Rotational Motion

Rotational motion is the motion of a body about an internal axis. In rotational motion the axis of motion is part of the moving object. All of the properties of linear motion which we have discussed so far this year have corresponding rotational (angular) properties.

Rotational Motion linear propertyangular property distance(d) = angular displacement (θ) velocity(v)= angular velocity (ω) acceleration(a)=ang. accel. ( α ) inertia (m) =rotational inertia (I) force (F)=torque ( τ )

Rotational Motion The motion of an object which moves in a straight line can only be described in terms of linear properties. The motion of an object which rotates can be described in terms of linear or rotational properties.

Angular Displacement Since all rotational quantities have linear equivalents, we can convert between them. Angular displacement is the rotational equivalent of distance. To find the distance a point on a rotating object has traveled (its arc length) we need to multiply the angular displacement by the radius.

Angular Velocity where angular displacement is measured in “radians”: angular velocity= (angular displacement /time) ω = (Δθ)/t where angular velocity can be measured in radians/sec, or revolutions/sec.

Angular Velocity Angular speed is the rotational equivalent of linear speed. To find the linear speed of a rotating object (its tangential speed) we need to multiply the angular speed by the radius.

Angular Acceleration Angular acceleration is also similar to linear acceleration. Angular acceleration=angular speed/ time α = ω / t

Angular Acceleration Angular acceleration is the rotational equivalent of linear acceleration. To find the linear acceleration of a rotating object (its tangential acceleration) we need to multiply the angular acceleration by the radius.

Rotational Motion While an object rotates, every point will have different velocities, but they will all have identical angular velocities. All of the equations of linear motion which we have discussed so far this year have corresponding rotational (angular) equations. All of the equations of linear motion which we have discussed so far this year have corresponding rotational (angular) equations.

Rotational Motion linear equation angular equation v = ∆x/∆t => ω =∆ θ /∆t v = ∆x/∆t => ω =∆ θ /∆t a = ∆v/∆t => α =∆ ω /∆t v f = v i + a∆t=> ω f = ω i + α ∆t ∆d = v i ∆t + 1/2a(∆t) 2 => ∆ θ = ω i ∆t + 1/2 α( ∆t) 2 v f = √(v i 2 + 2a∆x)=> ω f = √(ω i 2 + 2a∆d) F=ma=> τ =I α

Rotational Motion Practice problems p

An object moving at constant speed in a circular path will have a zero change in angular speed, and therefore a zero angular acceleration. That object is changing its direction, however, and therefore has a changing linear velocity and a non-zero linear acceleration. It has an acceleration directed toward the center of the circle causing it not to move in a straight line. This is a centripetal (center seeking) acceleration.

Centripetal Acceleration This acceleration is perpendicular to the tangential (linear) acceleration. All accelerations are caused by forces and centripetal acceleration is caused by centripetal force. A force directed towards the center of a circle which causes an object to move in a circular path.

Centrifugal Force A centripetal force pulls an object towards the center of a circle while its inertia (not a force) tries to maintain straight line motion. This interaction is felt by a rotating object to be a force pulling it outward. This "centrifugal" force does not exist as there is nothing to provide it. It is merely a sensation felt by the inertia of a rotating object

Rotational Motion Practice problems p

Rotational Inertia Force causes acceleration. Inertia resists acceleration. Torque causes rotational acceleration. Rotational inertia resists rotational acceleration. τ =I α = r F I = m r 2

Rotational Inertia Inertia is measured in terms of mass. Rotational inertia is measured in terms of mass and how far that mass is located from the axis. The greater the mass or the greater the distance of that mass from the axis, the greater the rotational inertia, and therefore the greater the resistance to rotational acceleration.