Group Work

1. Predict the motion of a mass acted on only by a Hooke’s law spring.

a. Express your prediction as a position-time graph.

b. Explain why you believe that the mass will move in the manner you predicted.

Boing!

Things that vibrate

Objectives

• Analyze force, acceleration, velocity, and position at any point in a vibration cycle.

• Trace the evolution of kinetic and potential energy, velocity, and displacement in an oscillation.

• Identify the factors determining the period of a spring and a pendulum.

What’s the Point?

• The Hooke’s law model describes the essential features of many types of oscillation.

Springs

• Hooke’s law: F = –kx

• F = force exerted by the spring

• k = spring constant (characteristic of the particular spring)

• x = distance spring is displaced from equilibrium

How does a spring mass move?

• Newton’s second law: F = ma

• Force F depends on position by Hooke’s law: F = –kx

Poll Question

The spring’s force on an oscillating object is

A. zero at the extreme positions

B. maximum at the extreme positions

C. minimum but not zero at the extreme positions

Poll Question

The acceleration of an oscillating object is

A. zero at the extreme positions

B. maximum at the extreme positions

C. minimum but not zero at the extreme positions

Poll Question

The velocity of an oscillating object is

A. maximum at the equilibrium position

B. maximum at the extreme positions

C. maximum midway between equilibrium and an extreme positions

Uniform Circular Motion

• Centripetal force F = mv2/r inwards

• Constant magnitude F0; direction depends on position

F0

F0

F0

F0

F0

F0

• Force in y-direction is proportional to –y

Uniform Circular Motion

• Angle changes at a steady rate.

• Projection on y-axis has Hooke’s law force.

• So, projection on y-axis must have Hooke’s law motion too!

• What is the projection of an angle on the y-axis?

Hooke’s Law Motion

acceleration

velocity

upward

downward

upward

downward

position

max

min

0

0

0

T 2T

Class Work

2. What is velocity (min, max, > 0, < 0, 0) of the oscillating mass at the top? At the equilibrium point? At the bottom?

3. What is acceleration (min, max, > 0, < 0, 0) of the oscillating mass at the top? At the equilibrium point? At the bottom?

Class Work

4. What is the object’s position when it is not accelerating?

5. What is the object’s velocity when it is not accelerating?

6. What is the object’s position when it is not moving?

7. What is the object’s acceleration when it is not moving?

Group Work

8. What is the object’s kinetic energy when it is not accelerating?

9. What is the object’s potential energy when it is not accelerating?

Energy

• Potential energy of a stretched spring :

PE = kx212

• Conservation of energy:

PE + KE = constant

(This of course ignores the nasty reality of energy dispersal by friction and drag.)

Group Work

10. Show that the total energy E of an oscillating Hooke’s law spring is always positive.

kx212 + mv21

2E = PE + KE = > 0

Energy

y

time

KE = 0

KE = 0

KE = max PE = 0

PE = max

PE = max

Period and Frequency

• Period T– time of one cycle (units: s)

• Frequency f– cycles per unit time (units: 1/s = Hz)

• f = 1/T

Poll Question

Increasing the spring constant k makes the period

A. shorter.

B. longer.

C. k has no effect on the period.

Poll Question

Increasing the mass m makes the period

A. shorter.

B. longer.

C. m has no effect on the period.

Period

• Period T = 2mk

So

• increased mass m longer period

• increased k shorter period

• period does not depend on amplitude

Effect of Gravity

• Less than you might expect:

• Changes equilibrium position x = 0

• Does not change k

Spring + Gravity

position

forc

e

0

0

spring alone

gravity

Spring + Gravity

position

forc

e

0

0

spring alone

gravity

0

spring + gravity

Spring + Gravity

position

net

forc

e

0

0

different equilibrium position

same k

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