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                                                        Quantities and Formulas for Rotary Motion

TORQUE is a force that produces rotation of a shaft. It is measured by the product of the force (F) and the perpendicular distance from the line of action of the force to the centerline of rotation (r).

T = F r

ROTATION results when an unbalanced torque acts on a body producing an angular acceleration. The torque to accelerate is the product of the body's moment of inertia about its axis of rotation (J) and the angular acceleration (a).

T = J a

MOMENT OF INERTIA of a body is determined by the distribution of its mass about the axis of rotation. It tends to resist any change in angular velocity.

J = m r²

ANGULAR ACCELERATION is the rate of change of angular velocity and is expressed in radians per second per second. If angular velocity changes from w_o at time o to w_t at time t in time (t), the average angular acceler- ation is:

a = (w_t - w_o)/t

ANGULAR VELOCITY is the rate of angular rotation about an axis and is expressed in radians per second. If a body moves through a rotation of q radians in a time of t seconds, the average angular velocity is:

a = q/t

ANGULAR ROTATION is the arc traveled in rotary motion and can be expressed in degrees, revolutions or radians. One radian is the angle defined from the center of a circle by an arc that is equal in length to the radius.

KINETIC ENERGY is the energy of a mass in motion. It is a function of the moment of inertia (J) and the square of the angular velocity (w) expressed as:

E_k = 1/2 J w²

EQUATIONS FOR ANGULAR MOTION are analogous to those for linear motion:

v = a t	s = 1/2 a t2	v2 = 2 a s
w = a t	f = 1/2 a t2	w2 = 2 w f

If v_o and w_o denote the initial linear and angular velocity, then

v = vo + a t	s = 1/2 j w2	v2 = vo2 + 2 a s
w = wo + a t	f = wo + 1/2 a t2	w2 = wo2 + 2 a f

Analogous Linear & Angular Quantities

Linear Displacement	s	s = q r	Angular Displacement	q
Linear Velocity	v	v = w r	Angular Velocity	w
Linear Acceleration	a	a = a r	Angular Acceleration	a
Mass (Inertia)	m	J = m r2	Moment of Inertia	J
Force	F	T = F r	Torque	T
Linear:	F = m a	Ek = 1/2 m v2	Work = F s	Power = F v
Angular:	T = J a	Ek = 1/2/ J 22	Work = T q	Power = T w