Angular acceleration is the time-rate of change of angular velocity. There is an angular acceleration whenever there is a change either in the magnitude of the angular velocity or in the direction of the axis of rotation. A body that in every equal time interval t changes in angular speed from wo to w, while the direction of the axis of rotation remains unchanged, has a constant angular acceleration about the fixed axis of the value
Commonly employed units of angular acceleration are te raian per second per second and the revolution per minute per second.
Any cause which either produces or tends to produce an angular acceleration of the motion of a body is called a torque or force couple. A torque is equivalent to two equal, oppositely directed forces, acting in parallel lines. The magnitude of a torque is measured by the product of one of the forces and the perpendicular distance between the lines of action of the two forces. The line around which a torque either produces or tends to produce angular acceleration is called the torque-axis. The magnitude of a torque is expressed in centimeter-dynes, pound-feet, etc.
That property of a body because of which a torque is needed to give to the body an angular acceleration is called moment of in- ertia. Moment of inertia is measured by the tendency of a body to keep its angular velocity of constant magnitude and about an invariable axis of rotation. The moment of inertia of a particle of mass m at a distance r from the axis of rotation is mr2. The moment of inertia of a body about a given axis is numerically equal to the sum of the products of the masses of the particles composing the body and the squares of their respective distances from the axis of rotation.
Units of moment of inertia are the gram-centimeter2, the poundfoot 2, and the slug-foot'. This last unit is the British engineering unit of moment of inertia.
When a body rotates against a constant opposing torque L through an angular displacement ¢ radians, it does an amount of work
The kinetic energy of a body rotating with angular velocity w is
where K represents the moment of inertia of the body with respect to the axis of rotation. If K is measured in feet and slugs, and w in radians per second, W,, will be expressed in foot-pounds.
If an angular displacement 0 be effected in time t with a constant angular velocity w, then the ower develoed bthe torue
The product of the moment of inertia of a body with respect to a given axis and the angular velocity of the body about the same axis is called the angular momentum of the body with respect to the given axis. Thus, in symbols, the angular momentum with respect to an axis throuh c is
It can be shown that the sum of the moments of the linear momenta of the elementary parts of a body equals the angular momentum of the body. For this reason, angular momentum is also called moment of momentum. It is sometimes called kinetic moment.
The units employed are the gram centimeter2-radian per second and the slug foot2-radian per second. They have no names but are referred to as the centimeter-gram-second absolute unit and the British engineering unit of angular momentum, respectively.
3. Composition and Resolution of Forces. - A force has both direction and magnitude. It can be completely represented by a straight line drawn in the direction in which the force acts, of a length proportional to the magnitude of the force, and having an arrow-head marked on the line pointing in the direction of the action of the force. Two quantities represented by lines that intersect are said to be concurrent. The motion of a body may be due to the effect of two or more simultaneous forces. Two sys-
Angular velocity is the time-rate of change of angular displacement in a given sense about a given axis. The magnitude of angular velocity is called angular speed. Thus, one might say that a top has an angular velocity of 200 revolutions per minute in the clockwise direction about an axis inclined 30 degrees to the vertical and 45 degrees west of the meridian plane. If the angular displacement in every equal time interval t be constant and represented by 0, then the magnitude of the constant angular velocity is
