1. Group Work
Predict the motion of a mass acted on only by a Hooke’s law spring.
Express your prediction as a position-time graph.
Explain why you believe that the mass will move in the manner you predicted.

2. Boing!
Things that vibrate
13.1–13.5

3. 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.

4. What’s the Point?
The Hooke’s law model describes the essential features of many types of oscillation.

5. 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

6. How does a spring mass move?
Newton’s second law: F = ma
Force F depends on position by Hooke’s law: F = –kx

7. Poll Question The spring’s force on an oscillating object is
zero at the extreme positions
maximum at the extreme positions
minimum but not zero at the extreme positions

8. Poll Question The acceleration of an oscillating object is
zero at the extreme positions
maximum at the extreme positions
minimum but not zero at the extreme positions

9. Poll Question The velocity of an oscillating object is
maximum at the equilibrium position
maximum at the extreme positions
maximum midway between equilibrium and an extreme positions

10. Uniform Circular Motion
Centripetal force F = mv2/r inwards
Constant magnitude F0; direction depends on position F0
Force in y-direction is proportional to –y

11. 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?

12. Hooke’s Law Motion

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

14. Class Work What is the object’s position when it is not accelerating?
What is the object’s velocity when it is not accelerating?
What is the object’s position when it is not moving?
What is the object’s acceleration when it is not moving?

15. Group Work
What is the object’s kinetic energy when it is not accelerating?
What is the object’s potential energy when it is not accelerating?

16. Energy Potential energy of a stretched spring : kx2 PE =1 2
Conservation of energy:
PE + KE = constant
(This of course ignores the nasty reality of energy dispersal by friction and drag.)

17. Group Work
Show that the total energy E of an oscillating Hooke’s law spring is always positive.
kx2 1 2 +
mv2
E = PE + KE =
> 0

18. Energy
KE = 0
PE = max
KE = max
PE = 0 y
time
PE = max

19. Period and Frequency Period T Frequency f f = 1/T
time of one cycle (units: s)
Frequency f
cycles per unit time (units: 1/s = Hz)
f = 1/T

20. Poll Question Increasing the spring constant k makes the period
shorter.
longer.
k has no effect on the period.

21. Poll Question Increasing the mass m makes the period shorter. longer.
m has no effect on the period.

22. Period m Period T = 2p k So increased mass m  longer period
increased k  shorter period
period does not depend on amplitude

23. Effect of Gravity Less than you might expect:
Changes equilibrium position x = 0
Does not change k

24. Spring + Gravity
spring alone
gravity
force
position

25. Spring + Gravity
spring alone
gravity
force
spring + gravity
position

26. Spring + Gravity different equilibrium position same k net force
position