a sudden spring to a bracket on the wall, in the same horizontal plane with the initial position of the animal and at a given distance from the rod, and that the angle through which the rod afterwards oscillates from the vertical is observed, to find what must have been the impulsive stress on the hinge at the instant when the animal started.

Let m, μ, be the respective masses of the rod and the animal; 2a the length of the rod, b the vertical height of the higher end of the rod above the bracket, c the initial distance of the bracket from the rod, and a the observed angle of oscillation. Then, X, Y, representing the horizontal and vertical impulses on the hinge,

(24) A rod AB (fig. 258), attached to a fixed horizontal rod Ox and a fixed vertical rod Oy by rings at its ends, is kept at rest by a man's hand, while a monkey is sitting upon a small platform C vertically above 0: the monkey then springs horizontally from C and alights on the middle point G of the rod, to which it clings tightly to determine the motion of the rod the instant after the monkey's arrival at G, supposing the man to have loosed his hold the instant before.

=

Let AG = a = = BG, ABO = 0, m = L the mass of the moveable rod, m' the mass of the monkey, u = the velocity with which the monkey springs from C, h = the vertical altitude of C above G: then the angular velocity of AB, the instant after the monkey's arrival at G, is equal to

(25) A little animal is resting on the middle point of a thin uniform quiescent bar, the ends of which are attached by

W. S.

41

small rings to two smooth fixed rods at right angles to each other in a horizontal plane: supposing the animal to start off along the bar with a given velocity, relatively to the bar, to find the angular velocity initially impressed upon the bar.

Let m be the mass of the animal, m' of the bar, 2a the length of the bar, the inclination of the bar to either rod, and V the given velocity of the animal: then the required angular velocity of the bar is equal to

CHAPTER XIV.

MISCELLANEOUS PROBLEMS.

(1) A PARTICLE, placed at a centre of attraction varying as the distance, is urged from rest by a constant force, which acts for one sixth of the time of a complete oscillation about the centre, ceases for the same period, and then acts as before: shew that the particle will then be retained at rest, and that the spaces moved through in the two periods are equal.

(2) A particle moves in a straight line under the action of a force directed towards a point in that line and varying inversely as the square of the distance from that point: if v, v', v", be the velocities of the particle when at distances x, x', x", from the centre of force, shew that

(3) Two ships are sailing uniformly with velocities u, v, along lines inclined at an angle : shew that, if a, b, be their distances at one time from the point of intersection of their courses, the least distance of the ships is equal to

(av - bu) sin

(u2 + v2 - 2uv cos 0,

(4) A particle is projected vertically upwards with a velocity u, in a medium the resistance of which varies as k times the square of the velocity: if t be the time which elapses before the particle returns to the point of projection, prove that

2u

(2 – t) is positive and is the greater the greater k is.

(5) Shew that a boat must be rowed with a velocity, through the water, one half greater than that of the stream, so that it may be taken a given distance up a river with the least possible expenditure of work. (The resistance to the boat is supposed to be proportional to the square of its velocity through the water.)

(6) If 00' be the horizontal range of the path OPO' of a projectile P, prove that the angular velocities of OP, O'P, are to each other as the squares of the cosines of the angles POO', ΡΟΟ.

(7) If the product of the velocities at two points P, Q, of the parabolic path of a body, acted on by gravity, be constant, shew that the locus of the pole of PQ is a circle, the centre of which is at the focus of the parabola.

(8) A particle is projected horizontally with a given velocity if the squares of the times from the instant of projection. to the instants at which the particle arrives at a certain series of points in its path be in arithmetical progression, prove that the angular velocities of the tangents at these points are in harmonical progression.

(9) A body, acted on by gravity, is projected from a given point; and, when it has reached its greatest height, another body is projected from the same point in such a manner that it shall strike the first body: shew that, u, v, being the horizontal and vertical components of the velocity of projection of the former, and u, v, those of the latter body,

(10) A perfectly elastic ball is thrown into a smooth cylindrical well from a point in the circumference of the circular mouth shew that, if the ball be reflected any number of times from the surface of the cylinder, the intervals between the reflections will be equal: shew also that, if the ball be projected horizontally in a direction making an angle with

π

n

the tangent to the mouth at the point of projection, it will reach the surface of the water at the instant of the nth reflection, if the space due to the velocity of projection be equal to

(11) A perfectly elastic ball impinges, with a velocity v and at an angle a to the horizon, on an inclined plane: the direction of impact is in a vertical plane parallel to the plane's intersection with the horizon: after rebounding it falls on this line of intersection: shew that

2v sin a sin λ = (gh)*,

A being the plane's inclination to the horizon, and h the distance of the first point of impact from the horizontal plane.

(12) A ball, thrown from any point in one of the walls of a rectangular room, returns, after striking the three others, to the point of projection, before it falls to the ground: shew that the space due to the velocity of projection is greater than the diagonal of the floor.

(13) A particle moves in one plane under the action of two forces, at right angles to each other, one of which tends towards a fixed point in the plane: supposing the centric force to vary as the time from a given instant, and the angular velocity of its direction to be constant, prove that, being the angle described by the particle about the fixed point, the other force is equal to ue + Be° + y, where a, ẞ, y, are constants.

(14) In a curve described by a particle about a centre of force, the angle between the radius vector and the tangent varies as the time: if a, b, c, be the radii vectores at any three points of the path, and a, ẞ, y, their inclinations to their corresponding tangents, prove that

(sin a)62-c2. (sin B)c2-a2. (sin y) a2-b2 = 1.

(15) In an ellipse of small eccentricity, described by a particle about a centre of force at the focus, the equation of the centre varies nearly as the velocity parallel to the axis major.