1. Hour 6 Conservation of Angular Momentum
Physics 321
Hour 6
Conservation of Angular Momentum

2. Linear and Angular Relationships
ℓ = 𝑟 × 𝑝
𝛤 = 𝑟 × 𝐹
𝑝=𝑚𝑣
𝐹=𝑚𝑎
𝐹= 𝑝
𝑣= 𝑥
𝑎= 𝑥
ℓ=𝐼𝜔
𝛤=𝐼𝛼
𝛤= ℓ
𝜔= 𝜃
𝛼= 𝜃
𝑥=𝑅𝜃
𝑣=𝑅𝜔
𝑎=𝑅𝛼

3. Angular Momentum in Linear MotionP
We must calculate angular momentum about a point P. b 𝑟 𝑝
ℓ = 𝑟 × 𝑝
ℓ=𝑏𝑝

4. Central Force
The torque is zero, so angular momentum is constant. 𝑟 𝐹

5. 𝑑𝐴 𝑑𝑡 = 1 2 | 𝑟 × 𝑣 𝑑𝑡| 𝑑𝑡 = ℓ 2𝑚 Kepler’s 2nd Law
The area of a trapezoid with sides 𝐴 and 𝐵 is Area = | 𝐴 × 𝐵 |
𝑣 𝑑𝑡 𝑟
𝑑𝐴 𝑑𝑡 = 1 2 | 𝑟 × 𝑣 𝑑𝑡| 𝑑𝑡 = ℓ 2𝑚

6. “Bomb” Problem
An artillery shell explodes just as the shell reaches its maximum height. It breaks into three pieces. Ignore drag forces and the mass of the explosive itself. Describe what happens.

7. Center of Mass
𝐹 𝑒𝑥𝑡= 𝑖 𝑚 𝑖 𝑟 𝑖 ≡𝑀 𝑅

8. Center of Mass
𝑅 ≡ 1 𝑀 𝑟 𝜌( 𝑟 )𝑑 3 𝑟
𝑅 ≡ 1 𝑀 𝑖 𝑚 𝑖 𝑟 𝑖
𝑀≡ 𝜌( 𝑟 )𝑑 3 𝑟

9. Moment of Inertia
𝑑 𝑖
𝐼≡ 𝑖 𝑚 𝑖 𝑑𝑖 2
𝑟 𝑖

10. Moment of Inertia
𝐼≡ 𝑖 𝑚 𝑖 𝑑𝑖 2
𝐼≡ 𝑑2 𝜌( 𝑟 )𝑑 3 𝑟

11. The Rocket Problem Newton’s 2nd Law: 𝐹 = 𝑝 =𝑚 𝑣 + 𝑚 𝑣
𝐹 = 𝑝 =𝑚 𝑣 + 𝑚 𝑣
This is true – but momentum conservation is more transparent:
Start with mass 𝑚 and 𝑣=0. Let 𝑣𝑒𝑥 be the exhaust velocity.
𝑚−|∆𝑚| ∆𝑣= ∆𝑚 𝑣𝑒𝑥−∆𝑣
𝑚∆𝑣= 𝑣 𝑒𝑥 |∆𝑚|
𝑚𝑎= 𝑣 𝑒𝑥 | 𝑚 |
Adding any external forces: 𝐹=𝑚 𝑣 = 𝐹 𝑒𝑥𝑡 + 𝑣 𝑒𝑥 | 𝑚 |
Thrust = 𝑣𝑒𝑥| 𝑚 |