below the position of the lower end of the plank at the instant he alights.

Determine also how he must jump so that he may alight on the lower end of the plank.

26. A stone is projected upwards with velocity √2gc from a point on the margin of a circular pond, radius c. If all directions of projection be equally probable, shew that the chance that the stone falls into the pond is

27. A solid smooth cylinder, of radius r, lies on a smooth horizontal plane, to which it is fastened, and an inelastic sphere, of radius 2r, moves along the plane in a direction at right angles to the axis of the cylinder; find the condition that it may pass over the cylinder.

If the sphere be elastic, and the coefficient of the elasticity be greater than 1/8, prove that it cannot in any case pass over the cylinder; and if e be less than 1/8, find the condition that the sphere may, after its first ascent, fall upon the top of the cylinder.

28. From a point A in one of two vertical lines a particle is projected with a velocity u at a given inclination to the horizon, and meets the other vertical line in B: it is then projected from B with a velocity v at the same inclination to the horizon and returns to A. Prove that the harmonic mean between u2 and v2 is constant.

29. Find the path of a particle acted on by a repulsive force always perpendicular to a given straight line and proportional to the distance from it, the velocity at any point being that which would be acquired by moving from rest on the given line to that point.

30. If a particle be acted upon by a force always parallel to the axis of y and proportional to the square of the radius of curvature at the point, prove that it will describe the curve

the particle moving parallel to the axis of x at the point (o, b).

31. The trochoid xa (0-e sin (), y=a (1-e cos 0), is described under the action of a force parallel to the axis of x;

32. If a particle be moving in a medium whose resistance varies as the velocity of the particle, shew that the equation of the trajectory referred to the vertical asymptote and a line parallel to the direction when the velocity is infinite as coordinate axes, is of the form

y=bloga.

33. A body describes the curve whose equation is

xn

n
y
+ = 1,
a b

under the action of a force to the origin. Shew that the central acceleration is λr. (xy)-3, and that when n is even, the periodic time

where A is equal to the area of the curve.

34. Two particles A and B, of masses 8m and m respectively, lie together at a point on a smooth horizontal plane, connected by a string which lies loose on the plane; B is projected at an elevation of 30° with velocity equal to g. If the string becomes tight the instant before B meets the plane again, and breaks when it has produced half the impulse it would have produced if it had not broken, and if the particle rebounds at an elevation of 30°, shew that the elasticity of B is equal to 5/9.

35. A parabola, having its vertex at A and its axis coincident with AB the diameter of a semicircle, is described so as to cut the semicircle in P; prove that, if a body move in the semicircle under the action of a force perpendicular to AB, the time of moving from A to P varies as the difference between AB and the latus rectum. Prove also, that if a

B. D.

8

second body move from A to P in the parabola in the same time under the action of a force perpendicular to its axis, and the velocities in the two curves at P be equal, the latus rectum of the parabola is AB.

36. Three equal particles, A, B, C, each of mass m, are connected by strings, B and C being nearly in the same straight line with A, and equidistant from it. B and C repel each other with a force varying as the distance (mur).

If the string BC be cut prove that the time of a small oscillation of the system is π/√6μ.

37. A particle is in equilibrium at x, y under forces X, Y parallel to the axes, if it be disturbed it will execute small oscillations in a time π/р, where

38. Two particles are projected in parallel directions from two points in a straight line passing through a centre of force, the attraction to which varies as the distance, with velocities proportional to their distances from the centre. Prove that all tangents, to the path of the inner, cut off, from that of the outer, arcs described in equal times.

39. OA is a smooth tube; OB a light rod perpendicular to it; B, a fixed point in OB, a centre of force attracting with force ur a particle P in the tube OA. The system being made to revolve with uniform angular velocity o on a horizontal plane about O, determine the motion of P; and shew that, if μ > w2, P will oscillate with period 2π/√μ- w2.

μ

40. A rod revolves about its middle point with uniform angular velocity and has at its extremities two centres of force varying as the distance one attractive and one repulsive of the same absolute intensity; supposing a particle placed in the plane of rotation in a line perpendicular to the rod through its centre, shew that its path will be cycloidal, the time from one cusp to another being 2π/w.

41. A smooth horizontal disc rotates with angular velocity √ about a vertical axis at which is placed a particle attracted to a certain point of the disc by a force whose acceleration is μx distance; prove that the path of the particle on the disc will be a cycloid.

42. A particle moves under the action of two constant forces in the ratio of nine to one, whose directions rotate in opposite directions with uniform angular velocities in the ratio of three to one: prove that, under certain initial conditions, the path of the particle will be a closed curve, of the same form as that represented by the equation, r = a cos 20.

43. The two ends of a smooth weightless rod are moveable on two fixed straight wires intersecting each other at right angles. A particle can move on the rod and is attracted to the point of intersection of the wires by a force varying as the distance. Prove that if the particle have initially no motion the angular velocity of the rod is given by an equation of the form

w2 = n2 {1 — sin2 2a cosec2 20}.

44. A particle describes a curve, y = f (x), under forces having the potential V; if the same curve can be described under forces having the potential

find the differential equation of the curve.

If n = 1, prove that the curve is a cycloid.

If n=2, prove that the curve is

y=Ae+Bε-8,

where s is the arc measured from some fixed point, and A, B, λ are constants.

CHAPTER VII.

RADIAL AND TRANSVERSAL ACCELERATIONS.

97. HAVING discussed, in the previous chapter, the use of the components of acceleration parallel to two coordinate axes, we now take into consideration the expressions for radial and transversal components, leading to the equations of motion,

mP and mQ being the radial and transversal forces acting on a particle of mass m.

For our first illustration we take the following case.

Motion of a particle in a smooth straight tube which revolves uniformly, in a horizontal plane, about a fixed point in the axis of the tube.

In this case the only force acting on the particle is the pressure R of the tube, and if w be the angular velocity, the equations are

r - w2r=0, 2mrw = R.

If the particle start from the distance a with no initial velocity along the tube, we obtain from the first equation,

α

r = = (ewt +e¬wt) = a cosh wt,

and from the second,

R=2maw2 sinh wt.