1. Torque and Angular Momentum
“Twisty-ness” and other rotational analogues

2. Review of linear and rotational analogues
x, y, z – position
v – velocity
a – acceleration
Kinematic equations:
𝑣= ∆𝑥 ∆𝑡
𝑎= ∆𝑣 ∆𝑡
∆𝑥= 1 2 𝑎 𝑡 2 + 𝑣 0 𝑡
θ – angle
ω – angular velocity
α – angular acceleration
Kinematic equations:
𝜔= ∆𝜃 ∆𝑡
𝛼= ∆𝜔 ∆𝑡
∆𝜃= 1 2 𝛼 𝑡 2 + 𝜔 0 𝑡
Linear ↔ Rotational
𝑠=𝑟𝜃
𝑣=𝑟𝜔
𝑎=𝑟𝛼

3. Preview of linear and rotational analogues
F – force
m – mass
p - momentum
τ – torque
I – moment of inertia
L – angular momentum

4. Torque
A rotating force; influence that causes changes in the rotational motion of an object.
𝝉=𝒓×𝑭=𝑭 𝒓 𝐬𝐢𝐧 𝜽

5. Example
A bolt is loosened when a 30.0 N force is applied perpendicularly to a 5.0 cm wrench. What is the torque on the bolt? [Answer in units of Nm.]

6. Example
What force is necessary to loosen the same bolt if the force is applied at a 45°(π/4) degree angle from the wrench?

7. Assignment
Complete the problems on the ½ sheet of paper. Be ready to white board any one of them tomorrow. (That means do all 7.)

8. Moment of Inertia
Rotational analog of mass - varies by object and rotational axis.
Generally:
𝐼= 𝑛 𝑚 𝑛 𝑟 𝑛 2 = 𝑚 1 𝑟 𝑚 2 𝑟 …
0 𝑀 𝑟 2 𝑑𝑚
Point mass: 𝑰=𝒎 𝒓 𝟐

9. I = moment of inertia
R = radius
L = length
M = mass
Moment of Inertia

10. Angular momentum Rotational analog of linear momentum (p = mv)
𝐿=𝑟×𝑝=𝑚 𝑣 𝑟 sin 𝜃
Kepler’s 2nd Law
The product of the moment of inertia and the angular velocity
𝑳=𝑰×𝝎
Conserved if there is no external torque on the object
A vector quantity

11. Angular momentum To determine direction, use the right hand rule:
Curl fingers in direction
of rotation
Thumb points in
direction of L.

12. Assignment
Create a double bubble map to compare linear and rotational motions. You need 2-3 colors.