1. Rotational Dynamics

2. Translational-Rotational Analogues & Connections Continue!
Translation Rotation
Displacement x θ
Velocity v ω
Acceleration a α
Force (Torque) F τ
Mass m ?
CONNECTIONS
v = rω
atan= rα
aR = (v2/r) = ω2 r
τ = rF

3. Analogous to Newton’s 2nd Law:
Goal: Newton’s 2nd Law for rotational motion.
We’ve just seen:
The angular acceleration is  the net torque or
α  τnet = ∑τ = sum of torques
Analogous to Newton’s 2nd Law:
a  Fnet = ∑F = sum of forces
Recall, Newton’s 2nd Law: ∑F = ma. a  α , F  τ
Question: What plays the role of mass m for rotational problems?

4. Simplest Possible Case
A mass m moving in a
circle of radius r, one force F
TANGENTIAL
to the circle τ = rF
Newton’s 2nd Law
+ relation (a = rα)
between tangential
& angular accelerations
 F = ma = mrα So
τ = mr2α  Newton’s 2nd Law for Rotations
Proportionality constant between τ & α is mr2 (point mass only!)

5.  Moment of Inertia of the body
 For one particle, moving in a circle, Newton’s 2nd Law for rotational motion is:
τ = mr2α
Extrapolate to a rigid body, made of many particles. Newton’s 2nd Law, rotational motion becomes
∑τ = ∑(mr2)α
Since α is the same for all points in the body, it comes out of the sum.
 Write ∑τ = Iα, where the proportionality constant I = ∑(mr2) = m1r12 + m2r22 + m3r32 + …
 Moment of Inertia of the body
sum over all point masses in the body  

6. Moment of Inertia of the body:
 Newton’s 2nd Law, rotational motion is:
The net torque =
(moment of inertia)  (angular acceleration)
Moment of Inertia of the body:
Moment of Inertia I = A measure of the rotational inertia of the body. Analogous to the mass m = A measure of translational inertia of the body.

7. Translational-Rotational Analogues & Connections
Translation Rotation
Displacement x θ
Velocity v ω
Acceleration a α
Force (Torque) F τ
Mass (moment of inertia) m I
CONNECTIONS
v = rω, atan= rα, aR = (v2/r) = ω2r
τ = rF , I = ∑(mr2)
NEWTON’S 2nd LAW: ∑F = ma; ∑τ = Iα

8. Obtained for simple
shaped bodies using
calculus.
Useful for problem
solving!

9. Idisk > Irod , even if their masses are the same!!!
Moment of Inertia:
I = ∑(mr2)
Depends on mass m AND on the distribution of mass (r2) in the object!
Idisk > Irod , even if their masses are the same!!!
  Disk or thin
cylinder
  Long, small
radius cylinder

10. I depends on the rotation
Example
I = ∑(mr2)
a) I = (5)(2)2 +(7)(2)2 = 48 kg m2
b) I = (5)(0.5)2 + (7)(4.5)2
= 143 kg m2
NOTE!
I depends on the rotation
axis and on the mass
distribution