Q1. Consider the mass and spring system shown in the figure. Both springs have the same spring constant k and all three strings have the same length l. All the masses are the same.
(a) Find the normal frequencies
(b) Find the normal coordinates
(c) Find the motion of the normal coordinates
(d) The system is started with right most mass displaced from equilibrium by an amount x0 and the other masses in their equilibrium positions. Find the physical motion of the system.
(e) Take x0 = 0.1, l = 1 and k = 1 and plot the motion of each mass as a function of time.
Q2. A mass M on a spring with spring constant k and damping coefficient b is driven by a time-dependent driving force.
The driving force is
and is repeated for later times as shown in the plot.
The initial conditions are x(0) = 0 and v(0) = 0. Take ω0 and F0 to be one and γ = 1/10 in the appropriate units. Find x(t) and plot your result as a function of time. Hint: expand F(t) in a Fourier series.
Q3. An observer is sitting in frame S at rest with respect to S. A second person is sitting in frame S' which is moving with β = 0.8 away from S. The person in S' who is at rest with respect to S' throws a ball with a velocity of β = 0.5 in frame S' toward the person in frame S. In the balls rest frame it emits a flash of light with frequency v.
(a) What is the velocity of the ball in frame S?
(b) What frequency is see by the observer in frame S'?
(c) What frequency is see by the observer in frame S?
(d) The observer in S uses a mirror to send the light back to the observor in S'. What frequency is see by the observer in frame S'?
Q4. The methods that we studied in class for solving the wave equation can be applied to other partial differential equations as well. Heat flow in one dimension is described by the one-dimensional heat equation
∂T/∂t = κ(∂2T/∂x2)
where κ is the thermal diffusivity and T(x, t) is the temperature.
(a) What are the units of κ?
(b) Find the general solution of the heat equation using separation of variables.
(c) Now assume that we have a one dimensional rod of length L whose ends are held at T = 0 so that T(0, t) = T(L, t) = 0. Apply these boundary conditions to your general solution.
(d) Suppose the initial temperature distribution is given by T(x, 0) = T0x(x-L)/L2. Find T(x, t). Hint: Remember Fourier analysis.
(e) P1lot your solution at t = 0 and show that you reproduce the initial condition. Make a second plot showing T(L/2,t) as a function of time. What do you get for the long time limit? Is this what you would expect?
Q5. A disk of radius R and mass M is attached to a spring as shown in the figure. Let x by the displacement of the spring from its equilibrium position. The disk rolls without slipping and there is no damping. The disk is not of uniform density but has a mass density
σ(r) = (M/2πR2)er/R
(a) Using energy methods derive the equation of motion for this system.
(b) What is the frequency of oscillation for this system?