out at the centre owing to the tendency to fly outwards from the centre of rotation. At a great speed the impression of the separate strips run together, and give to the eye the appearance of an oblate spheroid.

Examples of centripetal force may be found in astronomy. The moon's motion is due to a combination of two velocities, revolving round the earth in an orbit nearly circular because deflected from a natural rectilinear path by a centripetal force, viz. the earth's gravitative attraction. In fact, Newton showed that its path may be regarded as compounded of FIG. 21.- Explanation of the an original impulse in a rectilinear spheroidal form of the earth. direction, and a constant pull-the force of gravitation-towards the earth's centre, and that generally, were a planet projected with a certain velocity in a direction right angles to the radius connecting the planet and the sun, the planet would for ever describe a circle round the sun.

It is easy to understand that some modification of this velocity or direction of projection may lead to other closed curves than a circle, e.g. an ellipse, and the law of gravitation, according to which the force of attraction varies inversely as the square of the distance, combined with a certain specific initial

FIG. 22.

impulse, has been proved to explain the elliptical paths of the planets, and their varying velocity in different parts of the orbit.

37. Angular Velocity. -When a body is rotating about a fixed axis or fixed point, each particle describes a circumference whose centre is the axis.

Suppose the point O to be an axis, and P a point on a fixed line Ox; then the rate at which the angle POx increases is

called the angular velocity of P about the point O. Angular velocity is therefore the angle passed through by a rotating point in a unit of time. It is evident that every point on the line OP, when OP represents the rotating body, will describe the same angle in the same time, but that the linear velocities of the particles will increase as the distances of the particles from the fixed point O increase.

The unit of angular velocity is that which the point has when it describes unit angle in one second. The unit angle used in angular velocity is called a radian (the radian = 57'3°), that is, the unit of angular velocity is one radian per second. Angular velocity may also be expressed in degrees per second.

CHAPTER V.

PARALLEL FORCES AND MACHINES.

38. FORCES whose line of action are parallel are called parallel forces. If the parallel forces act in the same direction, they are known as like parallel forces; if they act in opposite directions, they are said to be unlike parallel forces.

We will first of all show that for two like parallel forces(1) The resultant acts in the same direction as the forces. (2) It is equal to their sum.

Experiment 17.-Take a light uniform bar of wood (about a metre long) and suspend it to the hook of a spring balance by a string passing through a hole at the centre of the rod, so that the latter can turn quite freely about

FIG. 23.

its point of suspension. Mark on the bar equal divisions from the centre, and at each division bore a small hole. Make several hooks of bent wire, and when necessary insert a hook into the hole from which you wish to suspend a weight. The rod will now swing freely and rest in any position, its weight being indicated by the spring balance. Attach in any way some small weights to one of the hooks, e.g. one of 3 units (pounds, ounces, or grams, or any other convenient unit) on the hook distant six divisions from the centre. That side of the rod will be drawn down, but on putting an equal weight at an equal distance on the other side, the counterpoise will be restored and the rod brought back to its original position. Notice the

pull indicated on the spring balance. Remove the two sets of weights, and suspend them both from the centre division. The spring balance shows that the pressure of the bar upon its support is exactly the same as in the former case. We therefore see that two forces of three units each acting in parallel directions-in this case vertically downwards-on a bar have a combined effect of six units acting in the same direction. Vary the experiment by attaching two or three sets of different weights to the hooks at different distances on each side of the centre so that the bar may keep horizontal. Note the pull on the spring balance, and then hang them all Again it will be found that their combined effect

on to the centre hook. is the same as before.

Therefore the resultant of a number of like parallel forces is numerically equal to their sum, and acts in the same direction as the forces themselves.

FIG. 24.

Let us now consider the case of unlike parallel forces, and endeavour to prove that—

(1) The resultant acts in the same direction as the greater of the two sets of forces.

(2) It is equal to their difference.

Experiment 18.-Fit up the apparatus as in the preceding experiment, but turn one of the hooks upside down. Attach a weight to this hook by a string passing over the pulley. The effect of the pulley, as we shall afterwards show, is merely to allow the weight to act upwards upon the rod, and does not alter its magnitude. Let a force of 2 units, for example, act in an upward direction on the sixth hook from the centre, and hang on to the other hooks any suitable weights so as to keep the bar horizontal, as shown in the figure. Subtract from the sum of the weights acting downwards, the weight whose force is directed upwards, and it will be found that their difference is their combined effect on the rod, as indicated by the balance, after the weight of the rod has been allowed for.

This proves that, as stated above, the resultant of unlike

E

parallel forces is numerically equal to the sum of those which act in one direction, less the sum of those which act in the opposite direction.

39. Machines. A machine is an instrument by means of which a force can be applied at one point so as to overcome a weight or resistance at another point. This is done either (1) by means of a solid body which is movable about a given point, or (2) by means of a flexible band or string, or (3) by means of a hard inclined surface.

The simple machines, therefore, are (1) the lever, (2) the pulley, and (3) the inclined plane.

A machine may be considered to be acted upon by two forces

(1) An external force applied to work the machine; this is called the power.

(2) A resistance to be overcome; this is called the weight.

FIG. 25.

It is usually the object of a machine that a small power shall overcome a large resistance or weight.

40. The Lever.-A lever is a rigid bar capable of turning about a fixed point called the fulcrum. The fulcrum may be an axle passing through the lever, or an edge upon which it rests. The portions of the lever measured from the fulcrum to the power and weight respectively are called the arms of the lever.

Experiment 19.--Make a lever by balancing a stiff lath upon the edge of a triangular prism, cutting a small grove on the under surface of the lath to enable it to rest more firmly on the edge of the prism. (A lever may also be made with a lath or ruler by boring a hole through its centre so that the rod will remain horizontal when suspended on a nail passing through the hole.)

Place or suspend a four-ounce weight, for example, on one side of the lever, and then restore the balance by placing or suspending an equal weight on the other side. It will be found that the two weights are at equal distances from the fulcrum, i.e. that the two arms of a lever are equal