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9. Define Hodograph.
Find the hodograph of a heavy particle projected in vacuo at a given angle to the vertical.
10. A smooth circular tube is fixed in a vertical plane and has a centre of repulsive force which varies inversely as the square of the distance at its lowest point. A heavy particle which is subject also to the action of the 'repulsive force is placed in the tube at its highest point. Find the least value of the repulsive force that this may be a position of stable equilibrium, and the time of a small oscillation about it.
FRIDAY, JUNE 8, from 9.30 A.M. to 12.30.
SECTION IV. Mathematics.
11. Dynamics. 1. Prove that the moment of inertia of a rigid body about any line is equal to the moment of inertia about a parallel line through the centre of mass + Mh?, M being the mass of the body and h the distance between the parallels.
A portion of a solid paraboloid of revolution cut off by a plane at right angles to the axis is such that the moment of inertia about a diameter of the circular base is equal to the moment of inertia about a line drawn through any point on the rim of the base parallel to the axis. Find the ratio between the length of the axis and the radius of the base.
2. Establish D'Alembert's Equations of Motion, and deduce the theorems that if no external forces act on a system,
(1) The momentum of the whole system in any fixed direction is constant;
(2) The angular momentum relative to any line which passes always through the centre of inertia but remains fixed in direction is constant.
3. If two bodies are smooth and slide on each other, or if they are rough and roll on each other without sliding, the kinetic energy of the system is not altered by the normal resistance in the former case nor by the rolling friction in the latter.
A heavy wedge lies at rest on a smooth horizontal plane on which it can move without rotation in a direction perpen
dicular to its lower edge. A heavy sphere is set rolling directly up the upper face which is rough. If the wedge is held at rest until the sphere has been so started, and is then let
go, find the greatest height to which the sphere will attain.
4. What is a Momental Ellipsoid of a Rigid Body ?
If a body is rotating about a fixed point in it, express in terms of the elements of the Momental Ellipsoid, the angular momentum of the body about any given line through the point; and if there is no' fixed point and the body is at rest, find the condition that the motion resulting from a blow shall be one of pure rotation.
A lamina in the form of an ellipse is struck perpendicularly to its plane. Shew that the spontaneous axis of rotation will be a tangent to the ellipse if the point of application of the blow lie anywhere on a certain concentric, coaxal, and similar ellipse.
5. A heavy hemisphere of mass 8m, having a heavy particle of mass 3m fixed to a point P of its circular rim, rests in equilibrium with its curved surface on a rough horizontal plane. Find the time of a small oscillation, if it is slightly disturbed so that its centre and the point P move in the same vertical plane.
6. A heavy rod of length 2 a has a small ring fixed at one end, which slides on a smooth circular wire of radius a fixed in a vertical plane. The rod is slightly disturbed in this vertical plane from its position of equilibrium. By means of Lagrange's process or otherwise find the differential equations necessary for determining its motion.
7. Define Conservative Forces, and T the kinetic and V the potential energy, of a moving system, and shew that if no external forces act, T + V = constant.
8. Find the velocity with which a heavy incompressible fluid runs out through a small orifice in the containing vessel.
If the vessel is one of revolution about a vertical axis and the orifice is at the lowest point, and if u be the velocity with which the surface of the fluid sinks and u, any constant, find the shape of the vessel in order that 22 — 4,2 may vary as the quantity of fluid not yet run out.
9. Find Euler's Equations for the angular motion of a rigid body.
A rigid body acted on by no forces is rotating about an axis very nearly coincident with its greatest Central MomentAxis. Shew that this moment-axis will describe a cone about a certain line (the Invariable Line) fixed in space, and determine the greatest and least angular distances between the Moment-Axis and the Invariable Line.
10. Find the differential equation for the oscillations of air in a smooth tube of small uniform section.
FRIDAY, JUNE 8, from 2.30 to 5.30 P.M.
SECTION IV. Mathematics.
12. Optics and Astronomy. 1. When a pencil is incident directly on a spherical refracting surface, find the point where the direction of a given ray after refraction cuts the axis, to the second power of small quantities.
2. A small pencil of parallel rays falls upon a sphere of water. Find the position of the point of incidence in order that the primary focus of the refracted pencil may be upon the surface of the sphere.
Shew that if a pencil so incident suffers one internal reflexion, the portion of it in the primary plane, as it emerges from the sphere, will consist of parallel rays; and state the connection of this theorem with the theory of the formation of the rainbow. 3. Why is an achromatic combination possible?
A pencil of light passes centrically with small obliquity through two thin lenses in contact, find the condition of achromatism.
4. If a system of parallel rays falls on a reflecting curve lying in the plane of the rays, the distance from a point of incidence P to the corresponding point on the caustic is equal to one-fourth of that chord of the circle of curvature at P which is parallel to the incident ray.
Rays parallel to the axis of x fall upon the curve cos y = 6*; draw the caustic,
5. On the principles of the undulatory theory explain the reflexion of light, and shew that two rays of light reaching a point do not always combine to produce an illumination greater than that of either separately.
6. A circular plane screen of which the centre is O, is the base of a hemispherical box. A small rectangular hole, of which the centre is also at 0, is cut in the screen, its length being small compared with the radius of the hemisphere, and its breadth small compared with a wavelength. A succession of plane waves whose front is parallel to the screen passes through the rectangular hole. Find the illumination at any point of the circle in which the plane through O perpendicular to the shorter sides of the rectangle cuts the hemispherical back of the box,
7. Shew how the deviation error of a transit instrument is corrected'; and determine the uncorrected error by observation of the transits of two known stars.
8. Prove that the declination of the sun on the day when (in latitude o) the twilight is shortest is
-sin-(tan 9° x sin o). 9. Determine the two parts of the equation of time' which arise from the obliquity of the ecliptic and from the excentricity of the earth's orbit.
If the sun were in perigee at the time of the vernal equinox, and if on that day the clock and sun were together, find the greatest value which the equation of time would have ; and shew that it would vanish very nearly at the other equinox and at the solstices.
10. Obtain the differential equation to the moon's radius vector, da u
P T du
h2u2 h2u3 do
SATURDAY, JUNE 9, from 9.30 A.M. to 1 P.M.
SECTION IV. Mathematics.
13. Problems. 1. Shew that the equation x2 + y2 = 23 can be solved in integers.
n.n I I -N.
+ XC + P
x + 2 P
3. If • (m, n) =
where a and n are positive integers, prove that
пр ♡ (, n) (x+p, n— ). 4. Find the locus of the centre of a given ellipse which passes through a fixed point and touches a fixed right line.
5. Shew that the envelope of the axis of a parabola which passes through three given points is a curve of the third class.
6. Prove that the spiral r = 2a0 can be placed so as to touch the parabola, y2 = 4ax, and to have its pole at the foot of the ordinate of the point of contact, and that if it roll from this position along the parabola its pole will describe the axis.
7. Two rough rods, without weight, are cross-jointed like the letter X, O being the joint. The two lower limbs are placed across a heavy rough sphere at rest, and touch it at A, A', the rods being in a vertical diametral plane of the sphere and the point vertically above the centre. The ends of a thread are fastened to points B, B' in the upper limbs, B, B', A, A' being all at the same given distance from 0. If the string is suspended by its middle point so that the lower limbs clip the sphere, the friction is sufficient to keep the sphere from falling. Find the greatest length which the thread can have, that this may
be 8. A heavy cylinder is closed at both ends, and a concentric circular hole is then cut in one end, and the cylinder placed with this end on a horizontal table. Through the upper end a small vertical tube is fitted which extends within the cylinder nearly to the bottom. Water is poured into the upper end of the tube. What will be the levels of the water in the tube and in the cylinder, when the internal pressure of the compressed air becomes just sufficient to upset the cylinder ? ..