2. Physics 1710—Warm-up Quiz Two 2.0 kg disks, both of radius 0.10 m are sliding (without friction) and rolling, respectively, down an incline. Which will reach the bottom first? 1.Rolling disk wins. 2.Sliding disk wins. 3.Tie.

3. Solution: Physics 1710—Chapter 11 Rotating Bodies Kinetic Energy of sliding disk: K 1 = ½ mv 2 = mg(h-z); v =√[2g(h-z)] Kinetic Energy of rolling disk: K 2 = ½ mv 2 + ½ I ω 2 = mg(h-z) = ½ mv 2 + ½ ( ½ mr 2 ) (v/r) 2 = 3/4 mv 2 ; v =√[4/3 g(h-z)] Slider wins!

4. Consider two spindles rolling down a ramp: a b c c Which one will win and why? Physics 1710—Chapter 11 Rotating Bodies

5. No Talking! Think! Confer! Peer Instruction Time Which one will win and why? Physics 1710—Chapter 11 Rotating Bodies

6. 1′ Lecture The energy of rotationThe energy of rotation K = ½ I ⍵ 2 Torque (“twist”) is the vector product of the “moment” and a force. τ = r x F Torque (“twist”) is the vector product of the “moment” and a force. τ = r x F τ = I ⍺ = I d⍵/dt τ = I ⍺ = I d⍵/dt Angular momentum L is the vector product of the moment arm and the linear momentum. L= r x p.Angular momentum L is the vector product of the moment arm and the linear momentum. L= r x p. τ = d L/dt τ = d L/dt Physics 1710—Chapter 11 Rotating Bodies

7. Moment of Inertia—sphere I = ∭ R 2 ρ d V N.B. : R 2 = r 2 – z 2 R 2 = r 2 – (r cos θ) 2 =r 2 (1– cos 2 θ) I = ∫ 0 2π dφ∫ 0 π (1– cos 2 θ) sin θdθ∫ 0 a r 4 ρdr = [2π][2- 1/3(2)][1/5 a 5 ] ρ = [4π][2/3][1/5 a 5 ] ρ =[4π/3][2/5 a 5 ] ρ = 2/5 M a 2 R r Physics 1710—Chapter 11 Rotating Bodies

8. Kinetic Energy of Rotation: K = ½ Σ i m i v i 2 K = ½ Σ i m i (R i ω i ) 2 K = ½ Σ i m i R i 2 ω i 2 For rigid body ω i = ω K = ½ [Σ i m i R i 2 ] ω 2 K = ½ I ω 2 With I =Σ i m i R i 2 = the moment of inertia. Physics 1710—Chapter 11 Rotating Bodies

9. Why do round bodies roll down slopes? The torque is the “twist.” θ Fsinθ θ FgFg F = m a = mr dω/dt rF = rF g sinθ = mr 2 dω/dt Torque = r x F = I α τ = r x F, | τ | = rFsinθ τ = r x F, | τ | = rFsinθ Physics 1710—Chapter 11 Rotating Bodies

10. Torque and the Right Hand Rule: Physics 1710—Chapter 11 Rotating Bodies r F r x F X

11. Physics 1710—Chapter 11 Rotating Bodies Vector Product: C = A x B C x = A y B z – A z B y Cyclically permute: (xyz), (yzx), (zxy) |C| =√[C x 2 + C y 2 + C z 2 ] = AB sin θ Directed by RH Rule.

12. Physics 1710—Chapter 11 Rotating Bodies Vector Product: A x B = - B x A A x ( B + C ) = A x B + A x C d/dt ( A x B ) = d A /dt x B + A x d B/dt i x i = j x j = k x k = 0 i x j = - j x i = k j x k = - k x j = i k x i = - i x k = j

13. Torque Bar: Physics 1710—Chapter 11 Rotating Bodies r F τ = r x F τ

14. A B C Teeter-totter: Physics 1710—Chapter 11 Rotating Bodies τ = r x F F2F2F2F2 F1F1F1F1 Where should the fulcrum be place to balance the teeter-totter?

15. A. A. B. B. C. C. Physics 1710—Chapter 11 Rotating Bodies

16. Torque Ladder Which way will the torque ladder move? ? Physics 1710—Chapter 11 Rotating Bodies

17. Which way will the torque ladder move? A.Clockwise B.Counterclockwise C.Will stay balanced Physics 1710—Chapter 11 Rotating Bodies

18. Torque Ladder Which way will the torque ladder move? ? r r sin θ Physics 1710—Chapter 11 Rotating Bodies

19. Second Law of Motion L = r x p L = r x p is the “angular momentum.” F = m a Or F = dp/dt Then: r x F = d (r x p)/dt Torque = τ = d L/dt Physics 1710—Chapter 11 Rotating Bodies

20. Angular Momentum: L = r x p The angular momentum is the vector product of the moment arm and the linear momentum. ∑ T = d L/dt The net torque is equal to the time rate of change in the angular momentum. Physics 1710—Chapter 11 Rotating Bodies

21. Angular Momentum: Proof: ∑ T = r x ∑F = r x d p/dt And d L/dt = d( r x p) /dt = d r/dt x p + r x d p/dt. But p = m d r/dt, therefore d r/dt x p = 0 But p = m d r/dt, therefore d r/dt x p = 0 d L/dt = r x d p/dt And thus ∑ T = d L/dt. Physics 1710—Chapter 11 Rotating Bodies

22. Second Law of Motion L = constant means L = constant means angular momentum si conserved. Torque = τ = d L/dt If τ = 0, then L is a constant. Physics 1710—Chapter 11 Rotating Bodies

23. Rotating Platform D emonstration Physics 1710—Chapter 11 Rotating Bodies

24. Analysis: Why does an ice skater increase her angular velocity without the benefit of a torque?Why does an ice skater increase her angular velocity without the benefit of a torque? L = r x p = r x ( m v) = r x ( m r x ⍵) L i = m i r i 2 ⍵ L z = (∑ i m i r i 2 ) ⍵ L z = I ⍵; & ⍵ = L z / I Therefore, a decrease in I ( by reducing r) will result in an increase in ⍵. Physics 1710—Chapter 11 Rotating Bodies

25. Summary: The total Kinetic energy of a rotating system is the sum of the rotational energy about the Center of Mass and the translational KE of the CM.The total Kinetic energy of a rotating system is the sum of the rotational energy about the Center of Mass and the translational KE of the CM. K = ½ I CM ⍵ 2 + ½ MR 2 ⍵ 2 τ = r x F τ = r x F Physics 1710—Chapter 11 Rotating Bodies

26. Summary: Angular momentum L is the vector product of the moment arm and the linear momentum.Angular momentum L is the vector product of the moment arm and the linear momentum. L = r x p The net externally applied torque is equal to the time rate of change in the angular momentum. The net externally applied torque is equal to the time rate of change in the angular momentum. ∑ τ z = d L z /dt = I z ⍺