Chapter 10Rotation

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Outline for today• Rotation of a solid body:

• Angular position• Angular displacement• Angular velocity• Angular acceleration

• Rotation with constant angular velocity

• Rotation with constant angular acceleration

• Relating linear and angular variables

Rigid body

• Definition: the distance between any twopoints of the body does not change.

A

B

AB

Fixed axis of rotation

• We will first consider rotation of a rigid bodyaround a fixed axis.

• Examples:• fixed axis- hinges of a door, BBQ pit

• varying axis- spinning top, football

Rotation• Rotation is described by an angle θ with respect

to a reference line that is perpendicular to therotation axis.

• θ(t) varies with t as the object rotates.

Top viewSide view

Angular position

• All points in the rigid body rotate by thesame angle although they travel differentdistances.

• Angular position θ: the angle between areference line fixed with the body and acoordinate system fixed in space.

s2s1

r1 r2θ

!

" =s1

r1

=s2

r2

Units: θ is dimensionless but ismeasured in radians.

Equations of Motion

• Translation • RotationPosition x(t) [m] θ(t) [rad]

Displacement Δx(t) [m] Δθ(t) [rad]

Velocity v(t)=dx/dt [m/s] ω(t) = dθ/dt [rad/s]

Acceleration a(t)=dv/dt [m/s2] α(t) = dω/dt [rad/s2]

x=x0+v0t+0.5at2 θ=θ0+ω0t+0.5αt2

If a=const. If α=const.

v=v0+at ω=ω0+αt

v2-v02=2a(x-x0) ω2-ω0

2=2α(θ−θ0)

Relating Linear & Angular Quantities

• A point a distance r from the axis of rotationmoves a distance s in time t.• Position s = rθ• Velocity v= ds/dt = d(rθ)/dt = rdθ/dt = rω• Acceleration

• tangent at= dv/dt = d(rω)/dt = rα• radial ar= v2/r = ω2r• total a2=at

2+ar2

srθ at

ar

Example problem:The new Airbus A380 engine fans extendfrom a central spool of radius 0.5m to amaximum radius of 1.5m. The engine has atop rotation speed of 3000 rpm. What arethe linear velocities of the fan tip / base?

Solutionω=3000 rpm = 3000 Rev./min x 1 min/ 60s = 50 Rev./s

1 Revolution = 2π radians

ω=3000 rpm = 50 Rev./s x 2π rad/Rev = 100π rad/s

v = ωr = (100π rad) x 1.5m = 150π m

Speed of fan tip = 150π m/s = 471 m/s

Calculating the speed ofthe fan base is left to you :)

Angular Velocity Vectors• Translation:

• Rotation: the direction of the velocityvector is defined by the axis of rotationand the right-hand-rule

!

r r = x i

^

+ y j^

+ z k^

!

r v = vx i

^

+ vy j^

+ vz k^

Period & Frequency

• Period: the time it takes for a rigid bodyto make one full revolution.

• Frequency: number of revolutions in 1 s.

!

T =s

v=2"r

#r=2"

#

!

f =1

T="

2#$ " = 2#f

Example problem:• An object rotates about a fixed axis, and the

angular position of a reference line on the objectis given by θ=0.4e2t, where θ is in radians and t isin seconds. For a point on the object 4 cm fromthe axis find the magnitude of the tangentialacceleration and the radial acceleration for t=0s.

!

" =d#

dt=d

dt(0.4e

2t) = 0.8e

2t

at=d"

dt=d

dt(0.8e

2t) =1.6e

2tat(t = 0s) =1.6 rad /s

2

ar

=" 2r = (0.8e

2t)2r = 0.64e

4 tr

ar(t = 0s) = 0.64 $ 0.04 = 0.0256 m /s

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