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A smooth circular wire of radius a and mass M has strung on it a ring of mass m, and swings in a vertical plane round a point on its circumference. Show that the periods of oscillation of the system are those of simple pendulums of lengths 2a and
9. Investigate the steady motion of a sphere moving in an horizontal circle on the inner surface of a right cone whose axis is vertical, the surfaces in contact being perfectly rough. Find the period of the oscillations when the motion is slightly disturbed.
10. A smooth rod of length 2a is smoothly hinged at one end to a point on a perfectly rough horizontal plane, and leans against a cylinder of equal mass lying on the plane, whose diameter is equal to the length of the rod. The vertical plane through the rod contains the centre of gravity of the cylinder and is perpendicular to its axis. If the system is released from rest when the rod is very nearly vertical, show that the velocity of the centre of the cylinder at the instant when the rod ceases to be a tangent to its surface is
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1. Find the distribution of electricity over an insulated spherical conductor of radius r due to a charge e placed outside the sphere at a distance d from its centre.
2. Assuming that a conductor is completely discharged when joined by a wire to another conductor enclosing it, show that the law of electric force is that of the inverse square of the distance.
3. Prove that a charged body placed in an electric field cannot be in stable equilibrium.
4. Determine the intensity of magnetisation of an ellipsoid of permeability μ placed in a uniform magnetic field.
5. Find the force of attraction between two coaxial circular circuits of radii r1, r2, and carrying currents in the same sense i, and i1⁄2, respectively, r, being very small compared with r. Show that the attraction is a maximum when the distance between the planes of the circuits is"
6. If the potential V is given arbitrarily over a closed surface, and AV = 0 inside and outside the surface, show that the potential at any point P is -ff VodS taken over the surface, where a denotes the electric density at a point on the surface induced by a unit charge of electricity at P, if the surface is a conductor connected to the earth.
7. In any network of conductors of resistances 71, 7;, 73, ... 7, and having internal electromotive forces E,, E1, E... En, respectively, show that the actual current strengths make the expression r1 C+ r2 C22 +
- 2 (E1 C1+ E2 C2+...) a minimum. Apply this theorem to determine the currents in the arms of a Wheatstone bridge.
8. Obtain an expression for the intensity of illumination at any point on a screen due to a parallel beam of light diffracted through an infinitely long narrow slit of uniform width.
9. Prove, by the method of Fresnel, that the direction of vibration at any point of a wave front in a crystal is parallel to the projection of the radius vector of the wave on the tangent plane at the corresponding point on the wave surface.
10. Discuss the general theory of equivalent lenses.
1. Find the conditions satisfied by a pure strain.
Show that any strain can be resolved into a pure strain and a rotation.
2. Show that, in the case of an isotropic body, the potential energy can be expressed as a quadratic function of the strains involving only two constants.
Express for such a body Young's modulus in terms of the moduli of compression and rigidity.
3. A uniform spherical shell of radii a and b contains gas at a pressure p. Investigate the components of stress at any point within the shell.
4. Examine the stress and strain produced at any point when one end of an elliptic cylinder is fixed, and a torsion couple applied at the other end.
5. A uniform and uniformly loaded horizontal beam rests on a number of supports. Establish the relation connecting. the bending moments at three consecutive supports and the distances between these supports. When such a beam rests on three supports, examine the position of the supports for which the beam can support the greatest uniform load.
6. Obtain the differential equations of fluid motion in Lagrange's form in terms of the coordinates of the separate particles.
In the equations
x = a + Aeb sin k (a + ct),
У b - Akb cos k (a + ct),
a and b refer to a particle in the Lagrangian system, and x and y are its coordinates at time t. Show that, when proper values are given to the other constants, the equations represent wave motion in a liquid of infinite depth.
7. Show that any continuous acyclic irrotational motion of a liquid can be regarded as due to a distribution of simple sources and doublets over the boundary.
When the liquid extends to infinity and is at rest there, show that the distribution of sources over the internal boundary only need be considered.
8. If u, v, w be the components of velocity at any point of
a fluid moving rotationally, show that in general they can be represented in the form
and show how to determine ø, F, G, H.
9. An infinitely long circular cylinder is moving with uniform velocity perpendicular to its length in an infinite mass of liquid at rest at infinity. Show that the only effect of the pressure of the liquid is an apparent increase of mass of the cylinder.
If in addition to the motion produced by the cylinder there is an independent circulation round it, show that, in order to keep the cylinder moving with uniform velocity, a force must be applied perpendicular to the direction of motion.
10. Distinguish between 'group' velocity and 'wave' velocity, and show that the former is one-half the latter in the case of gravity waves in deep water.
1. Give some account of the experimental methods by which the value of the constant of gravitation has been found.
2. Explain how the existence of surface tension can be accounted for on the assumption of inter-molecular forces.
8. What phenomena in sound are analogous to the diffraction of light?
4. What methods have been employed for studying experimentally the motion of a point of a vibrating string? 5. Discuss some of the principal researches by which the velocity of sound has been measured. How does the velocity through air confined in a pipe compare with that in free air?
6. Give some experimental evidence which tends to show that in the electro-magnetic theory of light the light vector (i.e. the vector which affects a photographic plate) (a) is perpendicular to the plane of polarization; (b) corresponds to the electric force.
7. Describe some optical experiments which have been undertaken to determine the effect of moving matter on the ether. State the results obtained, and explain how it has been attempted to reconcile them.
8. Give an account of some attempt which has been made to explain the phenomenon of optical dispersion.
9. Give a description of some of the methods, theoretical and practical, by which the specific heat of a gas at constant volume has been determined.
10. Show that the ratio of the energy of translation of the molecules to the total energy of a gas is (Y - 1) where y is the ratio of the specific heats. How has this been verified?
1. If the primary terminals of a transformer are connected to an alternating supply, explain what becomes of the electric energy supplied to the transformer when ( the secondary circuit is open,