Part III General Physics Examples Class 1999-2000
In each section, qu. 1 is revision of relevant bookwork, qu. 2 is short and/or straightforward, and qu. 3 is a longer problem. The last part of qu. 4.3 is a more challenging problem for enthusiasts.
1. Orbits
1.1Write down the relevant conservation laws for a point mass moving under the influence of a central, conservative force. Use them to derive a differential equation for the radial motion.
1.2Discuss the stability and closure of almost-circular orbits as a function of n for a central force . [Discuss at least the cases n=1, n=-1, and n=-2; and address more general n too, for example, n=-6.
1.3A lunar excursion module is initially in a circular orbit
at a height R/4 above the lunar surface, where R is the radius of the
moon. The objective is to land at point B by firing the module's rockets
briefly at points A and B as indicated. Find the required impulses 
and 
, in terms of the initial momentum P of the module. (Ignore
the rotation of the moon.)
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[Ans: 
, 
.]
2. Rigid body dynamics
2.1Define the inertia tensor, principal axes and principal moments of inertia of a rigid body, and explain their relevance to the angular velocity and angular momentum of the body.
2.2Recall that a spherical top is a rigid body
for which all the principal moments of inertia are equal. Show that
a uniform cone of mass M with height h equal to the diameter
of its base is a spherical top with moment of inertia
.
2.3Such a cone rolls freely without slipping
on a horizontal table, with its curved surface in contact
with the table. Show that this is only possible if the
angular velocity of the cone about its axis satisfies
What happens when this condition is violated?
3. Normal modes
3.1Explain what is meant by the normal modes of oscillation of a many-particle system, and how their frequencies can be found.
3.2Discuss why the specific heats of gases at
moderately high temperatures are in the sequence
H
< O < HO < CO.
3.3An aircraft taking off is (crudely!) represented by two identical thin rods joined rigidly in a T configuration, with landing wheels attached to the ends by identical springs, as illustrated below. truein
(i) Show that the normal mode frequencies are given by
where k is the spring constant of each spring
and M is the mass of the aircraft.
(ii) Describe the oscillations excited when (a) the front wheel, (b) a side wheel passes over a bump of height h in the runway. Assume that , where is the time taken to go over the bump.
4. Elasticity
4.1Define the bending moment B and the moment of area I for a bent beam. Derive the relation B=YI/R where Y is Young's modulus and R is the radius of curvature.
4.2A uniform steel ruler of width a and thickness
b is clamped at its lower end in a vertical position with a
length l protruding above the clamp. A small sideways force
F is applied at the upper end. Find the displacement y
as a function of the height x above the clamp.
4.3(i) Show more generally that when a distributed
transverse force f(x) per unit length is applied to a beam
the equilibrium displacement, when small, satisfies the
differential equation
(ii) Hence show that free transverse oscillations
of the ruler in qu. 4.2 satisfy the differential equation
where 
is the density.
(iii) Show that the possible angular frequencies of transverse
oscillation of the ruler are of the form
where is a
solution of the equation .
(The smallest is )