we have hitherto regarded, and may proceed to those more complex cases in which different bodies and different forces occur. 85. One case of the general principle of Art. 83 will be as follows; the weight of a body at a given place is proportional to the product of the mass moved into the velocity generated in a given time. Let the given time be one second and the unit of length one foot; then the velocity generated is denoted by g. Let M be the mass of a body, and W its weight; then W varies as Mg, so that by Algebra W=CMg, where C is some constant. It is convenient to have this constant equal to unity; this we can secure by making a suitable connexion between the units of mass and of weight which have not yet been fixed: then W=Mg. Suppose, for example, we resolve to have one lb. as the unit of weight: required to determine the unit of mass. Let M=1; then we obtain W=g, that is 322; so that the unit of mass is so much mass as weighs 32.2 lbs. Again, suppose, for example, we resolve to have the mass of one cubic foot of water as the unit of mass, required to determine the unit of weight. Let W=1; then we obtain M= : so that the unit of weight is such 32.2 1 1 a weight that its mass is that is, the mass of the unit 1 32.2 32.2' of weight is of the mass of a cubic foot of water. Now it is known by experiment that a cubic foot of water 1000 weighs 1000 ounces, so that the unit of weight is 32.2 ounces. 86. We may illustrate the preceding remarks by discussing the motion of a body sliding on a rough inclined plane. Suppose a plane inclined at an angle a to the horizon; let a body be placed on the plane. Let M denote the The The mass of the body, and therefore Mg its weight. resolved force of gravity down the plane is Mg sin a. pressure on the plane is Mg cos a. If μ denote the coefficient of friction, the friction will be μMg cos a. If the body is moving down the plane, the friction acts up the plane. Hence the resultant force down the plane is Mg(sin au cos a). Now when a body is acted on by its own weight, the velocity generated in a unit of time is g; that is, the force Mg generates in a body of mass M the velocity g in a unit of time: therefore, by Art. 81, the force Mg (sin a-u cos a) will generate the velocity g (sin a- μ cos a) in a unit of time. Thus the motion of a body sliding down a rough inclined plane is similar to that of a body sliding down a smooth inclined plane, or to that of a body falling freely: the acceleration is g(sin a-μ cos a) for the rough plane, g sina for the smooth plane, and g for the body falling freely. In the same manner it may be shewn that if a body is sliding up a rough inclined plane the acceleration is g(sin a+μ cos a) downwards. EXAMPLES. VII. 1. A body weighing n lbs. is moved by a constant force which generates in the body in one second a velocity of a feet per second: find the weight which the force could support. 2. Find in what time a force which would support a weight of 4lbs., would move a weight of 9 lbs. through 49 feet along a smooth horizontal plane: and find the velocity acquired. 3. Find how far a force which would support a weight of n lbs., would move a weight of m lbs. in t seconds: and find the velocity acquired. 4. Find the number of inches through which a force of one ounce constantly exerted will move a mass weighing one lb. in half a second. 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 ƒ, 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 15o: 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 m sin (a–e) 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 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 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' g. 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 |