1. Vibrations and Waves 13.01 Hooke’s Law 13.02 Elastic Potential Energy 13.03 Comparing SHM with Uniform Circular Motion 13.04 Position, Velocity and Acceleration as a Function of Time Topics 13.05 Motion of a Pendulum Vibrations and Waves (3 of 33)

2. Hooke’s Law If an object vibrates or oscillates back and forth over the same path, each cycle taking the same amount of time, the motion is called periodic (T). m We assume that the surface is frictionless. There is a point where the spring is neither stretched nor compressed; this is the equilibrium position. We measure displacement from that point (x = 0 ). x = 0 Vibrations and Waves (4 of 33)

3. m x = 0 Hooke’s Law The minus sign on the force indicates that it is a restoring force – it is directed to restore the mass to its equilibrium position. The restoring force exerted by the spring depends on the displacement: m x F Vibrations and Waves (5 of 33)

4. m x F (a) (k) is the spring constant (b) Displacement (x) is measured from the equilibrium point (c) Amplitude (A) is the maximum displacement (e) Period (T) is the time required to complete one cycle (f) Frequency (f) is the number of cycles completed per second (d) A cycle is a full to-and-fro motion Hooke’s Law Vibrations and Waves (6 of 33)

5. If the spring is hung vertically, the only change is in the equilibrium position, which is at the point where the spring force equals the gravitational force. m xoxo mg Equilibrium Position Hooke’s Law Vibrations and Waves (7 of 33)

6. Any vibrating system where the restoring force is proportional to the negative of the displacement moves with simple harmonic motion (SHM), and is often called a simple harmonic oscillator. Hooke’s Law Vibrations and Waves (8 of 33)

7. Potential energy of a spring is given by: The total mechanical energy is then: The total mechanical energy will be conserved Elastic Potential Energy Vibrations and Waves (9 of 33)

8. If the mass is at the limits of its motion, the energy is all potential. m A m x = 0 v ma x If the mass is at the equilibrium point, the energy is all kinetic. Elastic Potential Energy Vibrations and Waves (10 of 33)

9. This can be solved for the velocity as a function of position: The total energy is, therefore And we can write: where Elastic Potential Energy Vibrations and Waves (11 of 33)

10. The acceleration can be calculated as function of displacement m x F Elastic Potential Energy Vibrations and Waves (12 of 33)

11. If we look at the projection onto the x axis of an object moving in a circle of radius A at a constant speed v max, we find that the x component of its velocity varies as: This is identical to SHM. A v ma x  x v Comparing Simple Harmonic Motion with Circular Motion Vibrations and Waves (13 of 33)

12. Therefore, we can use the period and frequency of a particle moving in a circle to find the period and frequency of SHM: Comparing Simple Harmonic Motion with Circular Motion Vibrations and Waves (14 of 33)

13. A mass m at the end of a spring vibrates with a frequency of 0.88 Hz. When an additional 680 g mass is added to m, the frequency is 0.60 Hz. What is the value of m? Comparing SHM with Uniform Circular Motion (Problem) Frequency of oscillation k is constant regardless of the mass attached to the spring Vibrations and Waves (15 of 33)

14. Vibrations and Waves 11-01 ~M1~M2~M3~M4~M5~M6~M7~M8~M10 ~M9 ~M11~M12~M13~M14~M15~M16~M17~M18~M20 ~M19 ~M21~M22~M23~M24~M25~M26~M27~M28~M30 ~M29 ~M31~M32~M33~M34~M35~M36~M37~M38~M40 ~M39 ~M41~M42~M43~M44~M45~M46~M47~M48~M50 ~M49 ~M51~M52~M53~M54~M55~M56~M57~M58~M60 ~M59 A mass on a spring undergoes SHM. When the mass passes through the equilibrium position, its instantaneous velocity (A) is maximum. (B) is less than maximum, but not zero. (C) is zero. (D) cannot be determined from the information given.

15. Vibrations and Waves 11-02 ~M1~M2~M3~M4~M5~M6~M7~M8~M10 ~M9 ~M11~M12~M13~M14~M15~M16~M17~M18~M20 ~M19 ~M21~M22~M23~M24~M25~M26~M27~M28~M30 ~M29 ~M31~M32~M33~M34~M35~M36~M37~M38~M40 ~M39 ~M41~M42~M43~M44~M45~M46~M47~M48~M50 ~M49 ~M51~M52~M53~M54~M55~M56~M57~M58~M60 ~M59 A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its kinetic energy is a minimum? (A) at either A or B (B) midway between A and B (C) one-fourth of the way between A and B (D) none of the above

16. A 0.60 kg mass at the end of a spring vibrates 3.0 times per second with an amplitude of 0.13 m. Determine the velocity when it passes the equilibrium point, Comparing SHM with Uniform Circular Motion (Problem) Vibrations and Waves (18 of 33)

17. A 0.60 kg mass at the end of a spring vibrates 3.0 times per second with an amplitude of 0.13 m. Determine the velocity when it is 0.10 m from equilibrium, Comparing SHM with Uniform Circular Motion (Problem) con’t Vibrations and Waves (19 of 33)

18. A 0.60 kg mass at the end of a spring vibrates 3.0 times per second with an amplitude of 0.13 m. Determine the total energy of the system, Comparing SHM with Uniform Circular Motion (Problem) con’t Vibrations and Waves (20 of 33)

19. Vibrations and Waves 11-03 ~M1~M2~M3~M4~M5~M6~M7~M8~M10 ~M9 ~M11~M12~M13~M14~M15~M16~M17~M18~M20 ~M19 ~M21~M22~M23~M24~M25~M26~M27~M28~M30 ~M29 ~M31~M32~M33~M34~M35~M36~M37~M38~M40 ~M39 ~M41~M42~M43~M44~M45~M46~M47~M48~M50 ~M49 ~M51~M52~M53~M54~M55~M56~M57~M58~M60 ~M59 A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its potential energy is a minimum? (A) at either A or B (B) midway between A and B (C) one-fourth of the way between A and B (D) none of the above

20. Vibrations and Waves 11-04 ~M1~M2~M3~M4~M5~M6~M7~M8~M10 ~M9 ~M11~M12~M13~M14~M15~M16~M17~M18~M20 ~M19 ~M21~M22~M23~M24~M25~M26~M27~M28~M30 ~M29 ~M31~M32~M33~M34~M35~M36~M37~M38~M40 ~M39 ~M41~M42~M43~M44~M45~M46~M47~M48~M50 ~M49 ~M51~M52~M53~M54~M55~M56~M57~M58~M60 ~M59 Doubling only the amplitude of a vibrating mass-and- spring system produces what effect on the system's mechanical energy? (A) increases the energy by a factor of two (B) increases the energy by a factor of three (C) increases the energy by a factor of four (D) produces no change

21. A mass of 2.62 kg stretches a vertical spring 0.315 m. If the spring is stretched an additional 0.130 m and released, how long does it take to reach the (new) equilibrium position again? Comparing SHM with Uniform Circular Motion (Problem) Vibrations and Waves (23 of 33)

22. Vibrations and Waves 11-05 ~M1~M2~M3~M4~M5~M6~M7~M8~M10 ~M9 ~M11~M12~M13~M14~M15~M16~M17~M18~M20 ~M19 ~M21~M22~M23~M24~M25~M26~M27~M28~M30 ~M29 ~M31~M32~M33~M34~M35~M36~M37~M38~M40 ~M39 ~M41~M42~M43~M44~M45~M46~M47~M48~M50 ~M49 ~M51~M52~M53~M54~M55~M56~M57~M58~M60 ~M59 Doubling only the spring constant of a vibrating mass- and-spring system produces what effect on the system's mechanical energy? (A) increases the energy by a factor of three (B) increases he energy by a factor of four (C) produces no change (D) increases the energy by a factor of two

23. A simple pendulum consists of a mass at the end of a lightweight cord. We assume that the cord does not stretch, and that its mass is negligible. The Simple Pendulum Vibrations and Waves (25 of 33)

24. mg  F s x Small angles x  s k for SHM L m  The Simple Pendulum Vibrations and Waves (26 of 33)

25. Vibrations and Waves 11-06 ~M1~M2~M3~M4~M5~M6~M7~M8~M10 ~M9 ~M11~M12~M13~M14~M15~M16~M17~M18~M20 ~M19 ~M21~M22~M23~M24~M25~M26~M27~M28~M30 ~M29 ~M31~M32~M33~M34~M35~M36~M37~M38~M40 ~M39 ~M41~M42~M43~M44~M45~M46~M47~M48~M50 ~M49 ~M51~M52~M53~M54~M55~M56~M57~M58~M60 ~M59 A simple pendulum consists of a mass M attached to a weightless string of length L. For this system, when undergoing small oscillations (A) the frequency is proportional to the amplitude. (B) the period is proportional to the amplitude. (C) the frequency is independent of the length L. (D) the frequency is independent of the mass M.

26. The length of a simple pendulum is 0.760 m, the pendulum bob has a mass of 365 grams, and it is released at an angle of 12.0° to the vertical. With what frequency does it vibrate? Assume SHM. Comparing SHM with Uniform Circular Motion (Problem) Vibrations and Waves (28 of 33)

27. The length of a simple pendulum is 0.760 m, the pendulum bob has a mass of 365 grams, and it is released at an angle of 12.0° to the vertical. What is the pendulum bob’s speed when it passes through the lowest point of the swing? L 00 Comparing SHM with Uniform Circular Motion (Problem) con’t Conservation of energy Vibrations and Waves (29 of 33)

28. The length of a simple pendulum is 0.760 m, the pendulum bob has a mass of 365 grams, and it is released at an angle of 12.0° to the vertical. Assume SHM. What is the total energy stored in this oscillation, assuming no losses? Comparing SHM with Uniform Circular Motion (Problem) con’t L 00 Vibrations and Waves (30 of 33)

29. Summary of Chapter 11 For SHM, the restoring force is proportional to the displacement. The period is the time required for one cycle, and the frequency is the number of cycles per second. Period for a mass on a spring: During SHM, the total energy is continually changing from kinetic to potential and back. Vibrations and Waves (31 of 33)

30. A simple pendulum approximates SHM if its amplitude is not large. Its period in that case is: The kinematics of a mass/spring system: Velocity Acceleration Summary of Chapter 11 Vibrations and Waves (32 of 33)