CONTENTS. Definitions - Motion - Velocity - Acceleration - Graphical Methods-Velocity Diagrams-Falling Bodies-General Formulaæ- Rotation-Angular Velocity-Circular Measure-Angular Accelera- tion Composition and Resolution of Velocities - Parallelogram of Velocities-Triangle of Velocities-Polygon of Velocities-Rectangular Resolution-Composition and Resolution of Accelerations-The Hodo- graph-Hodograph for Motion in a Circle-Examples I., II., III., and IV.-Instantaneous Centre-Questions. DEFINITION. A body is said to be in Motion when it is con- tinually changing its position in space, and to be at Rest when it retains a fixed position in space. These are the definitions of absolute motion and absolute rest. We can never know the absolute motion of any body because we know no fixed bodies to which we may refer its positions at different times. We, therefore, can only deal with the relative DEFINITION. A body is said to have Relative Motion with respect to another body when it is continually changing its posi- Thus, take the case of a train moving on a railway. We always Motions of Translation and Rotation. The motion of a body A body is said to have a motion of simple translation when all points in the body move with the same velocity and in the same direction at the same instant, so that no line in the body changes its direction. Hence, the motion of the whole body is known when that of any point in it is known. A body is said to have a motion of simple rotation when the various points in the body describe circles about some fixed axis either within or without the body. Hence, the motion of the whole is known when that of any line in the body (other than the axis about which the motion takes place) is known. Thus, the The motion of a body may be complex; being composed or compounded of motions of translation and rotation. connecting-rod of an engine has a complex motion. motion of translation in a vertical plane containing the centre line of the engine, and a motion of rotation in the same plane about the crosshead pin. DEFINITION.—The Path of a moving point is the line, straight or curved, which passes through all the successive positions of the point. Direction of Motion.-The direction of motion of a body is, at any particular instant, the tangent to the path of the body at that instant, or the path itself if the motion is rectilinear. A Ꭱ B ILLUSTRATING DIRECTION OF MOTION. Thus, let A B be the path of a moving body. When the body occupies the position, P1, its direction of motion is along P1, T1, the tangent to the path at that point. Similarly, when the body occupies the position P2, its direction of motion is along the tangent P, To Hence, when a body moves in a circular path its direction of motion at any instant will be perpendicular to the radius drawn to its position on the circle at that instant. DEFINITION.-The Velocity of a body is the rate at which it changes its position. A velocity is completely specified when we know (1) its direction, and (2) its magnitude. Hence, a velocity can be completely represented by a straight line of finite length with a suitably-directed arrow head. DEFINITION. -- A body is said to be moving with Uniform Velocity when it is moving in a constant direction and passes over equal distances in equal intervals of time, however small these may be. The last clause in the above definition is necessary, because a body might describe equal distances in equal times, and yet its motion might not be uniform. Thus, a train may describe 20 miles in each of two consecutive hours, and yet its motion may have varied continuously during that time; sometimes its velocity may be 60 miles an hour, and at other times it may be nil. Uniform Velocity, how Measured. When uniform, the velocity of a body is measured by its displacement in unit time. Thus:Displacement Velocity = Time DEFINITION. A body is said to have Unit Velocity when it describes unit distance in unit time. The unit of distance in this country is the foot, and the unit of time is usually the second, although engineers often take the minute, or even the hour, as the unit of time. For example, the speed of a railway train is always spoken of as so many miles per hour, and that of the piston of an engine as so many feet per minute. Whatever units may be used, we get : Or, Where, 8 = And, Displacement, or distance described, in time, t. v = Velocity, supposed to be uniform. *[ From the above definition and equation it is evident that v must be the same however small t may be. Thus, let the displacement be very small, say As, then the time taken to describe it will be correspondingly small, say ▲ t, and we get:— This being true for the smallest fraction of time, it must also be true in the limit. ds v = d t (II)] DEFINITION. A body is said to be moving with Variable Velocity when it is either changing its direction of motion or passing over unequal distances in equal intervals of time. Students who have no knowledge of the notation of the Calculus, and those merely reading for examination in the Advanced Stage of this subject, may omit for the present the text within the brackets, thus [ ]. From this definition it appears that a body has a variable velocity when the direction or magnitude of its velocity is variable. Thus, a point on the rim of the flywheel of an engine has a variable velocity whether the rotary motion of the wheel be uniform or not. This follows at once from the fact that a velocity is only completely specified when we know its direction and magnitude, and a change in either the direction or in the magnitude causes a change in the velocity. It is usual, however, in most problems, to speak of the velocity as being uniform or variable, according as the magnitude of the velocity is uniform or variable. Variable Velocity, how Measured. When variable, the velocity of a body is measured at any particular instant by the displacement which the body would have received if it moved for a unit of time with the same velocity which it had at the instant under consideration. By Thus, we see a train approaching a station and say that its velocity is 10 miles an hour, although we at the same time observe that its velocity is diminishing rapidly, and will soon be zero. the expression "10 miles an hour we, therefore, do not mean that it will run 10 miles during the next hour, but simply that if the train continued to run for one hour with the same speed that it had at the instant the remark was made, it would travel a distance of 10 miles. Average Velocity. When the velocity of a body is variable, and we know its magnitudes for several positions of the body, then its average velocity can be found in the same way as we find the average of a series of numbers. Thus, let V1, V2, V3, points in its path; then : vn denote the velocities at n different Or it may be defined as follows: DEFINITION. When a body moves through a certain distance with a variable velocity, its average velocity is that uniform velocity which it would require to have in order to traverse the same distance in the same time. If the velocity increase or decrease uniformly, then the mean or average velocity is half the sum of the initial and final velocities. Where V1 and V2 respectively. denote the initial and final velocities DEFINITION. The acceleration of a body is its rate of change of velocity. Acceleration may be either uniform or variable. DEFINITION.-Acceleration is uniform when equal changes of velocity take place in equal intervals of time, however small these may be. Otherwise, the acceleration is variable. Acceleration, how Measured.-Uniform acceleration is measured by the change in the velocity in a unit of time. Variable acceleration is measured at any particular instant by what would be the change of velocity in a unit of time, on the supposition that during that unit of time the acceleration remained the same as at the instant under consideration. If the student thoroughly understands the method of measuring a variable velocity, he should have no difficulty in perceiving from the above statement how variable acceleration is measured. Uniformly Accelerated Motion. We shall now deduce the ordinary formulæ for the motion of a body uniformly accelerated in its line of motion. Let "1 V2 = = 8 = Distance described during interval (t2 – t1), a = Or, denoting the interval of time (t, – t1) by t, we get :— That is:-Final Velocity = Initial Velocity + Change of Velocity. Again, since the acceleration is uniform, we get: |