Uniform Angular Velocity.-Let the centre line, OP, of the crank in the above figure sweep out the angle, A O P = 0, in the interval of time, t; then the angular velocity of the body (usually denoted by the Greek letter w) is :— The angle,, is measured in circular units, and not in degrees. The unit angle in circular measure is called the radian, and may be defined as the angle subtended at the centre of a circle by an arc of its circumference, equal in length to the radius of the circle. Hence, if t is in seconds, the unit of angular velocity will be the radian per second. 2 Since the length of the arc subtending a right angle is xr, and, therefore, the circular measure of a right angle equal 2 to radians, we may easily determine the number of degrees in a radian. Thus : Degrees in 1 radian : Degrees in 1 right angle = 1: T In general, if be the circular measure of an angle of no, Hence, if the angle described in time, t, by OP, be no, we get:— When the linear velocity of any point, P, in the body, and its distance from the axis are known, the angular velocity of the body can be found. Thus :: Component of linear velocity of P perpendicular to r = Radius, O P. Then, v = Are described by P in unit time. Circular measure of angle described by OP in unit time. Variable Angular Velocity.-Angular Acceleration. When the angular velocity is variable, it is measured in a way similar to that of variable linear velocity. = [Let A small angle described by O P, in small interval of time, At; then we have : DEFINITION.--The angular acceleration of a rotating body is the rate of change of its angular velocity. Angular acceleration may be either uniform or variable according as equal changes of angular velocity take place in equal or unequal intervals of time. When uniform, angular acceleration is measured by the increase or decrease of angular velocity per unit time. Angular velocities at the beginning and end of interval of time, t. Angular displacements at the beginning and end of interval of time, t. Angular acceleration. From these equations and those previously deduced for uniformly accelerated linear motion, the student will notice the similarity of the relations between the terms s, v, and a, and 0, ∞, and a respectively. Hence, we get the remaining and corresponding equations for rotary motion, viz. :— *It is sometimes convenient to speak about the angular velocity of a point, such as P in the foregoing figure. Such a phrase is not strictly correct, and when used, it should be understood to mean the angle described in unit time by the radius drawn through the point, P. Composition and Resolution of Velocities.-A moving body may have at any instant two or more velocities in different directions, and it then becomes an important problem to be able to determine the resultant velocity, both in magnitude and in direction. Thus, the magnitude and direction of the motion of a man who walks across the deck of a moving ship is different from that of the ship and also from that of his motion relative to the deck. Similarly, the motion of a point on the rim of a carriage wheel in motion is, in general, different in magnitude and direction from its circular motion about the axle, and also from the onward motion of the wheel as a whole. The process of finding a single velocity equivalent in effect to two or more velocities is called the Composition of Velocities. The process of finding two or more velocities equivalent in effect to a single velocity is called the Resolution of Velocities. DEFINITIONS.-The single velocity which is equivalent to two or more velocities is called their Resultant, and these two or more velocities are called the Components. Parallelogram of Velocities.-If two component velocities be represented, in magnitude and direction, by two adjacent sides, OA, OB, of a parallelogram, their resultant velocity will be represented by the diagonal, O D, through their intersection. B Resultant Thus, if a moving point, O, possess simultaneously two velocities, P and Q, in directions OA and OB respectively, and, if O A and OB represent the magnitudes of these velocities, their resultant velocity, R, will be represented both in magnitude and in direction by the diagonal, OD, of the parallelogram constructed on O A, and O B, as adjacent sides. = Component 60° Component PA PARALLELOGRAM LAW. angle between the directions of the velocities, P and Q. a = ZA OD, and ß = <BOD, the angles between the direction of the resultant, R, and the components P and Q respectively. Then the student may easily prove from Euclid II., 13 and 14, or by trigonometry, that: From these equations the magnitude and direction of the resultant velocity can be calculated. It is not necessary to complete the parallelogram as explained above, it being quite sufficient to draw but one-half of the figure. Thus, A D is equal and parallel to OB; hence, as much can be determined from the triangle, O A D, as from the complete parallelogram, O A D B. Triangle of Velocities. If two component velocities be represented in magnitude and direction by two sides of a triangle taken in order, their resultant will be represented in magnitude and direction by the third side taken in the reverse direction. Hence, if there be simultaneously impressed on a point three velocities represented in magnitude and direction by the sides of a triangle taken in order, then the point will remain at rest.* Polygon of Velocities. If several component velocities be repre Ц POLYGON OF VELOCITIES. city will be represented in magnitude and direction by the side, A F, re quired to complete the polygon. * In setting out the Parallelogram, or Triangle of Velocities, it is not necessary to draw the sides parallel to the velocities represented. The sides may be drawn in directions perpendicular to the respective velocities, or, indeed, at any other angle, so long as the angle is the same for all the sides. In such cases the line representing the resultant will be equally inclined to its true direction. If the figure whose sides represent the component velocities be closed or completed when the last velocity has been represented, then there is no resultant velocity, and the point will remain at rest. ... It is equally important to be able to resolve a given velocity into two or more component velocities. Thus, the velocity, R (see the figure for Parallelogram of Velocities), can be resolved into two components, P and Q, in the directions O A, O B respectively. Or, the velocity, V (in the last figure), may be resolved into a number of components, v1, v2, in directions A B, BC, . . . Further, the directions of the component velocities may be anything we like. Thus, in resolving a given velocity, R, into two components, we can do so in an infinite number of ways, since an infinite number of parallelograms, such as O AD B, can be found having O D for one of their diagonals. When, however, the directions of the components are fixed, their magnitudes will be definite and easily determined. Referring to the figure for the Parallelogram of Velocities, let O D represent a velocity, R, which has to be resolved into two components in the directions OA and OB. From D draw DA parallel to BO and D B parallel to A O, meeting the lines OA and O B in the points A and B respectively. Then O A and O B represent the component velocities P and Q to the same scale that O D represents the velocity R. C The most important case of resolution is that wherein the given velocity has to be resolved into components whose directions are at right angles to each other. Thus, let it be required to resolve the velocity, v, whose direction is OC, into its Rectangular Components along O x and O y. B -U cos RECTANGULAR RESOLUTION. From C drop the perpendiculars CA, CB on the axes Ox and Oy. Then, OA, OB are the components in the required directions. Components of v in directions Ox, O y respectively. Composition and Resolution of Accelerations. tion is a rate of change of velocity, whether in direction, it follows that accelerations may be resolved according to the same rules as velocities. Since an acceleramagnitude or in compounded or |