Chapter 8

Rotational Kinematics

Radians

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θ rad =π

180θ o

Angular DisplacementAngle through which something is

rotated Counterclockwise => positive(+)

Units => radians

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θ =Arc length

Radius=s

r

Angular SpeedRate of Rotation

Counterclockwise => positive(+) Clockwise => negative(-)

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ωavg=ΔθΔt

Units => radians/secondAlso rev/min or rpm

Angular Acceleration

Units => radians/second2

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αavg=ω2 −ω1

Δt=

ΔωΔt

Rate of change of angular speed Counterclockwise => positive(+) Clockwise => negative(-)

Linear vs. Angular Quantities Linear

x

v

a

Angular

θ

•(m)

•(m/s)

•(m/s2)

• (rad)

• (rad/s)

• (rad/s2)

Linear vs. Angular Quantities Linear Angular

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f =ωi +αΔt

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Δθ = iΔt +1

2α (Δt)2

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f2 =ωi

2 + 2αΔθ

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Δθ =1

2(ωi +ω f )Δt

Warm-upA ceiling fan’s angular speed increases

from 5.2 rad/s to 20.9 rad/s. During this constant angular acceleration, the fan moves through an angular displacement of 216 rad. How long does it take the fan to reach its final angular speed?

Tangential Velocity Instantaneous linear speed of an object

tangent to a circular pathObjects with the same angular speed,

may have different tangential speedsm/s

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vt =rω

Tangential Acceleration Instantaneous linear acceleration of an

object tangent to a circular pathObjects with the same angular

acceleration, may have different tangential accelerations

m/s2

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at =rα

Centripetal AccelerationAlways directed towards the center of

the circle

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ac =vt

2

r

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ac =rω2

Total AccelerationTangential and centripetal acceleration

are perpendicular to one another

Use Pythagorean’s theorem to find the total acceleration.Angle ϕ is measured relative to the radius.

Rolling MotionAssuming that a wheel rolls without

slipping, then The Tangential speed of a point on the

outside of the wheel will equal the linear velocity.

The tangential acceleration of a point on the outside of the wheel will equal the linear acceleration.