Hence the required space, expressed in feet, is equal to

(3) Two bodies, the weights of which are W and W', hang from the extremities of a cord passing over a smooth peg; if, at the end of each second from the beginning of motion, W be suddenly diminished and W' suddenly increased, so as not to experience any impulse, byth of their original difference; shew

n

that, W being the greater weight, their velocity will be zero at the end of n + 1 seconds.

Since the acceleration of the weights varies as the ratio which their difference bears to their sum, it follows that, ƒ denoting their acceleration during the first second, the accelerations during the 2nd, 3rd, 4th, ..... (n+1)th seconds, will be

2n

ƒ (1–3), ƒ(1–4), ƒ(1-9),... ƒ(1–2), .

and therefore their velocity at the end of the (n+1)th second will be equal to

(4) A string, passing over a smooth pully, supports, at one end, a scale-pan of weight W, which contains a weight 5 W, and, at the other end, a weight 3 W: to find the tension of the string and the pressure of the weight on the pan.

Let R= the mutual pressure between the pan and the weight

5 W, and T= the tension of the string.

Then, the downward acceleration of the pan is equal to

and the upward acceleration of the weight 3 W, to

.(2),

(3).

Since these accelerations are all equal, we have, from (1) and (2),

(5) A particle is projected up a rough inclined plane of indefinite length; to compare the velocities at the beginning of the ascent and the end of the descent.

Let a = the inclination of the plane, tan e = the coefficient of friction, m = the mass of the particle.

The normal pressure of the particle on the plane is equal to mg cos a: hence the friction is equal to mg cos a tan e. Thus, while the particle is ascending, it is acted on by a retarding force equal to

g (sin a + cos a tan e),

and, while it is descending, by an accelerating force equal to

Hence, s being the length of the inclined plane traversed by the particle, and u, v, the velocities at the beginning of the ascent and the end of the descent, respectively,

and

u2 = 2gs (sin a + cos a tan e),

v2 = 2gs (sin a cos a tan e);

(6) Two equal particles, connected by a fine string, are placed upon two given inclined planes which have a common altitude: to determine the acceleration of the centre of gravity of the two particles.

Let AC, BC, fig. (119), be the two inclined planes. Let <BAC=σ, LABC= B. Produce BC indefinitely to B', and draw CD to bisect the angle ACB'.

The accelerations of the particles, in the directions CA, CB', respectively, are both equal to

it is evident therefore that their accelerations at right angles to CD are equal and in opposite directions, and that the acceleration of each parallel to CD is equal to

This is therefore, the particles being equal, also the acceleration of their centre of gravity.

(7) Find the weight of a body which, initially at rest, describes in a certain time, under a pressure of six pounds, onethird of the space through which it would in the same time fall freely.

The weight of the body is eighteen pounds.

(8) A force, which can support 50 lbs., acts for one minute on a body the weight of which is 200 lbs.: find the velocity and momentum acquired by the body.

The velocity acquired is 483 feet per second and the momentum is 3000.

(9) Two weights, one of 81 and the other of 80 ounces, are suspended by a fine string over a fixed pully: determine the

space described by each of them from rest in one second, and the velocity acquired.

The required space and velocity are respectively one tenth and one fifth of a foot.

(10) Two weights, each of 7 lbs., hang at rest by means of a string over a fixed pully, the mass of which may be neglected to one of them an additional weight of 2 lbs. is attached, which is removed after the weight has fallen through 8 feet: find the time occupied in falling through the 8 feet, and also the time in which 8 feet would be described after the removal of the additional weight, supposing 32 feet to be the measure of gravity.

The required times are respectively 2 seconds and 1 second.

(11) A weight of 4 lbs., hanging down vertically, draws another of 6 lbs. up a plane inclined at an angle of 30° to the horizon: find the space moved through in four seconds.

The required space is 25.76 feet.

(12) A body, weighing 10 pounds, is moved along a horizontal plane by a constant force, which generates in the body in a second a velocity of 1 foot per second; find what weight the force would support.

It would support rather less than 5 ounces.

(13) If a weight of ten pounds be placed upon a horizontal plane, which is made to descend with a uniform acceleration of 10 feet per second, determine the pressure on the plane.

The required pressure is a little greater than 6 lbs. 14 oz.

(14) Two bodies, weighing respectively 1 lb. and 2 lbs., are placed upon a horizontal plane, which descends with a uniform acceleration of 1 foot per second: determine the pressure which each exerts on the plane.

The required pressures are, approximately, 151⁄2 oz. and 31 oz.

(15) Two weights are connected by a fine string which passes over a pully: if the weights be 50 and 72 lbs., determine what stationary weight the string must be able to support, that it may just escape breaking during the motion.

It must be able to support a stationary weight of 59 lbs.

(16) Two equal bodies P and Q, connected by a string, six feet long, are placed on a horizontal table, three feet high: Pis gently pushed over the edge, and falls, drawing Q along the table find when each will fall to the ground.

Time being dated from the commencement of the motion, P and Q will reach the ground respectively in the times

(17) A weight P, descending vertically, draws a weight 2P up an inclined plane, by means of a string passing over a pully at the top of the plane: the ascending weight starts from the foot of the plane, and, when it has travelled half-way up the plane, the motion of the descending weight is stopped: find the inclination of the plane, that the ascending weight may just reach the top.

The required angle of inclination is equal to sin1

(18) A body, of weight 4P, is drawn up a plane, of length a, inclined at 30° to the horizon, by a weight 3P, connected with the former by a string passing over the upper edge of the plane: find the tension of the string and the time before the former weight arrives at the top of the plane.

The tension of the string is equal to

time is equal to (14a).

18

7

P, and the required

(19) A chord AB of a circle is vertical, and subtends at the centre an angle 2 cotμ: shew that the time down any chord AC, drawn in the smaller of the two segments into which AB divides the circle, is constant; AC being rough, and μ being the coefficient of friction.

(20) A heavy body is placed on a smooth inclined plane, which at the same instant begins to descend with a uniform acceleration of 5 feet in a second, the inclination of the plane to the horizon being invariable: determine the motion of the body and its pressure on the plane.