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Classical Mechanics Lecture 16 Today’s Concepts: a) Rolling Kinetic Energy b) Angular Acceleration Physics 211 Lecture 16, Slide 1

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Schedule Mechanics Lecture 14, Slide 2 One unit per lecture! I will rely on you watching and understanding pre-lecture videos!!!! Lectures will only contain summary, homework problems, clicker questions, Example exam problems…. Midterm 3 Wed Dec 11

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Main Points Mechanics Lecture 16, Slide 3

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Main Points Mechanics Lecture 16, Slide 4 Rolling without slipping

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Total Kinetic Energy = Translational Kinetic Energy + Rotational Kinetic Energy Energy Conservation Rotational Kinetic Energy Mechanics Lecture 16, Slide 5 H Rolling without slipping

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Newton’s Second Law Mg f a Acceleration of Rolling Ball Mechanics Lecture 16, Slide 6 Newton’s 2 nd Law for rotations Rolling without slipping

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Rolling Motion Objects of different I rolling down an inclined plane: v 0 0K 0v 0 0K 0 K U M g h R M h v = Rv = R Mechanics Lecture 16, Slide 7

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Rolling If there is no slipping: In the lab reference frame In the CM reference frame v v v Where v R Mechanics Lecture 16, Slide 8

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Rolling Use v R and I cMR 2. Doesn’t depend on M or R, just on c (the shape) Hoop: c 1 Disk: c 1/2 Sphere: c 2/5 etc... c c v So: c c Mechanics Lecture 16, Slide 9

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Clicker Question A. B. C. Mechanics Lecture 16, Slide 10 A) B) C) A hula-hoop rolls along the floor without slipping. What is the ratio of its rotational kinetic energy to its translational kinetic energy? Recall that I MR 2 for a hoop about an axis through its CM :

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A block and a ball have the same mass and move with the same initial velocity across a floor and then encounter identical ramps. The block slides without friction and the ball rolls without slipping. Which one makes it furthest up the ramp? A) Block B) Ball C) Both reach the same height. CheckPoint v v Mechanics Lecture 16, Slide 11

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The block slides without friction and the ball rolls without slipping. Which one makes it furthest up the ramp? A) Block B) Ball C) Same v v B) The ball has more total kinetic energy since it also has rotational kinetic energy. Therefore, it makes it higher up the ramp. Mechanics Lecture 16, Slide 12 CheckPoint

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Rolling vs Sliding Mechanics Lecture 16, Slide 13 Rolling Ball Sliding Block Ball goes 40% higher! Rolling without slipping

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CheckPoint A cylinder and a hoop have the same mass and radius. They are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first? A) Cylinder B) Hoop C) Both reach the bottom at the same time Mechanics Lecture 16, Slide 14

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A) Cylinder B) Hoop C) Both reach the bottom at the same time A) same PE but the hoop has a larger rotational inertia so more energy will turn into rotational kinetic energy, thus cylinder reaches it first. Which one reaches the bottom first? Mechanics Lecture 16, Slide 15

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CheckPoint A small light cylinder and a large heavy cylinder are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first? A) Small cylinder B) Large cylinder C) Both reach the bottom at the same time Mechanics Lecture 16, Slide 16

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A small light cylinder and a large heavy cylinder are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first? A) Small cylinder B) Large cylinder C) Both reach the bottom at the same time C) The mass is canceled out in the velocity equation and they are the same shape so they move at the same speed. Therefore, they reach the bottom at the same time. Mechanics Lecture 16, Slide 17 CheckPoint

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Mechanics Lecture 16, Slide 18 v f R M

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Mechanics Lecture 16, Slide 19 v f R M a

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v a R M t Once v R it rolls without slipping t v0v0 R Rt t Mechanics Lecture 16, Slide 20 t v v0v0

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v a R M Plug in a and t found in parts 2) & 3) Mechanics Lecture 16, Slide 21

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v a R M We can try this… Interesting aside: how v is related to v 0 : Mechanics Lecture 16, Slide 22 Doesn’t depend on

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v f R M Mechanics Lecture 16, Slide 23

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Atwood's Machine with Massive Pulley: A pair of masses are hung over a massive disk-shaped pulley as shown. Find the acceleration of the blocks. For the hanging masses use F ma m 1 g T 1 m 1 a m 2 g T 2 m 2 a For the pulley use I T 1 R T 2 R (Since for a disk) Mechanics Lecture 16, Slide 24 y x m2gm2g m1gm1g a T1T1 a T2T2 R M m1m1 m2m2

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We have three equations and three unknowns ( T 1, T 2, a ). Solve for a. m 1 g T 1 m 1 a (1) m 2 g T 2 m 2 a (2) T 1 T 2 (3) Mechanics Lecture 16, Slide 25 y x m2gm2g m1gm1g a T1T1 a T2T2 R M m1m1 m2m2 Atwood's Machine with Massive Pulley:

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Three Masses Mechanics Lecture 16, Slide 26

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Three Masses Mechanics Lecture 16, Slide 27

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Three Masses Mechanics Lecture 16, Slide 28

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Three Masses Mechanics Lecture 16, Slide 29

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Three Masses 2 Mechanics Lecture 16, Slide 30

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Three Masses 2 Mechanics Lecture 16, Slide 31

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Suppose a cylinder (radius R, mass M ) is used as a pulley. Two masses ( m 1 m 2 ) are attached to either end of a string that hangs over the pulley, and when the system is released it moves as shown. The string does not slip on the pulley. Compare the magnitudes of the acceleration of the two masses: A) a 1 a 2 B) a 1 a 2 C) a 1 a 2 Clicker Question a2a2 T1T1 a1a1 T2T2 R M m1m1 m2m2 Mechanics Lecture 16, Slide 32

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Clicker Question Suppose a cylinder (radius R, mass M ) is used as a pulley. Two masses ( m 1 m 2 ) are attached to either end of a string that hangs over the pulley, and when the system is released it moves as shown. The string does not slip on the pulley. How is the angular acceleration of the wheel related to the linear acceleration of the masses? A) Ra B) a/R C) R/a Mechanics Lecture 16, Slide 33 a2a2 T1T1 a1a1 T2T2 R M m1m1 m2m2

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Clicker Question Suppose a cylinder (radius R, mass M ) is used as a pulley. Two masses ( m 1 m 2 ) are attached to either end of a string that hangs over the pulley, and when the system is released it moves as shown. The string does not slip on the pulley. Compare the tension in the string on either side of the pulley: A) T 1 T 2 B) T 1 T 2 C) T 1 T 2 Mechanics Lecture 16, Slide 34 a2a2 T1T1 a1a1 T2T2 R M m1m1 m2m2

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