Oscillations 1. Different types of motion: Uniform motion 1D motion with constant acceleration Projectile motion Circular motion Oscillations 2. Different types of oscillations: Periodic oscillations A motion is called periodic when the system comes back to the same physical conditions every time interval T. Harmonic oscillations (the simplest, the most important type of periodic oscillations) Chaotic oscillations

3. Harmonic oscillations Relation between circular motion and simple harmonic oscillations y R=A x linear equation Period of SHO is independent from amplitude!

T x A=xmax t v t a t

4. Hook’s law and simple harmonic motion Newton’s second law: Simple harmonic motion: Question: Two identical masses hang from two identical springs. In case 1, the mass is pulled down 2 cm and released. In case 2, the mass is pulled down 4 cm and released. How do the periods of their motions compare? A. T1 < T2 B. T1 = T2 C. T1 > T2 Period is independent of amplitude! This is in fact general to all SHM (not only for springs)!

x t 5. The simple pendulum mg L If than Units: Example: Independent from amplitude! Units: Example:

Example: A person swings on a swing Example: A person swings on a swing. When the person sits still, the swing moves back and forth at its natural frequency. If, instead, the person stands on the swing, the new natural frequency of the swing is: A. Greater B. The same C. Smaller If the person stands, L becomes smaller Example: Grandpa decides to move to the Moon, and he naturally takes his old pendulum clock with him. But gravity on the Moon is approximately g/6... As a result, his clock is: A. Too fast B. Too slow C. Too fast until Noon, too slow after noon The period of the pendulum is longer on the Moon. So each “second according to this clock” is then longer than a real second. To tune the clock you can move the clock’s disk up.

Question: Mass m attached to a spring with a spring constant k Question: Mass m attached to a spring with a spring constant k. If the mass m increases by a factor of 4, the frequency of oscillation of the mass is doubled is multiplied by a factor of 4 is halved is multiplied by a factor of 1/4 Question: A 2.0 kg mass attached to a spring with a spring constant of 200 N/m. The angular frequency of oscillation of the mass is __ rad/s. A. 2 B. 10 C. 60 D. 100

6. Energy in the simple harmonic motion U x E –A A K t U t E t Total mechanical energy is constant through oscillation: conservation of energy!

7. Damped Harmonic Motion x(t) t

8. Resonance