Physics 450 - Problem Set 1 - due January 22 2001
Anjum Ansari, ansari@uic.edu     John Marko, jmarko@uic.edu

1. Statistical Fluctuations:
(a) In lecture we considered four particles moving around in a box, and we computed the probabilities of 1,2,3 and 4 particles to be found in the left half of the box.
For N particles, find a formula for the probability that n of them will be found in the left half of the box.
(Hint: Remember that the number of different ways to choose n objects from a set of N objects is N! / [n! (N-n)! ], where ! represents the factorial function)

(b) Plot the distribution for N = 10.

2. Boltzmann Distribution:
(a) Think about a simple harmonic oscillator, for which the energy is

E = m
2
 v2 + k
2
 x2
Find the probability distribution for the position x of the particle at temperature T.

(b) At temperature T, estimate the typical value of x2 excited thermally.

(c) Some commonly used force transducers are `cantilevers', or few-micron sized pieces of metal or glass, which bend in response to applied force. For cantilevers with k = 1 nN/Å, and k = 1 pN/mm, estimate the typical value of |x| generated by thermal fluctuation.

(the first k is similar to that of atomic-force-microscope tips used to study solid surfaces, while the latter k is similar to cantilevers used to do experiments on molecules)

3. Pressure of a gas:
For a gas in a container, use the fact that each translational degree of freedom has » kB T of energy to estimate (you may neglect numerical prefactors) the pressure exerted on the walls of the container, as a function of container volume V, number of molecules N, and temperature T.

(Hint: think about the number of collisions per unit time and the average momentum transferred per collision, and use Newton's 2nd law, force = Dmomentum / Dtime; also recall that pressure is force per area)

4. Drag and Sedimentation:
(a) For a small particle of 3 mm diameter, find the force that must be applied to move it through water with a velocity of 10 m/sec.

(b) If the particle weighs 1.5 times as much as the water which it displaces, find its terminal sedimentation velocity due to gravity.

(Hint: for (b), the terminal velocity is achieved when the gravitational force balances the drag force so that there is no acceleration)


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On 11 Jan 2001, 10:35.