Rotation Atwood Machine with Massive Pulley Energy of Rotation

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1 Rotation Atwood Machine with Massive Pulley Energy of Rotation Lana Sheridan De Anza College Nov 21, 2017

2 Last time calculating moments of inertia the parallel axis theorem

3 Overview applications of moments of inertia Atwood machine with massive pulley work, kinetic energy, and power of rotation

4 Applying Moments of Inertia Now that we can find moments of inertia of various objects, we can use them to calculate angular accelerations from torques, and vice versa. We can solve for the motion of systems with rotating parts. τ = Iα

5 is the angular speed of the rod and ball? (c) What is the linear speed of the center of mass of the ball? (d) How Example - Unwinding does it compare with from the speed Pulley had the ball -fallen Page freely 329, #55 A Figure P10.51 ir-pollution problem, runs over its citywide e, rapidly rotating fly- The flywheel is spun f rev/min by an l. Every time the bus n slightly. The bus is g so that the flywheel own. The flywheel is s kg and radius against air resistance e rate of 25.0 hp as it eed of 35.0 km/h. ct has a mass of m 1 5 ass of m kg; R 1 through the same distance of 28 cm? 55. Review. An object with a mass of m kg is M attached to the free end of a light string wrapped around a reel of radius R m and mass M kg. The reel is a solid disk, free to rotate in a vertical plane about the horizontal axis passing through its center as shown in Figure P The suspended object is released from rest 6.00 m above the floor. Determine (a) the tension in the string, (b) the acceleration of the object, and (c) the speed with which the object hits the floor. (d) Verify your answer to part (c) by using the isolated system (energy) model. M R m 1 m

6 slight nudge. (a) After the combination rotates through 90 degrees, what is its rotational kinetic energy? (b) What is the angular speed of the rod and ball? (c) What is the linear speed of the center of mass of the ball? (d) How does it compare with the speed had the ball fallen freely through the same distance of 28 cm? Example - Unwinding from Pulley lem, wide g flyspun y an bus us is heel el is dius ance as it. Page 329, # Review. An object with a mass of m kg is M attached to the free end of a light string wrapped around a reel of radius R m and mass M kg. The reel is a solid disk, free to rotate in a vertical plane about the horizontal axis passing through its center as shown in Figure P The suspended object is released from rest 6.00 m above the floor. Determine (a) the tension in the string, (b) the acceleration of the object, and (c) the speed with which the object hits the floor. (d) Verify your answer to part (c) by using the isolated system (energy) model. M

7 is el is s e it eration of the object, and (c) the speed with which the object hits the floor. (d) Verify your answer to part (c) by using the isolated system (energy) model. Example - Unwinding from Pulley Page 329, #55 M 5 g; R m

8 Example - Unwinding from Pulley (a) Tension in string?

9 Example - Unwinding from Pulley (a) Tension in string? T = mg ( ) m 1 = Mmg m + M/2 2m + M = 11.4 N (b) Acceleration?

10 Example - Unwinding from Pulley (a) Tension in string? T = mg ( ) m 1 = Mmg m + M/2 2m + M = 11.4 N (b) Acceleration? a = mg = 7.57 m s 2 m + M/2 (c) & (d) Speed at impact?

11 Example - Unwinding from Pulley (a) Tension in string? T = mg ( ) m 1 = Mmg m + M/2 2m + M = 11.4 N (b) Acceleration? a = mg = 7.57 m s 2 m + M/2 (c) & (d) Speed at impact? acceleration is constant kinematics. v = 9.53 m s 1

12 The Atwood Machine Revisited a Fixed Axis Remember that previously we studied the Atwood machine, assuming the pulley was massless. What if it s not? he 22 R tes se ce, ue igof in re- Figure (Example 10.12) An Atwood machine with a massive pulley. h m 1 m 2 h al potential energy as that which exists when the system is k 2 is associated 1 See the with slides a decrease from lecture in system 8. potential energy

13 The Atwood Machine Revisited Suppose the pulley has mass M and we model it as a cylinder, radius R. Then I = 1 2 MR2 for the pulley. Torque is needed to accelerate it. pulley of machine. g. Detern in the + m 1 m 2 T 1 T 2 T S m 1 T S α T 1 T 2 : as one objects of equal + S m 1 g m 2 itational a b S m 2 g

14 The Atwood Machine Revisited Find the acceleration of the masses?

15 The Atwood Machine Revisited Find the acceleration of the masses? Forces on object 1: F net,1 = m 1 a = T 1 m 1 g Forces on object 2: F net,2 = m 2 a = m 2 g T 2 Torque on pulley: τ net = Iα = T 2 R T 1 R

16 The Atwood Machine Revisited From the torque equation: Iα = T 2 R T 1 R I R α = T 2 T 1

17 The Atwood Machine Revisited From the torque equation: Iα = T 2 R T 1 R I R α = T 2 T 1 I a R R = m 2(g a) m 1 (g + a) ( ) I R 2 + m 1 + m 2 a = (m 2 m 1 )g

18 The Atwood Machine Revisited From the torque equation: Iα = T 2 R T 1 R I R α = T 2 T 1 I a R R = m 2(g a) m 1 (g + a) ( ) I R 2 + m 1 + m 2 a = (m 2 m 1 )g a = (m 2 m 1 )g ( I ) + m R m 2

19 The Atwood Machine Revisited From the torque equation: Iα = T 2 R T 1 R I R α = T 2 T 1 I a R R = m 2(g a) m 1 (g + a) ( ) I R 2 + m 1 + m 2 a = (m 2 m 1 )g Putting in I = 1 2 MR2 : a = (m 2 m 1 )g ( I ) + m R m 2 a = (m 2 m 1 )g ( M 2 + m 1 + m 2 )

20 Rotational Kinetic Energy When a massive object rotates there is kinetic energy associated with the motion of each particle.

21 Rotational Kinetic Energy When a massive object rotates there is kinetic energy associated with the motion of each particle. Imagine an object made up of a collection of particles, mass m i, radius r i. The kinetic energy or each particle is K i = 1 2 m iv 2 i = 1 2 m ir 2 i ω 2

22 Rotational Kinetic Energy When a massive object rotates there is kinetic energy associated with the motion of each particle. Imagine an object made up of a collection of particles, mass m i, radius r i. The kinetic energy or each particle is K i = 1 2 m iv 2 i = 1 2 m ir 2 i ω 2 And the total kinetic energy of all the particles together would be the sum: K = K i i ( ) = 1 m i ri 2 2 i ω 2

23 Rotational Kinetic Energy When a massive object rotates there is kinetic energy associated with the motion of each particle. Imagine an object made up of a collection of particles, mass m i, radius r i. The kinetic energy or each particle is K i = 1 2 m iv 2 i = 1 2 m ir 2 i ω 2 And the total kinetic energy of all the particles together would be the sum: Notice that I = i m ir 2 i. K = K i i ( ) = 1 m i ri 2 2 i ω 2

24 Rotational Kinetic Energy Kinetic energy of a rigid object rotating at an angular speed ω is K = 1 2 Iω2

25 is the angular speed of the rod and ball? (c) What is the linear speed of the center of mass of the ball? (d) How Example - Unwinding does it compare with from the speed Pulley had the ball -fallen Page freely 329, #55 A Figure P10.51 ir-pollution problem, runs over its citywide e, rapidly rotating fly- The flywheel is spun f rev/min by an l. Every time the bus n slightly. The bus is g so that the flywheel own. The flywheel is s kg and radius against air resistance e rate of 25.0 hp as it eed of 35.0 km/h. ct has a mass of m 1 5 ass of m kg; R 1 through the same distance of 28 cm? 55. Review. An object with a mass of m kg is M attached to the free end of a light string wrapped around a reel of radius R m and mass M kg. The reel is a solid disk, free to rotate in a vertical plane about the horizontal axis passing through its center as shown in Figure P The suspended object is released from rest 6.00 m above the floor. Determine (a) the tension in the string, (b) the acceleration of the object, and (c) the speed with which the object hits the floor. (d) Verify your answer to part (c) by using the isolated system (energy) model. M R m 1 m

26 Summary Atwood machine revisited kinetic energy of rotation Next test Monday, Nov 27. (Uncollected) Homework Serway & Jewett, Ch 10, onward from page 288. Probs: 45, 47, 49, 51, 53, 69

Rotation Moment of Inertia and Applications

Rotation Moment of Inertia and Applications Lana Sheridan De Anza College Nov 20, 2016 Last time net torque Newton s second law for rotation moments of inertia calculating moments of inertia Overview calculating