as to reach C in one second, is PQ. In the same way the velocity which has to be combined with OQ when the particle reaches C is QR.

Now any curve may be considered as built up of an infinite number of very small straight lines, i.e. to be a polygon; hence we should expect that the results obtained above would be applicable to such a curve and its hodograph.

Suppose ABCD (Fig. 26) to be the path of a particle, and PQRS its hodograph with reference to the origin 0. By this we mean that if from o lines are drawn to represent the velocity of the particle in its path ABCD in magnitude and direction at every instant of its motion the extremities of all these lines will lie on the curve PQRS.

S

Since at any instant the direction of motion of the particle is along the tangent to its path, it follows that OP must be parallel to the tangent of the curve ABCD at A, and os parallel to the tangent at D; also the velocity at A is equal to OP, and that at D to OS. To find the velocity at any other point, E, of the path, we draw a tangent BT at the given point, then in the hodograph through O draw a line OQ parallel to BT. OQ will then repre

sent the velocity at B. In the same way OR represents the velocity at C. Suppose the particle has taken a time d1 to travel from B to C, then

->

during this time the velocity has changed from OQ to OR. If 8t is excessively small, so that B and C, as well as Q and R, are very close together, we may regard the portion of the hodograph QR as being a straight line coincident with the tangent. Then, from what has been said with reference to the case of a polygon, the velocity which has to be compounded with OQ to give OR is represented in magnitude and direction by QR.

1 See note on p. 28.

Hence QR represents the change in velocity during the time dt. Н the acceleration is defined as the change in velocity during a given ti divided by the time, so that the acceleration with which the body moving between B and C is given by QR/8t; and, further, this accelerati

acts along QR, i.e. along the tangent at that point of the hodograph whi corresponds to the point on the path at which the acceleration is bei considered. If we look upon the hodograph as being traced out by point at the same time as the particle traces out the path ABCD, then t portion QR will be traced out during the time of that the particle spen in travelling from B to C. Hence we might call the quotient QR/dt t speed of the tracing-point of the hodograph.

We may therefore sum up the results by saying that (1) the directi of the tangent to the hodograph at any point represents the direction the acceleration of the motion of the particle at the corresponding poi of its path, while (2) the speed of the tracing-point of the hodograph any instant represents the magnitude of the acceleration of the partic at that instant.

42. Motion in a Circle.-The only case of curvilinear motion wi

QQ

FIG. 27.

which we shall deal is that uniform motion in a circle. Sin the speed is constant the hod graph is a circle, for the lines draw from the centre o (Fig. 26) to r present the velocity of a partic moving along the circle BCD magnitude and direction will all l of the same length, the speed bein constant, and hence their extr mities must all lie on a circle.

Suppose the particle to move with a uniform speed in a circle radius r. Then the hodograph will be a circle of radius v units of lengt Let A (Fig. 27) be the centre of the circle along which the partic moves, and B any point on the circumference. At B the particle will moving with a speed v in the direction of the tangent BT. Hence t line OP in the hodograph represents the velocity at B, and the acceler tion at B is in the direction of the tangent at P, i.e. in the direction P But since both the path and the hodograph are circles, the tangents a at right angles to the radii passing through the points of contact. Hen UP is at right angles to OP, and BT to AB. But OP and BT are by co struction parallel. Hence PU is parallel to BA. The acceleration of th particle at any point is therefore along the radius of the circle at th point, the sense of the acceleration being towards the centre.

1 The speed, if variable, is defined as in § 31.

To find the magnitude of the acceleration, we require to know the speed in the hodograph. Since the tracing-point of the hodograph will make a whole revolution in the same time that the particle describes the circle BCD, it follows that the speed in the hodograph is to the speed of the particle (7) as the radius of the hodograph is to the radius of the circle, since the circumferences of two circles are to one another as their radii, therefore the speed (u) in the hodograph is given by

where is the radius of the hodograph and is equal to v,

Hence the magnitude of the acceleration acting on the particle is vr, and the direction of the acceleration is towards the centre of the circle in which the particle is moving.

CHAPTER VI

MOTION OF A RIGID BODY

48. Definition of a Rigid Body.-A rigid body is an extended piece of matter which can move as a whole with reference to surrounding objects, but whose component particles are incapable of any displacement relative one to the other. Hence a rigid body is incapable of having a strain (§ 122) impressed upon it. Although a rigid body is an ideal which cannot be realised in practice, the consideration of the dynamics of a rigid body is useful as an introduction to the study of the more complex problems which arise when we have to deal with such substances as exist in nature, besides which, in many problems, such bodies as steel and glass may be taken as rigid.

44. Motion of a Rigid Body.—Any displacement of a rigid body can be produced by a pure translation of the body, and a pure rotation of the body round a certain point called the centre of figure1 of the body. Thus to consider the case of the displacement of a straight line AB

A

FIG. 28.

B'

(Fig. 28) in a plane, from the position AB to the position A'B'. In the case of a straight line the centre of figure is at B" the middle point. Hence in the first position C is the centre of figure, and in the second c'. The line AB may be displaced to the position A"B" by a pure translation (§ 29), since all the particles will move along equal parallel straight lines. Then it can be rotated about C into the required position A'B'. In this

case all the particles will move in circles with C' as a centre. The motion of a rigid body may thus be resolved into a motion of translation obeying the laws considered in Chapter V., and a motion of rotation.

45. Motion of Rotation.-When a rigid body moves so that the paths of all the particles of which it may be regarded as built up are circles having their centres on a fixed line, called the axis of rotation, the body is said to undergo a pure rotation about this axis.

Since the distances of the different particles from the axis of rotation

1 When we come to consider the action of gravity on bodies, we shall find that the centre of figure is what is better known as the centre of gravity.

may not all be the same, but as they are all rigidly attached together they must complete a revolution in exactly the same time, it follows that the speed of the different particles is different. Thus in the case of a fly-wheel, consider the motion of two points, one on a spoke at a distance of 50 centimetres from the axis of rotation (the axle), and the other on the rim, say at a distance of 100 centimetres from the axis. Suppose the fly-wheel to make one turn per second, then the first particle will in one second traverse the circumference of a circle of 50 centimetres radius, i.e. will travel through 100 centimetres, and hence its speed will be 100 centimetres per second. The speed of the other particle will in the same way be 2007 centimetres per second.

The velocity of rotation of a body cannot therefore be measured by the speed of any unspecified particle, but is measured either by the number of turns made in a given time divided by that time, i.e. the number of turns per second, or by the speed of a particle at unit distance from the axis of rotation, this speed being called the angular velocity of the body.

Suppose the angular velocity of a body is w, that is the linear speed of a particle at unit distance from the axis is w, so that the space passed over by such a particle in a second is a units of length. Now the length of the arc of the circle, along which the particle travels, passed over in one second being w, the angle subtended by this arc at the centre of the circle is w/1 in circular measure ($ 14), since the radius of the circle is unity. Since the angle swept out by the radius joining any particle of the body to the axis of motion is the same for all particles in the body, we may say that the angular velocity of a body represents the angle (measured in circular measure) through which the body turns in one second.

ய

If the rotating body turns through an angle w in one second, then the space traversed by a particle at a distance from the axis of motion is or, since this is the length of an arc of a circle of radius that subtends an angle at the centre.

The relation between the angular velocity of a body and the number of turns per second can be obtained as follows: Let the body make n turns per second, then since in one complete turn the angle turned through is 27 ($14), the angle turned through in one second, or the angular velocity, is 2πn. Since the body makes n turns per second, the time it takes to make one turn is = 7 say, and the angular velocity=2′′/T.

I

n

A rotation is said to be positive if, when looking along the axis in the positive direction, it appears to take place in the opposite direction to that of the hands of a watch; while if it takes place in the same direction it is negative.

As in the case of linear velocity, angular velocity may be uniform or variable. In the case of variable angular velocity, the change of angular