= 2 5 MR2. I sphere = MR 2. I hoop = 1 2 MR2. I disk

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1 A sphere (green), a disk (blue), and a hoop (red0, each with mass M and radius R, all start from rest at the top of an inclined plane and roll to the bottom. Which object reaches the bottom first? (Use the moments of inertia given below to guide your answer.) I sphere = 5 MR I hoop = MR I disk = 1 MR A) Sphere B) Hoop C) Disk D) Need more information. The one with the smallest moment of inertia will resist rolling the least. L34 W 11/1/14 a*er lecture 1

2 τ rf = Iα I point = mr up A light rod of length L has two heavy masses (each with mass m) attached at the end and middle. The axis of rotation is at one end. What is the moment of inertia about the axis? mg mg down A) ml B) ml C) 4mL D) 5mL E) 9mL I = 5mL What is the net torque due to gravity when it s released? A) mgl B) -mgl C) 3mgL D) -3mgL E) 4mgL τ = τ 1 + τ = -r 1 F r F = L(mg) L(mg) = 3Lmg τ = τ = 3mgL because all torques are CW. L34 W 11/1/14 a*er lecture

3 τ rf = Iα I point = mr up A light rod of length L has two heavy masses (each with mass m) attached at the end and middle. The axis of rotation is at one end. down What is the moment of inertia about the axis? I = 5mL What is the net torque due to gravity when it s released? τ = 3mgL If the bar s released from rest, what s the magnitude of its angular acceleration? A) 3g 5L B) 5g 3L C) 7L 3g D) 3L 5g α = τ I = 3mgL 5mL α = 3g 5L = 3g 5L L34 W 11/1/14 a*er lecture 3

4 Assignments Announcements: HW11 is due this week. CAPA 1 is now live. You should have read Ch. 10. Now move on to Ch. 11. Read about Center-of-Mass in Ch. 9. Not lecturing on this. This week s tutorial is based on a review packet designed to help you start studying for next week s exam. Print it prior to coming to your recitation. There are links to it on the course web site and it is also on DL. L34 W 11/1/14 a*er lecture 4

5 Assignments Announcements: HW11 is due this week. CAPA 1 is now live. You should have read Ch. 10. Now move on to Ch. 11. Read about Center-of-Mass in Ch. 9. Not lecturing on this. This week s tutorial is based on a review packet designed to help you start studying for next week s exam. Print it prior to coming to your recitation. There are links to it on the course web site and it is also on DL. Coming Up: Midterm exam 3 is NEXT week (Thurs Nov 0, same time, same place as the previous two midterms). Primary focus will be on Chs An old exam 3 has been placed on DL. L34 W 11/1/14 a*er lecture 5

6 Assignments Today: Continuing with rotational motion, found in Ch. 10. Continue with rotational kinematics and move on to rotational dynamics: torque, moment of inertia, rotational kinetic energy, and conservation of energy including rotational motion. L34 W 11/1/14 a*er lecture 6

7 Torque and Moment of Inertia Torque: τ rf = rf sinθ = r x F Moment of Inertia: = (lever arm) x (Component of force perpendicular to lever arm) I = m i r i = r dm i L34 W 11/1/14 a*er lecture 7

8 Torque and Moment of Inertia Torque: τ rf = rf sinθ = r x F Moment of Inertia: = (lever arm) x (Component of force perpendicular to lever arm) I point = mr L34 W 11/1/14 a*er lecture 8

9 Torque and Moment of Inertia Torque: τ rf = rf sinθ = r x F Moment of Inertia: = (lever arm) x (Component of force perpendicular to lever arm) I hoop = m i r i = i i m i R = MR L34 W 11/1/14 a*er lecture 9

10 Torque and Moment of Inertia I = m i r i = r dm i (disk) (i) Hollow Sphere: (/3)MR may also come in handy L34 W 11/1/14 a*er lecture 10

11 Torque and Moment of Inertia Two wheels have the same radius R and total mass M. They are rotating about their fixed axes. Which has the larger moment of inertia? τ rf = Iα i I = m i r i I hoop = MR A) Hoop B) Disk C) Same The hoop s mass is concentrated at its rim, while the disk s is distributed from its center to its rim. So, the hoop will have the larger moment of inertia. L34 W 11/1/14 a*er lecture 11

12 Torque and Moment of Inertia i I = m i r i Consider a uniform rod with an axis of rotation through is center and an identical rod with an axis of rotation through one end. Which has a larger moment of inertia? A) I C > I E B) I C < I E C) I C = I E D) Impossible to tell. If more mass is further from the axis, the moment of inertia increases. L34 W 11/1/14 a*er lecture 1

13 Torque and Moment of Inertia A force F is applied to a hoop of mass M and radius R. What s the resulting magnitude of the angular acceleration? τ rf = Iα i I = m i r i I hoop = MR A) RF/M B) F/MR C) MR D) F/MR τ = I hoop α = RF α = RF I hoop = RF MR = F MR L34 W 11/1/14 a*er lecture 13

14 Torque and Moment of Inertia i I = m i r i Two light (massless) rods, labeled A and B, each are connected to the ceiling by a frictionless pivot as shown. Both rods are released from a horizontal position. Which one experiences the larger torque? A) A B) B C) Same θ F = mgsinθ τ A = F L = Lmgsinθ τ B = L mgsinθ = Lmgsinθ θ F g = mg L34 W 11/1/14 a*er lecture 14

15 Torque and Moment of Inertia i I = m i r i Two light (massless) rods, labeled A and B, each are connected to the ceiling by a frictionless pivot as shown. Both rods are released from a horizontal position. Which one experiences the larger torque? A) A B) B C) Same Which one has the larger moment of inertia? A) A B) B C) Same I A = ml L I B = (m) = 1 ml L34 W 11/1/14 a*er lecture 15

16 Torque and Moment of Inertia i I = m i r i Two light (massless) rods, labeled A and B, each are connected to the ceiling by a frictionless pivot as shown. Both rods are released from a horizontal position. Which one experiences the larger torque? A) A B) B C) Same Which one has the larger moment of inertia? A) A B) B C) Same I A = ml I B = 1 ml Which one falls to vertical position fastest? A) A B) B C) Same Because the torques are the same, the one with the smaller moment of inertia, will rotate fastest. L34 W 11/1/14 a*er lecture α = τ I 16

17 Rotational Kinetic Energy and Conservation of Energy M ω v v ω (v/r) (same units: J) L34 W 11/1/14 a*er lecture 17

18 Rotational Kinetic Energy and Conservation of Energy KE tot = 1 Mv + 1 Iω I sphere = 5 MR I hoop = MR I disk = 1 MR A sphere, a disk, and a hoop, each with mass M and radius R, all start from rest at the top of an inclined plane and roll to the bottom. Which object reaches the bottom first? L34 W 11/1/14 a*er lecture 18

19 Rotational Kinetic Energy and Conservation of Energy KE tot = 1 Mv + 1 Iω I sphere = 5 MR I hoop = MR I disk = 1 MR KE i + PE i = KE f + PE f 0 + MgH = 1 Mv + 1 Iω MgH = 1 Mv + 1 I v R + 0 = 1 M + I L34 W 11/1/14 a*er lecture 19 R v

20 Rotational Kinetic Energy and Conservation of Energy KE tot = 1 Mv + 1 Iω I sphere = 5 MR I hoop = MR I disk = 1 MR KE i + PE i = KE f + PE f 0 + MgH = 1 Mv + 1 Iω MgH = 1 Mv + 1 I v R + 0 Which object will have the largest total kinetic energy at the bottom of the ramp? = 1 A) M + I Sphere B) Disk C) Hoop v D) All the same. L34 W 11/1/14 a*er lecture 0 R

21 Rotational Kinetic Energy and Conservation of Energy KE tot = 1 Mv + 1 Iω I sphere = 5 MR I hoop = MR I disk = 1 MR KE i + PE i = KE f + PE f 0 + MgH = 1 Mv + 1 Iω MgH = 1 Mv + 1 I v R + 0 = 1 M + I L34 W 11/1/14 a*er lecture 1 R v v = v = R Mgh I R + M MgH I + MR

22 Rotational Kinetic Energy and Conservation of Energy KE tot = 1 Mv + 1 Iω I sphere = 5 MR I hoop = MR I disk = 1 MR Sphere: v sphere = R MgH I + MR = R MgH 5 MR + MR = R MgH = 7 5 MR 10 7 gh > v disk > v hoop Hoop: v hoop = Disk: v disk = gh 4 3 gh L34 W 11/1/14 a*er lecture

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