Rotational MotionRotational Motion

Rotational MotionRotational Motion

Rotational motion is the motion of a Rotational motion is the motion of a body about an body about an internalinternal axis. In axis. In rotational motion the axis of motion rotational motion the axis of motion is part of the moving object.is part of the moving object.

All of the properties of linear motion All of the properties of linear motion which we have discussed so far this which we have discussed so far this year have corresponding rotational year have corresponding rotational (angular) properties.(angular) properties.

Rotational MotionRotational Motionlinear propertylinear property angular propertyangular property

distance(d) = angular displacement distance(d) = angular displacement ((θθ))

velocity(v)velocity(v) = angular velocity = angular velocity ((ωω))

acceleration(a)acceleration(a) == ang. accel. (ang. accel. (αα))

inertia (m) =inertia (m) = rotational inertia (I)rotational inertia (I)

force (F)force (F) == torque (torque (ττ))

Rotational MotionRotational Motion

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

Angular DisplacementAngular DisplacementSince all rotational quantities have linear Since all rotational quantities have linear equivalents, we can convert between them.equivalents, we can convert between them.

Angular displacement is the rotational Angular displacement is the rotational equivalent of distance. To find the distance equivalent of distance. To find the distance a point on a rotating object has traveled (its a point on a rotating object has traveled (its arc length) we need to multiply the angular arc length) we need to multiply the angular displacement by the radius.displacement by the radius.

dt()(r)

Angular VelocityAngular Velocity

where angular displacement is where angular displacement is measured in “radians”:measured in “radians”:

angular velocity= (angular angular velocity= (angular displacement /time)displacement /time)

ωω = ( = (ΔθΔθ)/t)/t

where angular velocity can be where angular velocity can be measured in radians/sec, or measured in radians/sec, or revolutions/sec. revolutions/sec.

Angular VelocityAngular Velocity

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

vt(r)( )

Angular AccelerationAngular Acceleration

Angular acceleration is also similar to Angular acceleration is also similar to linear acceleration.linear acceleration.

Angular acceleration=angular speed/ Angular acceleration=angular speed/ timetime

αα = = ωω / t / t

Angular AccelerationAngular Acceleration

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

at(r)( )

Rotational MotionRotational Motion

While an object rotates, every point While an object rotates, every point will have different velocities, but they will have different velocities, but they will all have identical angular will all have identical angular velocities.velocities.

All of the equations of linear motion All of the equations of linear motion which we have discussed so far this which we have discussed so far this year have corresponding rotational year have corresponding rotational (angular) equations.(angular) equations.

Rotational MotionRotational Motionlinear equation linear equation angular equationangular equation

v = ∆x/∆t =>v = ∆x/∆t => ωω =∆ =∆ θθ /∆t /∆t

a = ∆v/∆ta = ∆v/∆t => => αα=∆ =∆ ωω /∆t /∆t

vvff = v = vii + a∆t + a∆t =>=> ωω ff = = ωω ii + +αα∆t∆t

∆∆d = vd = vii∆t + 1/2a(∆t)∆t + 1/2a(∆t)22 =>=>∆ ∆ θθ = = ωω ii∆t + 1/2∆t + 1/2αα((∆t) ∆t) 22

vvff = √(v = √(vii22 + 2a∆x) + 2a∆x) => =>

ωω ff = √( = √(ωω ii22 + 2a∆d) + 2a∆d)

F=maF=ma => => ττ=I =I αα

Rotational MotionRotational Motion

Practice problems p. 145-147Practice problems p. 145-147

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

acvt

2

r(r)( 2 )

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

Fcma

cm(

vt

2

r)mr 2

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

Rotational MotionRotational Motion

Practice problems p. 149-150Practice problems p. 149-150

Rotational InertiaRotational Inertia

Force causes acceleration. Force causes acceleration.

Inertia resists acceleration. Inertia resists acceleration.

Torque causes rotational Torque causes rotational acceleration. acceleration.

Rotational inertia resists rotational Rotational inertia resists rotational acceleration.acceleration.

ττ=I =I αα = r F = r F

I = m rI = m r22

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