Rotational MechanicsRotational Mechanics

Rotary MotionRotary Motion

Rotation about internal axis (spinning)Rate of rotation can be constant or variableUse angular variables to describe rotationAll 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 AnglesRadian Measure of Angles

Radian is ratio of arc distance subtended by angle to the radius: = darc/r

Radian is dimensionless (meters/meters) One complete rotation equals 2 radiansOne radian equals 57.3 degrees

Angular DisplacementAngular Displacement

Angular displacement (), angle of rotation measured in radians

Linear displacement equals angular displacement times the radiusAll parts of rotating object have same angular displacement

Angular VelocityAngular 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 VelocityLinear velocity = angular velocity times radius, v = r

All parts of rotating object have same angular velocity

Angular AccelerationAngular Acceleration

Change in angular velocity with respect to time: t; units are rad/s2

Vector whose direction is found using right hand ruleIf angular acceleration is constant, constant acceleration equations are used.

Comparing Linear and Angular VariablesComparing Linear and Angular Variables

Quantity Linear Angular Conversion Displacement d (x, y) d = r

Velocity v v = r Acceleration a a = r

Constant Acceleration EquationsConstant Acceleration Equations

Linearvf = vi + at

d = vit + 1/2(at2)

vf2 = vi

2 + 2ad

Angularf = i + t

it + 1/2(t2)f

2 = i2 + 2

Center of GravityThe 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 objectCan produce rotationConcurrent 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 objectActually are the sum of an infinite number of parallel weight vectors from an infinite number of mass unitsThe effect is as if all the weight was concentrated at the center of gravity

TorqueThe 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 InertiaRotational Inertia

Resistance of an object to any change in angular velocityDepends on mass and its orientation with respect to axis of rotation.Is rotational analogue to mass; symbol I, units kg m2

Sometimes called moment of inertia

Rotational InertiaFor an object rotating about an external point, I = mr2

For objects rotating about an internal axis, inertia must be calculated using calculusUse rotational inertia equations for general type of regularly shaped solid bodies

Newton’s Second Law for RotationNewton’s Second Law for Rotation

Substitute angular variables for linearF = ma becomes = Iwhere is the net torque and I is the rotational inertia of the body.

Work in Rotary MotionWork in Rotary Motion

Work done by torqueW = = Fr is angular displacement in radiansAssumes force is perpendicular to radius

Power in Rotary MotionPower in Rotary Motion

Power is rate of doing workP = /t

/t = , so P =

Kinetic Energy in Rotary MotionKinetic Energy in Rotary Motion

Energy possessed by rotating objectKErot = 1/2(I2)

Rolling objects have both linear and rotational kinetic energy

Kinetic Energy in Rotary MotionWhen 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 rotatingA combination of the rotational inertia and angular velocityFor a rotating object, L = IA 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 pointRadial distance is perpendicular distance form path of object to the point

Conservation of Angular MomentumExternal net torque is required to change angular momentumIf no net external torque is present, angular momentum of a system will remain constantTotal 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 MomentumAlways true, from atomic to galactic interactionsIf rotational inertia changes, angular velocity must change to conserve angular momentum

PrecessionPrecession

A secondary rotation of the axis of rotationDue to torque produced by weight of rotating objectCauses angular acceleration that changes direction of angular velocity of rotating objectEarth precesses on its axis with a secondary rotation period of 26,000 years