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.
Validity of the small-angle approximation
Effect of amplitude on a real pendulum
Vertical mass–spring equilibrium
Energy in SHM
Phase constant from initial conditions
Angular frequency vs frequency
Physical pendulum
Energy method for finding ω
The condition for simple harmonic motion
Period of a vertical spring
Phase relationships
Defining condition for SHM
Short answer 1. Define or explain: Underdamped, critically damped, overdamped
3 ptsShort answer 2. Define or explain: Damped oscillation solution and its envelope
3 ptsShort answer 3. Define or explain: Maximum acceleration in SHM
3 ptsShort answer 4. Define or explain: Simple pendulum period
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.