5. Two bodies urged from rest by the same uniform force describe the same space, the one in half the time the other does: compare their final velocities and their momenta.

6. If a weight of 8 lbs. be placed on a plane which is made to descend vertically with an acceleration of 12 feet per second, find the pressure on the plane.

7. If a weight of n lbs. be placed on a plane which is made to ascend vertically with an acceleration f, find the pressure on the plane.

8. Find the unit of time when the unit of space is two feet, and the unit of weight is the weight of a unit of mass; assuming the equation W = Mg.

9. A body is projected up a rough inclined plane, with the velocity which would be acquired in falling freely through 12 feet, and just reaches the top of the plane; the inclination of the plane to the horizon is 60°, and the coefficient of friction is equal to tan 30°: find the height of the plane.

10. A body is projected up a rough inclined plane with the velocity 2g; the inclination of the plane to the horizon is 30°, and the coefficient of friction is equal to tan 15° find the distance along the plane which the body will describe.

11. A body is projected up a rough inclined plane; the inclination of the plane to the horizon is a, and the coefficient of friction is tane: if m be the time of ascending, 2 sin (a- €) and n the time of descending, shew that

n

=

sin (a + €)*

12. Find the locus of points in a given vertical plane from which the times of descent down equally rough inclined planes to a fixed point in the vertical plane vary as the lengths of the planes.

87.

VIII. Third Law of Motion.

Newton's third law of motion is thus enunciated:

To every action there is always an equal and contrary reaction: or the mutual actions of any two bodies are always equal and oppositely directed in the same straight

line.

Newton gives three illustrations of this law:

If any one presses a stone with his finger, his finger is also pressed by the stone.

If a horse draws a stone fastened to a rope, the horse is drawn backwards, so to speak, equally towards the stone.

If one body impinges on another and changes the motion of the other body, its own motion experiences an equal change in the opposite direction. Motion here is to be understood in the sense explained in Art. 84.

88. The first of Newton's illustrations relates to forces in Statics; and the law of the equality of action and reaction in the sense of this illustration has been already assumed in this work; see Statics, Art. 283. The second illustration applies to a class of cases of motion which we shall consider in the present Chapter. The third illustration applies to what are called impulsive forces, which we shall consider in the next Chapter.

89. Two heary bodies are connected by a string which passes over a fixed smooth pully: required to determine the motion.

Let m be the mass of the heavier body, and m' the mass of the other. Let T be the tension of the string, which is the same throughout by the Third Law of Motion, the weight of the string being neglected as usual.

The forces which act on each body are its weight and the tension of the string; and these forces act in opposite

directions. Thus the resultant force on the heavier body is mg-T downwards, and on the lighter body T-m'g upwards. Therefore mg-T

the acceleration on the heavier body is

m

Now as the string is supposed to be inextensible, the two bodies have at any instant equal velocities; and therefore the accelerations must be equal. Thus

ठ

m

This is a constant quantity. Hence the motion of the descending body is like that of a body falling freely, but

is not so rapid; for instead of g we have now

m-m'

m + m2

9.

90. In the investigation of the preceding Article no notice is taken of the motion of the pully: thus the result is not absolutely true. But it may be readily supposed that if the mass of the pully be small compared with that of the two bodies, the error is very slight; and this supposition is shewn to be correct in the higher parts of Dynamics. Theoretically instead of a pully, we might have a smooth peg for the string to pass round, but practically it is found that owing to friction this arrangement is not so suitable: see Statics, Arts. 191 and 278.

91. The system of two bodies considered in Art. 89 forms the essential part of a machine devised by Atwood, for testing experimentally the results obtained with respect

to rectilinear motion under the action of uniform forces. Atwood's machine contains some contrivances for diminishing friction, and some for assisting in the arrangement and observation of the experiments; but the principle is not affected by these contrivances.

make

m+m'

The chief advantage secured by Atwood's machine is that by taking two bodies of nearly equal weight we can m-m' g as small as we please, and thus render the motion slow enough to be observed without difficulty. The results of experiments with Atwood's machine are found to agree with those assigned by the investigations already given.

92. Two bodies are connected by a string, which passes over a small smooth pully fixed at the top of two inclined planes having a common height: required to determine the motion, supposing one body placed on each plane.

Let m and m' be the masses of the two bodies; a and a' the inclinations of the planes on which they are respectively placed. Let T denote the tension of the string.

Suppose the body of mass m to be descending. The weight of this body is mg; the resolved part of the weight along the plane is mg sin a; hence the resultant force down the plane is mg sin a- T, and therefore the acceleramg sin α- T tion is

m

Similarly, for the other body, the resultant force up the plane on which it moves is T-m'g sin a', and the T-m'g sin a'

acceleration is

Now as the string is supposed to be inextensible, the two bodies have at any instant equal velocities: and therefore the accelerations must be equal. Thus

Thus we see, that in order that this may be positive, and so the body of mass m be acquiring downward velocity, we must have m sin a greater than m' sin a'.

EXAMPLES. VIII.

1. If the two weights in Art. 89 are 15 ounces and 17 ounces respectively, find the space described and the velocity acquired in five seconds from rest.

2. If the string in Art. 89 were cut at the instant when the velocity of each body is u, find the distance between the two bodies after a time t.

3. In the system of Art. 89 shew that if the sum of the weights be given, the tension is greater the less the acceleration is.

4. A weight P is drawn along a smooth horizontal table by a weight Q which descends vertically, the weights being connected by a string passing over a smooth pully at the edge of the table: determine the acceleration.

5. A weight P is drawn up a smooth plane inclined at an angle of 30° to the horizon, by means of a weight Q which descends vertically, the weights being connected