Physics 310 (Midterm I)
Midterm I, Fall 1999
Problem 1: 25 points
A boat with initial speed $v_0$ is launched on a
lake. The boat is slowed by the water with a force $F= -\alpha
~ exp^{\beta v}$.
Find the velocity $v(t)$.
Problem 2: 25 points
A particle moves under the influence of a {\bf force}
$F(x) = - k x + k x^3/\alpha^2$.
a) Find $U(x)$, the potential energy.
b) Find the equilibrium points, and determine whether
they are stable or unstable.
c) Sketch U(x).
Problem 3: 25 points
A rocket has initial mass $M$ and a fuel burn rate of
$\alpha$ (i.e. $dm/dt$).
What is the minimum exhaust velocity
that will allow the rocket to lift off immediately
after firing ?
Problem 4: 25 points
(a) A simple pendulum is suspended by a weightless rod
of length l. If there is a retarding force of $m \sqrt{gl}
d{\theta}/dt $, write down the equation of motion for small
oscillations.
(b) Sketch the phase diagrams for a simple harmonic oscillator
and the underdamped harmonic oscillator ?
(c) How does the period of a simple harmonic oscillator depend on
its amplitude ?
(d) Consider a forced damped harmonic oscillator with very
small damping.
If the oscillator has mass m,
natural frequency $\omega_0$, and the
amplitude of the forcing
function is $F_0$, {\it at what frequency does resonance occur} ?
University of Hawaii/ teb@uhhepi.phys.hawaii.edu