1. Oscillations

2. Definitions

3. Frequency

4. 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. The mass and spring system is a useful model for a periodic system. Oscillations of a Spring

5. Effect of mass and the force constant on the period For a spring that exerts a linear restoring force the period of a mass-spring oscillator increases with mass and decreases with spring stiffness.

6. Period and frequency of mass- spring system with SHM

7. Oscillations of a Spring

9. The restoring force is directed opposite to the direction of motion of the mass to restore the mass to its equilibrium position. k is the spring constant. The force is not constant, so the acceleration is not constant either. Oscillations of a Spring

10. 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. Oscillations of a Spring

11. Any motion that repeats itself is periodic or harmonic. If the motion is a sinusoidal function of time, it is called simple harmonic motion (SHM). Simple Harmonic Motion

12. Position as a function of time x = A cos (  t) Where: A = the amplitude (maximum displacement of the system) t = time  = angular frequency  = 2  f

13. Velocity and Acceleration as a function of time

14. Maximum velocity

15. Maximum acceleration

16. Velocity as a function of position

17. Acceleration

18. Maximum velocity

19. Pendulums In a simple pendulum, a particle of mass m is suspended from one end of an unstretchable massless string of length L that is fixed at the other end. The restoring torque acting on the mass when its angular displacement is , is:  is the angular acceleration of the mass. Finally, This is true for small angular displacements, .

20. Effect of length on the period of the pendulum For a simple pendulum oscillating the period increases with the length of the pendulum.

21. In the small-angle approximation we can assume that  << 1 and use the approximation sin   . Let us investigate up to what angle  is the approximation reasonably accurate?  (degrees)  (radians)sin  50.0870.087 100.1740.174 150.2620.259 (1% off) 200.3490.342 (2% off) Conclusion: If we keep  < 10 ° we make less than 1 % error. Pendulums