1. SCI 340 L25 SHM Things that vibrate
Boing!
Things that vibrate
Chapter 10

2. 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.

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

4. 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

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

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
SCI 340 L25 SHM
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 B

8. Poll Question The acceleration of an oscillating object is
SCI 340 L25 SHM
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 B

9. Poll Question The speed 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 q 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 a circular path on the y-axis?

12. Uniform Circular Motion
q = wt w is angular velocity Projected onto the Cartesian axes: x = cos(q) = cos(wt) y = sin(q) = sin(wt) w is angular frequency = 2pf = 2p/T

13. Hooke’s Law Motion

14. SCI 340 L25 SHM
Class Work
What is velocity (min, max, +, −, 0) of the oscillating mass at the top? At the equilibrium point? At the bottom?
What is acceleration (min, max, +, −, 0) of the oscillating mass at the top? At the equilibrium point? At the bottom?
2. 0, ±max, 0
3. −max, 0, +max

15. Class Work What is the object’s position when it is not accelerating?
SCI 340 L25 SHM
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?
4. Center (equilibrium)
5. ±max
6. Extremes (±max)
7. ±max (at the extremes)

16. 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

17. Poll Question Increasing the spring constant k makes the period
SCI 340 L25 SHM
Poll Question
Increasing the spring constant k makes the period
shorter.
longer.
k has no effect on the period. A

18. Poll Question Increasing the mass m makes the period shorter. longer.
SCI 340 L25 SHM
Poll Question
Increasing the mass m makes the period
shorter.
longer.
m has no effect on the period. B

19. Reference Circle Radius = oscillation amplitude A
SCI 340 L25 SHM
Reference Circle
Radius = oscillation amplitude A
Frequency = oscillation frequency f
Angular speed w = angular frequency w = 2pf
Period T = 1/f = period
Tangential speed v = 2pAf = max speed
2pAf = Aw
Acceleration = v2/A = Aw2 = max acc.
Force = mv2/A = mAw2 = kA = max force

20. Now, this is neat mAw2 = kA w2 = k/m w 2p = 2p 1 k/m
Oscillation frequency f =
Oscillation period T = 1/f = 2p
m/k

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

22. SCI 340 L25 SHM
Group Work
What is a spring mass’s kinetic energy when it is not accelerating?
What is a spring mass’s potential energy when it is not accelerating?
What is a spring mass’s kinetic energy when it is not moving?
What is a spring mass’s potential energy when it is not moving?
8. K = Kmax = ½ mvmax2
9. At the center, U = 0.
When v = 0, K = 0.
11. At extremes, U = ½ kA2.

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

24. SCI 340 L25 SHM
Group Work
Show that the total energy E of an oscillating Hooke’s law spring is always positive.
m and k are positive. x2 and v2 are always non-negative; when one is zero, the other is positive.
kx2 1 2 +
mv2
E = U+ K =
> 0

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

26. Total Energy At extremes, K = 0, U = ½ kA2
At center, U = 0, K = ½ m(Aw)2 = ½ kA2

27. Total Energy x = A cos(wt) So U = ½ kx2 = ½ kA2 cos2(wt)
v = −Aw sin(wt)
So K = ½ mv2 = ½ mA2w2 sin2(wt) = ½ kA2 sin2(wt)
E = U + K = ½ kA2 [cos2(wt) + sin2(wt)]
E = ½ kA2 = ½ mvmax2