Rotational Mechanics

Rotary Motion Rotation about internal axis (spinning) Rate of rotation can be constant or variable Use angular variables to describe rotation All parts of a rigid rotating object have same angular displacement, velocity and acceleration Linear displacement and velocity depend on distance from rotation axis.

Radian Measure of Angles Radian is ratio of arc distance subtended by angle to the radius:  =  d arc /r Radian is dimensionless (meters/meters) One complete rotation equals 2  radians One radian equals 57.3 degrees

Angular Displacement Angular displacement (  ), angle of rotation measured in radians Linear displacement equals angular displacement times the radius All parts of rotating object have same angular displacement

Angular Velocity Change in angular displacement with respect to time  =  t; fundamental units are rad/s, but usually measured in rev/s, or rev/min (rpm) Vector with direction found using right hand rule: Curl fingers of right hand in direction of rotation. Extended thumb points in direction of vector

Angular Velocity Linear velocity = angular velocity times radius, v =  r All parts of rotating object have same angular velocity

Angular Acceleration Change in angular velocity with respect to time:  t; units are rad/s 2 Vector whose direction is found using right hand rule If angular acceleration is constant, constant acceleration equations are used.

Comparing Linear and Angular Variables

Constant Acceleration Equations Linear v f = v i + a  t  d = v i  t + 1/2(a  t 2 ) v f 2 = v i 2 + 2a  d Angular  f =  i +  t  i  t + 1/2(  t 2 )  f 2 =  i 

Center of Gravity The point at which all object’s weight can be considered to be concentrated. For symmetrical bodies with uniform density, c.o.g. will be at geometric center. May be located outside the body of some objects. Bodies or systems rotate about their center of gravity. Similar to center of mass but not always the same

Parallel Forces Forces acting in the same or opposite directions at different points on an object Can produce rotation Concurrent forces act at the same point (often the center of gravity) at the same time on an object

Weight Vectors Drawn from center of gravity of object Actually are the sum of an infinite number of parallel weight vectors from an infinite number of mass units The effect is as if all the weight was concentrated at the center of gravity

Torque The result of a force that produces rotation, a vector The product of the force and its lever arm, Lever arm (or moment arm) is a vector whose magnitude is the distance from the point of rotation to the point of application of the force  = r x F A product of two vectors that produces a third perpendicular vector

Torque units are meters x newtons signs: ccw torques are considered +, cw torques are - direction of net torque is direction of resulting rotation

Rotational Inertia Resistance of an object to any change in angular velocity Depends on mass and its orientation with respect to axis of rotation. Is rotational analogue to mass; symbol I, units kg m 2 Sometimes called moment of inertia

Rotational Inertia For an object rotating about an external point, I = mr 2 For objects rotating about an internal axis, inertia must be calculated using calculus Use rotational inertia equations for general type of regularly shaped solid bodies

Newton’s Second Law for Rotation Substitute angular variables for linear F = ma becomes  = I  where  is the net torque and I is the rotational inertia of the body.

Work in Rotary Motion Work done by torque W =  = Fr   is angular displacement in radians Assumes force is perpendicular to radius

Power in Rotary Motion Power is rate of doing work P =  /  t  /  t = , so P = 

Kinetic Energy in Rotary Motion Energy possessed by rotating object KE rot = 1/2(I  2 ) Rolling objects have both linear and rotational kinetic energy

Kinetic Energy in Rotary Motion When object rolls downhill, potential energy is converted to both types of kinetic energy; amount of each depends on rotational inertia of object.

Angular Momentum The tendency of a rotating object to continue rotating A combination of the rotational inertia and angular velocity For a rotating object, L = I  A vector

Angular Momentum Objects in circular motion also have angular momentum: L = mvr Angular momentum can be applied to any moving object with respect to an external point Radial distance is perpendicular distance form path of object to the point

Conservation of Angular Momentum External net torque is required to change angular momentum If no net external torque is present, angular momentum of a system will remain constant Total angular momentum before the interaction equals total angular momentum after the interaction as long as no net external torque acts on the system

Conservation of Angular Momentum Always true, from atomic to galactic interactions If rotational inertia changes, angular velocity must change to conserve angular momentum

Precession A secondary rotation of the axis of rotation Due to torque produced by weight of rotating object Causes angular acceleration that changes direction of angular velocity of rotating object Earth precesses on its axis with a secondary rotation period of 26,000 years