Physics 310 (Midterm II)

Midterm II, Fall 1999

Problem 1: 25 points
An R-L-C circuit contains an inductor of 0.01 H and a resistor of 100 $\Omega$. The oscillation frequency is 1 kHz.

At $t=0$ the voltage across the capacitor is 10 V and the current is 0, find the charge as a function of time.

Problem 2: 25 points
A bead on a rotating circle has the potential energy as a function of $\theta$ shown below.

(a) Sketch the phase portrait of the system.

(b)Identify the equilibrium positions. Which are stable and unstable ?

Problem 3: 25 points
A {\it uniform solid sphere} of mass M and radius R is fixed a distance h above a {\it thin infinite sheet} of mass density $\sigma$ (mass/area). (a) What is the gravitational field of the sheet ? (either use Gauss' Law or integrate directly)

(b) With what force does the sphere attract the sheet ?

Problem 4: 25 points
(a) What is period doubling/bifurcation ? What is the Lyapunov exponent ? What does it measure ?

(b) What is the relation between the gravitational field and the gravitational force ?

(c)Why are there two high tides a day ?
(d) What is a Fourier series ? What is a practical application ?

(e) What electrical circuit is the equivalent of a forced damped harmonic oscillator ?


University of Hawaii/ teb@uhhepi.phys.hawaii.edu