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Physics 101 Lecture 22Simple Harmonic Motion

- Todays lecture will cover Textbook Sections 10.4

- 10.6

Review Ideal Springs

- Hookes Law The force exerted by a spring is

proportional to the distance the spring is

stretched or compressed from its relaxed

position. - FX -k x Where x is the displacement from

the relaxed position and k is the constant

of proportionality. (often called spring

constant)

relaxed position

FX - kx lt 0

x

x gt 0

x0

Review Simple Harmonic Motion

Uniform circular motion lt-gt Motion of an object

attached to an ideal spring

Period T (seconds per cycle) Frequency f

1/T (cycles per second) Angular frequency ?

2?f 2?/T

Simple Harmonic MotionQuick Review

x(t) Acos(?t) v(t) -A?sin(?t) a(t)

-A?2cos(?t)

x(t) Asin(?t) v(t) A?cos(?t) a(t)

-A?2sin(?t)

OR

Period T (seconds per cycle) Frequency f

1/T (cycles per second) Angular frequency ?

2?f 2?/T

xmax A vmax A? amax A?2

Review Period of a Spring

For simple harmonic oscillator ? 2?f 2?/T

For mass M on spring with spring constant k

Demos A,m,k dependence

Concept Question

- If the amplitude of the oscillation (same block

and same spring) was doubled, how would the

period of the oscillation change? (The period is

the time it takes to make one complete

oscillation) - 1. The period of the oscillation would double.2.

The period of the oscillation would be halved3.

The period of the oscillation would stay the same

x

2A

t

-2A

Potential Energy of a Spring

Where x is measured fromthe equilibrium position

PES

x

0

Same thing for a vertical spring

y

y0

Where y is measured fromthe equilibrium position

PES

y

0

In either case...

Etotal 1/2 Mv2 1/2 kx2 constant

KE PE

KEmax 1/2Mv2max1/2M?2A2 1/2kA2 PEmax

1/2kA2 Etotal 1/2kA2

Concept Question

- In Case 1 a mass on a spring oscillates back and

forth. In Case 2, the mass is doubled but the

spring and the amplitude of the oscillation is

the same as in Case 1. In which case is the

maximum kinetic energy of the mass the biggest? - 1. Case 12. Case 23. Same

Concept Question

PE 0 KE KEMAX

PE 1/2kx2 KE 0

same for both

samefor both

xA

x0

xA

x-A

x0

x-A

Concept Question

- The same would be true for vertical springs...

PE 1/2k y2

Y0

PE 1/2k y2

Y0

Pendulum

- For small oscillation, period does not depend

on - mass
- amplitude

Demos M,A,L dependence

Concept Question

- Suppose a grandfather clock (a simple pendulum)

runs slow. In order to make it run on time you

should - 1. Make the pendulum shorter
- 2. Make the pendulum longer

Concept Question

- A pendulum is hanging vertically from the ceiling

of an elevator. Initially - the elevator is at rest and the period of the

pendulum is T. Now the pendulum accelerates

upward. The period of the pendulum will now be - 1. greater than T
- 2. equal to T
- 3. less than T

Effective g is larger when accelerating

upward (you feel heavier)

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