Kinematics of Rotation of Rigid Bodie Angle of rotatio lar displacement Δθ = θ – θ 0 0 if rotation is counterclockwise 0 if rotation is clockwise z ngle of rotation θ is dimensionless quantity, ut it is a vector z i r s R adius length Arc radians in ) ( If θ is in radians, then s=rθ and s’=r’θ with the same r s r’ s’ 0 0 0 3 . 57 2 360 1 360 2 2 1 rad rad r r revolution
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Kinematics of Rotation of Rigid Bodies Angle of rotation Angular displacement Δθ = θ – θ 0 Δθ > 0 if rotation is counterclockwise Δθ < 0 if rotation is.
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Kinematics of Rotation of Rigid Bodies
Angle of rotationAngular displacement Δθ = θ – θ0
Δθ > 0 if rotation is counterclockwiseΔθ < 0 if rotation is clockwise
z
Angle of rotation θ is a dimensionless quantity, but it is a vector
zi
r
s
Radius
lengthArcradiansin )( If θ is in radians, then
s=rθ and s’=r’θ with the same θ.
r
s
r’
s’
00
0 3.572
36013602
21
radradr
rrevolution
Example:A total eclipseof the sun
Average angular velocityAngular Velocity and Acceleration
rpmrev
ors
rad
ttt zz
min
,0
0
Instantaneous angular velocity
zz idt
d
t
,lim0t
z z
ωz>0 ωz<0
Average angular acceleration
2
0
0 ,s
rad
ttt zzzz
z
Instantaneous angular acceleration
zzzz
z idt
d
dt
d
t
,lim2
2
0t
Instantaneous angular speed is a scalar .||
Kinematic Equations of Linear and Angular Motionwith Constant Acceleration
(1)
(3)
(4)
(2)
Rotational motion Quantity Linear motion
θ Displacement x
ω0z Initial velocity v0x
ωz Final velocity vx
αz Acceleration ax
t Time t
Relations between Angular and Tangential Kinematic Quantities
Ice-skating stunt“crack-the-whip”
dt
dr
dt
dvdt
dr
dt
ds
rs
Centripetal and Tangential Accelerations in Rotational Kinematics
Exam Example 22: Throwing a Discus (example 9.4)
Rotational Kinetic Energy and Moment of Inertia
Kinetic energy of one particle222
2
1
2
1 mrmvK T Rotational kinetic energy is the kinetic energy of the entire rigid body rotating with the angular speed ω
222
2
1
2
1 IKrmK Ri
iiR
Definition of the moment of inertia i V
ii dVrrmI 22
Total mechanical energy
UIMvE cmcm 22
2
1
2
1
Translationalkinetic energy
Rotational kinetic energy cmr
r
'r
cmv
0cm
Parallel-Axis Theorem 2cmcm MrII
Proof:
i
iicmi i
cmii
iicmi rmrrmrmrrmI '2)(')'( 222 =0
Potentialenergy
ω
vh
Rotation about an Axis Shifted from a Center of Mass Position
Exam Example 23: Blocks descending over a massive pulley (problem 9.83)