Simple Harmonic Motion and Uniform Circular Motion

A ball is attached to the rim of a turntable of radius A

The focus is on the shadow that the ball casts on the screen

When the turntable rotates with a constant angular speed, the shadow moves in simple harmonic motion

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Recall circular motion from PHF02 Velocity of object in circular motion is:

In this case our r is the amplitude A which leads to the max. velocity:

Then the max acceleration must be the circular motion acceleration:

Remember the acceleration is always directed to the center, so incase of the shadow, it is directed towards the equilibrium point.

T

rv

2

T

Av

2max

2

24

T

Aa

For SHM we can say: Negative because a always acts

opposite the displacement. Combining the equation for the spring-

mass system we obtain:

2

24

T

xa

k

mT

xm

k

T

xa

2

42

2

Effective Spring Mass

A graph of T2 versus m does not pass through the origin

The spring has mass and oscillates For a cylindrical spring, the

effective additional mass of a light spring is 1/3 the mass of the spring

Frequency and Angular frequency

m

k

fm

kf

Tf

2and

2

1

1

Study example 4.3 in course book.

Motion as a Function of Time Use of a reference

circle allows a description of the motion

x = A cos (2ƒt) x is the position at

time t x varies between

+A and -A

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Example 1

a) Find the amplitude, frequency, and period of motion for an object vibrating at the end of a horizontal spring if the equation for its position as a function of time is:

tx00.8

cos250.0

a) done

b) Find the maximum magnitude of the velocity and acceleration.

c) What is the position, velocity and acceleration of the object after 1.00s has elapsed?

Simple Pendulum The simple

pendulum is another example of simple harmonic motion

The force is the component of the weight tangent to the path of motion Ft = - m g sin θ a.k.a. restoring

force

x

Using Newton's second law: F = m a = - m g sin θ Assumption: θ is small < 15o then; sin θ ~ tan θ = x/L~

g

LT

L

xg

T

xa

T

xarecall

L

xga

L

xmgma

2

4

4:

2

2

2

2

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Quiz!! A pendulum clock depends on the period of a

pendulum to keep correct time. Suppose a pendulum clock is keeping correct time and then a naughty PHF03 student slides the bob of the pendulum downward on the oscillating rod. Does the clock run:

a) Slow,b) Fast orc) Correctly?

Study example 4.5 in your course book

Damped Oscillation Only ideal systems oscillate

indefinitely In real systems, friction retards the

motion Friction reduces the total energy of

the system and the oscillation is said to be damped

One application is the use of shock absorbers in motor vehicles

Summary