2. Photo by Mark Tippens A TRAMPOLINE exerts a restoring force on the jumper that is directly proportional to the average force required to displace the mat. Such restoring forces provide the driving forces necessary for objects that oscillate with simple harmonic motion.

3. Objectives: After finishing this unit, you should be able to: Write and apply Hooke’s Law for objects moving with simple harmonic motion.Write and apply Hooke’s Law for objects moving with simple harmonic motion. Describe the motion of pendulums and calculate the length required to produce a given frequency.Describe the motion of pendulums and calculate the length required to produce a given frequency. Write and apply formulas for finding the frequency f, period T, velocity v, or acceleration a in terms of displacement x or time t.Write and apply formulas for finding the frequency f, period T, velocity v, or acceleration a in terms of displacement x or time t.

4. Periodic Motion Simple periodic motion is that motion in which a body moves back and forth over a fixed path, returning to each position and velocity after a definite interval of time. Amplitude A Period (seconds,s) Period, T, is the time for one complete oscillation. (seconds,s) Frequency Hertz (s -1 ) Frequency, f, is the number of complete oscillations per second. Hertz (s -1 )

5. Example 1: The suspended mass makes 30 complete oscillations in 15 s. What is the period and frequency of the motion? xF Period: T = 0.500 s Frequency: f = 2.00 Hz

6. Simple Harmonic Motion, SHM Simple harmonic motion is periodic motion in the absence of friction and produced by a restoring force that is directly proportional to the displacement and oppositely directed. A restoring force, F, acts in the direction opposite the displacement of the oscillating body. F = -kx A restoring force, F, acts in the direction opposite the displacement of the oscillating body. F = -kx xF

7. Hooke’s Law When a spring is stretched, there is a restoring force that is proportional to the displacement. F = -kx The spring constant k is a property of the spring given by: k = FxFx F x m

8. Displacement in SHM m x = 0x = +Ax = -A x Displacement is positive when the position is to the right of the equilibrium position (x = 0) and negative when located to the left. The maximum displacement is called the amplitude A.

9. Period and Frequency as a Function of Mass and Spring Constant. For a vibrating body with an elastic restoring force: Recall that F = ma = -kx Recall that F = ma = -kx: The frequency f and the period T can be found if the spring constant k and mass m of the vibrating body are known. Use consistent SI units.

10. The Simple Pendulum The period of a simple pendulum is given by: mg L For small angles 

11. Summary Simple harmonic motion (SHM) is that motion in which a body moves back and forth over a fixed path, returning to each position and velocity after a definite interval of time. F x m The frequency (rev/s) is the reciprocal of the period (time for one revolution).

12. Summary (Cont.) F x m Hooke’s Law: In a spring, there is a restoring force that is proportional to the displacement. The spring constant k is defined by:

13. Summary (SHM) m x = 0x = +Ax = -A x v a ½mv A 2 + ½kx A 2 = ½mv B 2 + ½kx B 2 Conservation of Energy:

14. Summary (SHM)

15. Summary: Period and Frequency for Vibrating Spring. m x = 0x = +Ax = -A x v a

16. Summary: Simple Pendulum and Torsion Pendulum L