1

Pt 1B Advanced Physics Lent 2004

Classical Dynamics: Question Sheet

J. Ellis Questions are graded A to C in increasing order of difficulty. Review of Newtonian Mechanics and Non-Inertial Frames 1 (C) A circular coin of radius a falls at speed u without rotating onto a smooth horizontal table. The perpendicular to its face makes an angle with the vertical. Determine the state of motion of the coin just after it strikes the table, assuming that the collision is elastic. Show

that when is small the coin strikes the table a second time at an angle of 115 . [Neglect the

effect of gravity.] 2(A) Paraboloidal telescope mirrors can be made by spin casting, which involves rotating the molten glass and its container about a vertical axis as the glass solidifies. By considering the equilibrium of an element of the molten surface show that

xgdx

dy 2=

where y is the height of the surface, x is the distance from the axis of rotation and is the angular velocity of rotation. For a mirror of focal length 2m, what angular velocity is required? [Ans: 1.57 radians s-1] 3(B) A train at latitude in the northern hemisphere is moving due north with a speed v along a straight and level track. Which rail experiences the larger vertical force? Show that the ratio R of the vertical forces on the rails is given approximately by

ga

vhR

sin81

+=

where h is the height of the centre of mass of the train above the rails which are a distance a apart, g is the acceleration of free fall, and is the angular velocity of the Earths rotation. 4(B) A stone is dropped from a stationary helicopter 500m above the ground at the equator. How far from the point vertically beneath the helicopter does it land and in what direction? You should try to solve this problem in two ways: (a) by consideration of the angular momentum of the stone, and (b) by Coriolis force. [Ans: 24cm to the East] 5(C) A weather map shows a shallow, stationary depression centred over a point on the Earth's surface at latitude 50N. The isobars corresponding to pressures of 0.998, 1.000 and

510002.1 Pa are concentric circles of radius 50, 200 and 350 km respectively and the temperature is 300 K. Estimate the wind speed on the middle of these isobars, neglecting the effects of friction between air and the ground. [Ans: 27 km/h]

2

Normal Modes 6(A) When a diatomic molecule is adsorbed onto the surface of a metal the frequency of its internal vibrational mode is changed. If we consider a horizontal surface with the axis of the molecule vertical, a simple model which might describe this phenomenon is as follows. The molecule consists of two point masses, m, separated by a light spring of spring constant k. The mass nearer to the metal is attached to a fixed point (the metal surface) with a light spring of constant K. The two springs are collinear and the motion of the masses is regarded as confined to the line of the springs. Find the normal modes and how their frequencies vary with K. Sketch the results and comment on their physical significance for large and small K.

[Ans. 42222 KkKkm ++= ]

7(C) A simple model of a jet engine comprises three identical thin rigid discs mounted equidistant on a uniform light shaft. Describe the normal modes of oscillation of the system. A small object entering the engine produces an abrupt change in the angular velocity of the first disk. Obtain an expression for the maximum resultant angle of twist of the shaft between the discs, given that the angular frequency of the lowest vibrational normal mode is

. [Ans:

+ 23

11 - an upper bound from the difference between two non-

harmonically related sine-waves] 8(B) A uniform rod of length a hangs vertically on the end of an inelastic string of length a, the string being attached to the upper end of the rod. What are the frequencies of the normal

modes of oscillation in a vertical plane? [Ans: ( ) ag1952 = ] 9(A) The ringing note produced by a tea cup when it is tapped on the rim with a spoon is liable to vary in pitch depending on whereabouts in relation to the handle the cup is tapped. Predict this variation. 10(C) A block of mass M can move along a smooth horizontal track. Hanging from the block is a mass m on a string of length l that is free to move in a vertical plane that includes the line of motion of the block. Find the frequency and displacement patterns of the normal modes of oscillation of the system. (You may assume that to first order the tension in the string is equal to the weight of the mass m).

M

m

l

3

Orbits 11(A) A satellite travelling round the Earth in a circular orbit with centre O experiences, at the point P, a sudden impulse which deflects it into a new orbit. Sketch the orbits to be expected if the impulse acts:

(a) in the direction of the motion of the satellite; (b) in the reverse direction; (c) outwards along the line OP.

On each sketch show clearly, in relation to O and P, the points A and B at which the new orbit is furthest from and nearest to O. 12(B) Two stars of unequal mass are in circular orbit about one another. One day the more massive star suffers a spherically symmetrical loss of matter (it explodes). After explosion the masses of the stars are equal. Show that the binary system will be disrupted if > 3, where is the ratio of the original masses. 13(B) In an attempt to place a satellite into a geostationary orbit, the correct speed and radial distance are achieved but the direction of motion has an angular error in the plane of the orbit. Show that the maximum and minimum radii of the orbit are ( )sin10 r where 0r is the radius of the geostationary orbit. By a short burst of its booster while at maximum radius, the satellite is able to put itself into a circular orbit. If the initial error is 1 minute of arc, calculate the period of this new orbit. [Ans: 1 day + 38 seconds.] 14(B) particles with energy E are scattered through an angle by nuclei of atomic number Z. Assuming that the only interaction between the particles and the nucleus is the electrostatic force, using the formulae derived in lectures show that the distance of closest approach is given by the Rutherford formula:

+12cosec4 0

2 EZe

particles are scattered by lead (Z = 82) through an angle of 60degrees . As the energy of the particles is increased the scattering is found to be in agreement with the Rutherford formula up to an energy E0 = 25 MeV. Estimate an upper limit for the range of the nuclear force in lead. [Ans: 151014 m]

4

15(B) A point mass m on the end of a light string of length l is free to swing as a conical pendulum. Show that, in terms of the (constant) angular momentum J of the bob about the vertical axis, the energy of the pendulum may be written as

( ) 222

1 mlVE +=

where

( ) ( )

22

2

sin2cos1

ml

JmglV +=

is the effective potential that determines the motion in . Sketch the form of ( )V and by differentiating V twice with respect to , show (a) that the bob can move steadily round a circle, with 0 = and angular velocity given by

cos

2

l

g=

(b) that, if the pendulum is given a little extra energy without changing its angular momentum, oscillates about 0 with an angular frequency given by ( )0222 cos31 +=

5

Rigid body dynamics 16(A) A uniform disc of radius 0.1 m and mass 0.4 kg is rotating with angular velocity 1rad/s about an axis at 45degees to its plane through its centre of mass. What is (a) its angular momentum, and (b) its kinetic energy? [Assume the centre of mass is stationary. Ans: (0.7, 0, 1.4)x10-3 kg m2 s1 w.r.t. obvious axes; 3/4mJ.] 17(B) A uniform rectangular tile drops without spinning until its corners reach positions (0, 0, 0), (2a, 0, 0), (2a, 2b, 0), (0, 2b, 0), when it strikes the top of a vertical pole at a point very close to the (0, 0, 0) corner. Just before impact the velocity of the tile was (0, 0, -u). Assuming that the tile does not break, and that the impact is elastic (i.e. the kinetic energy of the tile is conserved), find immediately after impact

(a) the velocity of its centre; (b) the angular momentum about its centre; (c) its angular velocity;

Show that the velocity of the corner at (0, 0, 0) becomes (0, 0, +u) immediately after impact. 18(C) A uniform smooth disc of mass m and radius a is initially at rest. A small particle of the same mass m with an initial velocity v normal to the plane of the disc makes an elastic collision with the disc, striking it at a point midway between the centre and the rim. Find the velocity of the centre of mass of the disc and the angular momentum of the disc about its centre after the collision.

Suppose now that before the collision the disc was rotating about its own axis with angular velocity 2v/3a. Describe the motion after the collision as completely as you can, showing in particular that the disc returns to its initial orientation in the time taken for its

centre to move through a distance 2a . 19(B) The moments of inertia of the Earth about polar and equatorial axes differ by 1 part in 300. Estimate the instantaneous rate of precession of the Earths axis in mid-summer due to the couple exerted by the Sun alone, assuming the Earths equatorial bulge to be concen- trated in a ring round the equator. The Earths axis is inclined at 23.5degrees to the normal to its orbit. [Ans: 1.6x10-4 radians year-1 .] 20(C) A coin of radius a is spun on a perfectly rough table is such a way that its centre is stationary, while its axis precesses steadily about the vertical at a fixed inclination with

angular velocity . Show that 03 = and hence that sin42

a

g= .

Show also that the head of the coin, viewed from above, appears to rotate with angular

velocity ( ) ag /cos1 3 =

6

Elastic Media 21(A) A straight tube 1m long, of radius 10mm and wall thickness 100m, is closed at both ends. It is found that when the pressure inside is increased from 1 to 10 atm the tube lengthens by 100m. What is the bulk modulus of the material of the tube? [Ans: 1.5x1011Nm-2] 22(B) A horizontal cantilever of uniform cross section is such that when a weight of 1 kg is hung from its free end a point half way along the cantilever is displaced downwards by 10 mm. What is the displacement of the end when the same weight is hung from the point half way along? [Ans: 10 mm.] Re-examine the quesion, but for any two points on a cantilever whose cross section varies arbitrarily along its length. Show that the deflection at A due to a load at B is always equal to the deflection at B due to the same load at A. [Hint: remember the proof that M12 = M21 for mutual inductance.] 23(B) A long straight cylindrical wire is fitted with universal couplings at its ends. Through these couplings a torque is applied, and the design is such that the wire is free to choose between remaining straight but twisted or coiling up into a spring. Show that the energy stored in these two configurations for a given torque are in the ratio Y/2n, and hence that the wire will coil up if > 0 (i.e. in practice it will always coil up). 24(B) A pillar has length H when lying horizontally; it is of uniform isotropic material and has uniform diameter