Chapter 11– Harmonic Motion Peter Anderson ander324@purdue.edu Office: PHYS 221 Office Hours: Tuesday 12pm to 2pm

The prototypical oscillator involves a mass and a spring and is called the “simple harmonic oscillator”. k m k m where k m

Quiz 1 A particle undergoing simple harmonic oscillation of period T is at –A at time t=0. At time t = 2.00T is it at -A A Between –A and 0 Between 0 and A

Quiz 2 A particle undergoing simple harmonic oscillation of period T is at –A at time t=0. At time t = 3.50T is it at -A A Between –A and 0 Between 0 and A

Quiz 3 A particle undergoing simple harmonic oscillation of period T is at –A at time t=0. At time t = 5.25T is it at -A A Between –A and 0 Between 0 and A

SHM and energy The hallmark equation for a SHO is 𝑎 𝑡 =− 𝜔 2 𝑥(𝑡) Anything that follows that equation is a SHO The Total Energy of the SHO is 𝐸= 1 2 𝑘 𝐴 2

Quiz 4 Which of the following relationships between the force F on a particle and the particle’s position x implies SHO? 𝐹 = −5𝑥 𝐹 = −400 𝑥 2 𝐹 = 10𝑥 𝐹 = 3 𝑥 2

Sample Problem 1 In the figure, a penguin (obviously skilled in aquatic sports) dives from a uniform board that is hinged at the left and attached to a spring at the right. The board has a length L = 2.0m and mass m = 12 k; the spring constant is k=1300N/m. When the penguin dives, it leave the board and spring oscillating with a small amplitude. Assume that the board is stiff enough not to bend, and find the period T of the oscillations.

Damped Motion With damped motion all three cases eventually lose all of their energy due to frictional forces Under Damped Still oscillates but eventually dies Over Damped Does not oscillate Critically Damped Approaches 0 energy in the fastest possible manner

Damped Motion

Quiz 5 If you were to design a bridge and found it had a resonance frequency. How would you want to damp this frequency? Over Under Critically

Quiz 6 Will this bowling ball hit me in the face? Yes No I hope so