PART II-KINEMATICS

CHAPTER IV

POSITION

24. Province and Subdivisions of Mechanics.-The title mechanics is generally given to that part of physics which deals with the effects of force on matter, without in any way considering how the force originates. For the present we may regard force as typified by muscular exertion. When we exert our muscular powers to overcome some obstacle we derive, by means of our sense organs, a certain sensation which we describe as due to the fact that we are exerting a force. When any inanimate agency produces effects on bodies which are similar to those which we produce by muscular exertion, it is in the same way said to exert force.

As far as mechanics is concerned, the effects of force on matter are of two kinds―(1) change of motion, and (2) change of size or shape.

Before studying the effects of force on the motion of bodies, which constitutes the branch of mechanics called Dynamics, it is advantageous to study motion in the abstract, i.e. without reference to the cause of the motion. This branch of mechanics is called Kinematics.

25. Material Particle.-A portion of matter so small that, for the purposes of the discussion in hand, the distances between its different parts may be neglected, compared to the other lengths we are considering, is called a material particle.

The limiting size of a material particle varies very much in different investigations. Thus in some astronomical problems the earth and the other planets are treated as material particles, while if we attempt to account for the different kinds of light emitted by glowing gases, by a consideration of the vibrations of the molecules or even of the atoms, it is no longer permissible to regard an atom as a material particle.

26. Position. The definition of a material particle amounts to a statement that the position of such a material particle can be represented by a geometrical point, which has position but not magnitude. This at once leads to the question of position.

In order to define the position of a point, we require to know its distance from some fixed point of reference, called the origin, and also the

direction in which we must go in order to pass from the origin to given point. In order to be able to specify this direction, it is neces

tion. Then it is evident that if we know the angle , which the strai line joining P to the origin makes with Ox, and also the distance (~) have to travel along this line from 0 to reach P, then the position of completely defined. The quantities r and 6, which serve to define position of P, are called the co-ordinates of P.

Another method of defining the position of a point in a plane i

P is defined. For if we measure off from o along ox a distance equal to NP, and through N draw a line parallel to YY', the point P m lie somewhere on this line. In the same way P must lie somewhe on the line NP, and hence must lie at the only point which is comm to the two, that is at their point of intersection. It is usual to i dicate the distance OM or NP by the symbol r, and ON or MP by t symboly, so that the co-ordinates of the point P are x and y. In almo all practical applications of this method of defining the position of point (called the Cartesian method) the two axes are taken at rig angles to one another. In order to define the position of a point in spa we require three co-ordinates. In the Cartesian method three axes a taken which are at right angles to each other, and the co-ordinates of

point are then the distances from the origin of the feet of the perpendiculars drawn from the point to the three axes.

27. Vectors and Scalars.-Suppose we have the positions of two points (0 and P) given. Then the position of P relative to o is given by the length and the direction of the straight line OP drawn from O to P. That is, starting from O you will arrive at P if you go in the direction of the line OP for a distance equal to the length of this line.

In geometry the expression OP is used simply to designate a line. When, however, it is used to designate the operation by which the line is drawn, i.e. the motion of a tracing point in a certain definite direction for a certain definite length, it forms an example of a quantity called a vector. To emphasise this fact we shall indicate a line such as OP, when it is used as a vector, by an arrow placed over the letters which define the ends of the line, thus OP. The arrow will here remind us of the distinctive property of a vector, namely, that in addition to a definite magnitude, it has also a definite direction, for we are constantly in the habit of indicating a direction by means of an arrow-head. The expressions OP and PO represent two different vectors, for although the distance is the same in the two cases, yet in one the tracing point is supposed to move from 0 to P, and in the other from P to O. Where we use a single symbol to represent a vector quantity, and we want to emphasise that it is a vector, we shall use a thick fount of type, while for scalar quantities the ordinary type will be employed. Thus will represent a vector of which the magnitude is v units in some definite direction.

A quantity which has only magnitude and not direction is called a scalar. Thus mass and density are scalars, but velocity and force, as we shall see, are vectors, for they have not only magnitude, but have associated with this magnitude a certain direction.

28. Motion. If the position of a material particle is changed, then if we only consider its state before and after the process of change, and take no account of the time during which this change takes place, we are said to study the displacement of the particle. When a particle is displaced, however, from one point to another, it must travel over a continuous path from one position to the other; and further, it must take a certain time in travelling over this path, so that it has occupied in succession every point along this path. When we consider the actual process of change of position as occurring during a certain time, we are said to study the motion of the particle, while that branch of mechanics which is concerned with the motion of bodies treated in the abstract, i.e. without considering what causes the motion or change of motion, is called Kinematics.

29. Different Kinds of Motion.-The motion of a material particle, taken with reference to some fixed point as origin, can consist either in change in the distance of the particle from the origin, the

direction of the straight line joining the particle to the origin remain fixed, i.e. motion can take place along this straight line either away f or towards the origin, or in a change in the direction of the line join the particle to the origin, the length of this line remaining fixed, motion along the circumference of a circle having the origin as cen or in a combination of these two. In the case of a material particle, si it has no parts, the above are the only kinds of motion possible, and form of motion is called motion of translation. If, however, instea dealing with a material particle, we are dealing with a body of appr able size, so that its different parts can have different motions, we hav further kind of motion possible. Thus in addition to a motion of tran tion, in which the body moves so that the line joining any two point: the body is always parallel to some fixed line, the body may spin rotate. In the case of a pure translation, the motion of all the partic of which we may consider the body to be built up, is exactly the sa while when the body rotates the motions of the different parts of the bo are different. The most general kind of motion of which an extend body is capable is a combination of a rotation with a translation.

As an example of a motion of translation, if we neglect the curvat of the earth's surface, we may take the case of a boat sailing in a straig line. The fly-wheel of a stationary engine is an example of a motion pure rotation. The motion of the screw propeller of a ship, the wheel a locomotive, and a ball rolling along the ground are obvious examp of the combination of a motion of translation with one of rotation.

CHAPTER V

MOTION OF TRANSLATION

30. Velocity, Speed. The rate at which a point changes its position is called its velocity. From what has been said in § 27 it is evident that the change in the position of a particle must not only have magnitude, ie. there must be a certain distance measured along the path traversed by the particle between its first and last positions, but also the motion of the particle must have been in some direction, although not necessarily along a straight line, so that velocity is a vector. Velocity, therefore, may vary both in regard to its magnitude and also in regard to its direction. This may be illustrated by the motion of a train going round a curve. Here, although the magnitude of the velocity may be constant, ¿e. the train may travel along the rails for equal distances in each successive second, yet the direction of the motion is continually varying, since at any given point it is along the tangent to the curve at that point.

Hence, to measure the velocity of a particle two things have to be determined: (1) the space which the particle has moved over in a given time, and (2) the change in the direction of motion during this time. In ordinary language, and in very many books on mechanics, the word velocity is used to indicate the first of these rates, i.e. the space passed over in a given time, without taking any account of any change in direction which may take place. Thus the end of the hand of a watch is said to move with uniform (i.e. constant) velocity, since it moves over equal spaces in successive equal times. It is, however, evident that the direction of the velocity is continually altering, and hence from this point of view the velocity is variable. It therefore saves confusion if we use, at any rate wherever ambiguity may arise, a separate word to denote the rate at which a particle describes its path, without reference to the direction, and for this purpose the word speed is generally used. Hence, if a particle moves in a straight line (so that the direction of motion remains constant), and passes over equal spaces in successive equal times, its velocity is said to be constant. If, however, a particle moves in a curve, so that its direction of motion continually changes, but passes over equal

1 The ratio of the total change in any quantity which occurs during a given time to that time is called the rate at which that quantity is changing.