Understanding Harmonic Motion: Vector & Acceleration Components, Spring Oscillations, Slides of Physics

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Harmonic Motion

Vector Components 

Circular motion can bedescribed by components.

x

r

cos

y

r

sin



For uniform circular motionthe angle is related to theangular velocity.

t



The motion can be describedas a function of time.

x

r

cos

t 

y

r

sin

t

r 

r^

sin

r^

cos

Acceleration Components



For uniform circular motion theacceleration vector points inward.

a

x^

a

cos

a

y^

-a

sin



The acceleration is also related tothe angular frequency.

a

-a

sin

-a

cos

t

r

a

t

r

a

x y









cos sin 2 2

 

 

Changing Angle to Position 

If only one component is viewed the motion issinusoidal in time.

This is called harmonic motion.

Springs and pendulums also have harmonic motion.

x

=

A

cos



t

1 period

Spring Oscilations 

From the law of action theforce is proportional to theacceleration.



Harmonic motion has aposition-dependent force.

Force is negative

Restoring force

x

m

ma

F

x

2

m

k

x

m

kx

F

2

Spring Constant Curve 

The spring force has a potential energy

U

= ½

kx

2

.

U

x

U

x

Near the minimum all curves are approximately a spring force.

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