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12. Prove that a first integral of the equations to a geodesic on a surface of revolution (about Oz) is given by
1. Show that any system of forces can be reduced to a wrench in one definite manner.
2. Enunciate and prove the Principle of Virtual Work for a system of bodies.
Show how the stability of a system can be determined from a knowledge of the work function.
3. Investigate the form assumed by a uniform inextensible string under the action of a central repulsive force varying inversely as the square of the distance.
4. A gravitating spherical shell of radii a and b, and density p, is divided into two parts by a plane through the centre. Find the force required to separate the parts.
5. If two different masses have the same external equipotential surfaces, show that their potentials at any external point are proportional to their masses.
Show also that the two masses have the same centre of mass and the same principal axes at that point.
6. Prove that the work required to scatter the particles of a gravitating system to infinite distances from each other is equal to m V, where V is the potential at the place where the element of mass m is situated.
Find the work required in changing a uniform sphere of attracting matter into two uniform spheres, each half the original mass and volume, and at an infinite distance apart.
7. State and prove the geometrical relations between the positions of the metacentres of a floating body and the surfaces of buoyancy and flotation. A homogeneous right circular cylinder floats in a liquid. Find the condition that the equilibrium may be stable whatever the liquid.
8. A cone of height h and radius of base a floats vertex downwards in water, the specific gravity of the cone being σ.
The water is set in rotation about a vertical axis. Determine what angular velocity will be sufficient to make the water just reach the edge of the base of the cone.
9. A homogenous solid floats completely immersed in a liquid, the density of which varies as the depth. Determine the time of a small vertical oscillation of the solid in terms of the depth of its centre of gravity.
10. Investigate as accurate a working formula as you can for determining the heights of mountains by means of a barometer.
1. Establish the differential equation of a central orbit in polar coordinates.
Find the law of force for an orbit whose equation is u = a cos (no + a). Consider the case when n 1.
2. Find what alteration would be made in the length of the year if the mass of the Earth were equal to that of the Sun, instead of being negligible in comparison with it.
3. A particle is projected in any manner in the presence of a number of centres of force, situated at any points in
space and having different strengths, the attraction to each centre varying directly as the distance. Show that the path of the particle will be an ellipse, and find an expression for the periodic time.
4. A right cone stands on an elliptic base. Find the principal moments of inertia at its centre of gravity.
to the law are situated at distance 2a apart.
5. Two equal centres of force, each attracting according Show that a particle placed midway between them is in unstable equilibrium, and will, if slightly displaced towards one of the centres of force, reach this point after infinite time.
6. A smooth pulley is at height a above the edge of a table. A string is coiled up on the table, rises vertically, passes over the pulley, and hangs with the free end at the level of the table. If it is slightly displaced from this position, show that when the length hanging freely is x, the velocity is
7. A uniform rod of mass M and length 2a lies on a smooth table. A particle of mass m impinges on the rod with velocity u in a direction perpendicular to the length of the rod. If the coefficient of restitution between rod and particle is e, find the point of impact such that m is brought to rest.
that the problem is impossible if M< e
8. A disc is moving in any manner in its own plane. Show that there is a straight line in the disc such that if any point on it is arrested, the disc is stopped dead.
9. A uniform rod of mass M and length has fastened to it a particle of mass m, at distance e from one end of the rod. If the whole swing in a vertical plane about a horizontal axis through this end, find the time of a small oscillation. Show that a certain position of the particle can be found which will make this time a minimum.
10. Fine weightless thread is wound on a cylindrical reel, and the free end of the thread is fastened to a point of a smooth inclined plane. Find the acceleration of the reel down the plane, if the thread as it unwinds lies along a line of greatest slope of the plane, and the centre of gravity of the reel is in the vertical plane through this line.
1. If a body moves in a semicircle under the action of a force directed to an infinitely distant point, find, by Newtonian methods, the law of force.
2. If equal areas were described in equal times by the line joining the Earth to the Moon, what could be deduced about the forces exerted on the Earth and Moon by the remainder of the solar system? In the actual case, how does the rate of describing area vary at different parts of the Moon's orbit?
3. Show that the curvature of a ray of light depends on the variation of the optical index of the medium in the direction of the principal normal to the ray, and investigate the law of this dependence.
4. Find an expression for the dispersion of a ray of white light in passing through a prism, and show that if the angle of the prism is less than the critical angle for the substance of the prism, a minimum value may be obtained for the dispersion.
5. Why is the brightness of the outer part of the field of view of a telescope less than that of the central part when no stop is used? Find expressions for the angular size of (a) the central region of uniform brightness, (b) the whole visible field.
6. Find what is the least height to which a fountain must rise that a rainbow may be seen at mid-day in the spray by a person looking from a tower, 100 feet high and
300 feet to the north of the fountain in latitude 54° north, when the declination of the Sun is 21°, given the deviation of the red ray in the ordinary rainbow to be 138°. Find the result also when the Sun's declination is -9°.
7. Explain fully how you would find your position at sea from simultaneous observations of the altitudes of two known stars with the means available on board a ship.
8. Account for the variations in the apparent positions of the fixed stars, and obtain expressions for the total apparent displacement of a star at any time.
9. How are eclipses (a) of the Sun, (b) of the Moon produced?
Obtain the conditions necessary for an eclipse in each
10. Explain fully how the mass of the Sun has been determined, taking the mass of the Earth as unit.
1. Give an account of the method of determining surface tension by means of the measurement of ripples.
2. On the kinetic theory of gases, find an expression for the coefficient of viscosity, and deduce a method of determining the mean molecular free path.
3. Give a description of Kundt's experiments on the velocity of sound in gases, and mention the deductions that have been drawn from them relative to the velocity of sound in straight tubes.