Assignment 2, solutions

Problem 1:

A system consists of two blocks of known masses M and m (M > m) connected by a light spring.  The system slides toward a wall along a frictionless floor at the speed v as shown.  What is the maximum potential energy U of the system after it bounces off the wall?



Problem 2:

Two point sources emit spherical waves with wave number k and frequency ω into 3D space.  One source is located at the origin and one source a small distance r away from the origin.   A detector is located far away at R, with R >> r.
For each source, let Ψi = A exp(i(kri - ωti + φ))/ri, where ri is the distance from the source and A is real.  For the source at the origin choose φ = 0, while for the source at r choose a non-zero φ.
(a)  Write down an expression |Ψtotal|2 (proportional to the intensity) at the detector.  Make the far-field approximation, i.e. lines from each source to the detector are parallel to each other.
(b)  Let k = kR/R.  What condition must hold for k·r, so that the average intensity at the detector is maximized?
(c)  What happens if another point source is added at position 2r with phase 2φ.
(d)  For case (c), what condition must hold for k·r, so that the average intensity at the detector zero?


Problem 3:

Moisture condenses at the constant rate λ units of mass per unit time on a falling raindrop.  If the drop falls from rest and has initial mass M, find the distance it has fallen in time t.  Neglect air resistance.


Problem 4:

A flexible chain of length 3L and mass m hangs over a frictionless pulley so that the length of the vertical parts of the chain on either side of the pulley is L.  After a slight disturbance, the chain begins to slide to the right. What is the force exerted by the chain on the pulley at the instant the length of the vertical part of the chain on the right is 3L/2?
Hint:  What is the acceleration of the CM of the chain?


Problem 5:

Use the uncertainty principle, ΔxΔp ≥ ħ/2, to estimate the ground state energy of a particle in a one-dimensional well of the form
(a)  U(x) = U0,  -a/2 < x < a/2,  U(x) = infinite for all other values of x,
(b)  U(x) = k|x|.