Ch. 10: Rotational Dynamics
Arc Length
θ=s/R is in radians
Angular Displacement
∆θ = θf – θi
s
τ=rFsinθ
R
I≡
i
mi ri2
Q1-2
Rotational Kinetic Energy: ½Iω2
Calculation:
Force times lever arm d = rsinθ
F||
τ = rFsinθ
θ
F
r
analog to force for rotational
motion
distinct from force
units: N·m (not Joules)
tendency for a force to rotate
an object about an axis
must specify the rotation axis
Alternative Calculation:
τ = rFsinθ
τ = Fd
θ
F
θ
Instantaneous Angular Speed
ω = dθ/dt
Angular Acceleration
α = dω/dt
Moment of Inertia:
Torque
r
d
τ = F⊥r
θ
F
r
F⊥
F⊥ = Fsinθ
1
Net Torque:
τnet = Σ τi = Σ Fidi
use right-hand-rule for signs:
If τ tends to rotate the object
clockwise (-)
counter-clockwise (+)
use same pivot point for all torques
Work, Power, Energy
Work:
Power:
dWR = τ dθ
PR =
WR = τ dθ
dWR
dθ
=τ
= τω
dt
dt
Work-Energy Theorem
WR =
τ dθ = ∆K R = 1 2 Iω 2f − 1 2 Iωi2
Newton’s Second Law
for Rotation:
Σ τi = Ια
net torque causes a change in angular
acceleration
see text for derivation
Mechanical Energy
Conservation
Ei + Wnc = Ef
Ki +KRi + Ui + Wnc = Kf + KRf + Uf
Translational kinetic energy
Rotational kinetic energy
2
Rolling without slipping
ω
R
θ
s
Pure Rolling
∆xcm = s = R∆θ
vcm = ds/dt
= Rdθ/dt
= Rω
vcm = Rω
Example: Rolling down an incline
Which object gets to the bottom first?
– Solid sphere?
– Spherical shell?
– Hoop?
– Disk?
Does mass matter?
Does radius matter?
Transp 11.3-4
Now, Ktot=½IPω2
= ½Icmω2 + ½MR2ω2
= ½Icmω2 + ½Mv2
∆x = s
acm = Rα
vcm = Rω
vpoint = vcm + vrot
Rotational contribution
Q3
Why?
Center-of-mass
contribution
Rolling Race
Let the object start from rest and roll without
slipping.
Note: Icm = cMR2
c = shape factor
Time depends on the acceleration, but
in this case, acm ∝ v2cm
(study examples in Ch. 10)
Winner has largest vcm at the bottom.
3
Race (cont)
initial
final
v
h
Ei=Ef
Ki + KRi + Ui = Kf + KRf + Uf
0 + 0 + mgh = ½mv2 + ½Icmω2 + 0
y=0
mgh = ½mv2 + ½(cmr2)(v/r)2
mgh = ½m(1+c)v2
v=
2 gh
1+ c
1/ 2
4