If a be the inclination of the plane to the horizon, and W the weight of the body, the body will descend in a straight line inclined to the vertical at an angle

(21) A balloon ascends with a uniformly accelerated velocity, so that a weight of one pound produces, on the hand of the aeronaut sustaining it, a downward pressure equal to that which a pound weight, augmented by the hundredth part of an ounce, would produce at the Earth's surface: find the height which the balloon will have attained in ten minutes from the time of starting, not taking into account the variation of the accelerating effect of the Earth's attraction.

The required height, supposing gravity to be represented by 32 feet, will be twelve hundred yards.

(22) Two weights, one of which is double of the other, are placed upon two smooth inclined planes which have a common vertex, the inclination of each being 30o, and are connected by means of a fine string passing over a small pully at the highest point of the planes: prove that, if the length of the string be equal to that of either plane, and the heavier weight start from the highest point, the two weights will reach the ground at the same time, if the string be cut when one sixth of it has passed over the pully.

Shew also that the times of motion, before and after cutting the string, are each equal to the time of a body's falling freely under the action of gravity through a height equal to the length of either plane.

(23) A bucket of water, weighing 32 lbs., is attached to a rope, which passes round a fixed pully, and is pulled by a man: supposing the man to raise the bucket two feet each stroke, which occupies half a second, and the force to be so applied that the velocity is uniformly generated during the first half of the stroke, and uniformly destroyed during the second half, find how

W. S.

23

the tension of the rope alters, and express in pounds the greatest tension, 32 feet being assumed as the measure of the force of gravity.

The tension is 64 lbs. during the first half, and zero during the latter half of each stroke.

(24) A string, charged with m+n+1 equal weights, fixed at equal intervals along it, is placed within a smooth bent tube, curved at the angle, and just wide enough to admit the weights, the two branches of the tube being rectilinear, one inclined to the horizon and the other vertical: the string, which would rest with m of the weights within the vertical branch, is so placed that the (m+1)th weight is just within this branch: shew that, if a be the distance between each two adjacent weights, the velocity which the string will have acquired, at the instant the last weight enters the vertical branch, will be (nag)*.

Campion and Walton's Solutions of the Cambridge Problems of 1857.

(25) A weight, descending vertically, draws another weight, n times as heavy as itself, up a rough inclined plane, by means of a fine string passing over a pully fixed at the top of the plane: the ascending weight starts from the foot of the plane, and, when it has travelled a space x, the connecting string is cut: shew that the ascending weight will just reach the top of the plane if

where a is the inclination of the plane, a its length, and tan e the coefficient of friction between the plane and the weight.

(26) If weights W, n W, move on two inclined planes and be connected by a fine string passing over the common vertex of the planes, the angles of inclination of the planes to the horizon being a, ß, respectively, prove that their centre of gravity describes a straight line with a uniform acceleration equal to

SECT. 8. Centrifugal Force.

(1) Two unequal weights are connected by a string of given length, which passes through a small ring: to find how many times in a second the lighter must revolve, in order that the heavier may be at rest at a given distance from the ring.

Let P, P', fig. (120), be the positions of the two weights, O the ring, being the inclination of OP' to the vertical.

Let OP'c', and let m, m', denote the masses of P, P', respectively.

The forces acting on P' are the tension of the string, which is equal to mg, the weight m'g, and the centrifugal force m'w'c' sin 0, acting horizontally.

For the equilibrium of P' we have, resolving horizontally,

Hence, n denoting the number of revolutions performed by P' in a second,

(2) A string will just bear a weight of 16 lbs. without breaking: if a weight of 1b. be attached to it, and whirled round in a horizontal circle, the radius of which is two feet, find the number of revolutions the weight must make in a second, in order to break the string.

The required number of revolutions is equal to 2√g.

π

CHAPTER IV.

IMPACT.

(1) A PARTICLE is projected, with a given velocity, from one extremity of a diameter of a horizontal circle, and, after reflection at the curve, passes through the other extremity: to find the elasticity, in order that the time of motion may be n times that of describing the diameter with the velocity of projection.

Let AB, fig. (121), be the diameter, APB the path of the particle, C the centre of the circle.

Join CP: let r = the radius, < PAC=a, V= the velocity of projection.

Since the component of the velocity, at right angles to CP, is the same before and after impact, and since A, B, are equidistant from the diameter PCP', it follows that the time of movement from A to P is equal to that from P to B: let t denote each of

these times.

Since the line AP is described, in the time t, with the velocity V,

Considering the component of the motion from P to B, parallel to PC, we have, since eV cos a is the normal component of the velocity after impact on the curve at P,

r (1― cos 2a) = eV cos a. t,

or

a=

2r sin2 a = eV cos a. t

(2).

Again, the time of motion being equal to that of describing AB with the velocity V, we have

From (4) and (5), by the elimination of a, we find that

(3).

(4),

(5).

From the equation (4), we see that n must be less than 2, and, from (6), that 4-n2 must be less than n2, or n greater than √2. Thus, that the problem may be possible, it is necessary that n be less than 2 and greater than √2.

(2) An imperfectly elastic ball is projected in a given direction within a fixed horizontal hoop, so as to go on rebounding from the surface of the hoop: to find the limit to which the velocity of the ball will approach; and to shew that it will attain this limit at the end of a finite time.

Let V, V, V, V,,... be the velocities of the successive impacts; a, a,,,, a, ... the acute angles between the successive directions of impact and the tangents to the circle at the respective points of impact.

Then, the tangential components of the velocities of impact and rebound being equal at each point of impact, we have

Again, resolving normally the velocities of impact and rebound,