpring Constant Curve The spring force has a potential energy The spring force has a potential energy UU = ½ = ½ kxkx22 . .

U

x

U

x

Near the minimum all curves are approximately a spring force.

SpringboardSpringboard

A diving board oscillates with a A diving board oscillates with a frequency of 5.0 cycles per frequency of 5.0 cycles per second with a person of mass second with a person of mass 70. kg. What is the spring 70. kg. What is the spring constant of the board?constant of the board?

mfk

fmk

f

22

222

4

4/

2

Find the spring constant Find the spring constant from the mass and from the mass and frequency.frequency.

With values:With values:• kk = 4 = 422(5.0 /s)(5.0 /s)22(70. kg)(70. kg)• K = 6.9 x 10K = 6.9 x 1044 N/m N/m

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