Oscillations unit test
A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.
Damped oscillation solution and its envelope
Phase relationships
Resonance
Small-angle approximation
Underdamped, critically damped, overdamped
Where kinetic and potential energy peak
Period independent of amplitude
Deriving the pendulum period
Defining condition for SHM
Physical pendulum period
Validity of the small-angle approximation
Amplitude dependence of a real pendulum
Short answer 1. Define or explain: Energy in SHM
3 ptsShort answer 2. Define or explain: Physical pendulum
3 ptsShort answer 3. Define or explain: Displacement, velocity and acceleration in SHM
3 ptsShort answer 4. Define or explain: Energy method for finding ω
3 ptsFree response
10 ptsA uniform rod of mass M and length L is pivoted at one end and is free to swing in a vertical plane about a horizontal axis through the pivot. The rod is displaced by a small angle from vertical and released. The rotational inertia of a uniform rod about one end is I = ML²/3.
A. Derive an expression for the torque exerted on the rod by gravity when the rod makes an angle θ with the vertical.
B. Show that for small angular displacements the rod undergoes simple harmonic motion, and derive an expression for the angular frequency of that motion in terms of L and physical constants.
C. Derive an expression for the period of the rod’s oscillation.
D. A second rod of the same length but twice the mass replaces the first. Indicate whether the period increases, decreases, or stays the same, and justify your answer.