Chapter 10Rotation
par 1
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
2
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