Download Harmonic Motion

Harmonic Motion
Vector Components

Circular motion can be
described by components.
• x = r cos q
• y = r sin q

For uniform circular motion
the angle is related to the
angular velocity.
•
q=wt

The motion can be described
as a function of time.
• x = r cos wt
• y = r sin wt
r sin q
r
q
r cos q
Velocity Components

v
The velocity vector can also be
described by components.
• vx = -v sin q
• vy = v cos q
q v cos q
-v sin q
q

This velocity is related to the
angular frequency.
vx  rw sin wt
v y  rw cos wt
Acceleration Components

-a cos q
q
a
q
For uniform circular motion the
acceleration vector points inward.
• ax = -a cos q
• ay = -a sin q
-a sin q

The acceleration is also related to
the angular frequency.
a x   rw 2 cos wt
a y   rw 2 sin wt
Changing Angle to Position

If only one component is viewed the motion is
sinusoidal in time.

This is called harmonic motion.
Springs and pendulums also have harmonic motion.

1 period
x = A cos wt
Acceleration and Position

In uniform circular motion acceleration is opposite to
the position from the center .

In harmonic motion the acceleration is also opposite
to the position.
a x  rw 2 cos wt  w 2 x
This is true for all small oscillations
Spring Oscilations

From the law of action the
force is proportional to the
acceleration.
F  max   mw 2 x

Harmonic motion has a
position-dependent force.
• Force is negative
• Restoring force
F  kx  mw 2 x
w  k/m
Spring Constant Curve

The spring force has a potential energy U = ½ kx2 .
U
U
x
Near the minimum all curves are approximately a spring force.
x
Springboard

Find the spring constant
from the mass and
frequency.
w  2f
k / m  w 2  4 2 f
k  4 2 f 2 m

A diving board oscillates with a
frequency of 5.0 cycles per
second with a person of mass
70. kg. What is the spring
constant of the board?

With values:
• k = 42(5.0 /s)2(70. kg)
• K = 6.9 x 104 N/m
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