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Exactly one turn of a flexible rope with mass m is wrapped

University Physics | 13th Edition | ISBN: 9780321675460 | Authors: Hugh D. Young, Roger A. Freedman ISBN: 9780321675460 31

Solution for problem 82P Chapter 9

University Physics | 13th Edition

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University Physics | 13th Edition | ISBN: 9780321675460 | Authors: Hugh D. Young, Roger A. Freedman

University Physics | 13th Edition

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Problem 82P

Exactly one turn of a flexible rope with mass ?m? is wrapped around a uniform cylinder with mass ?M? and radius ?R?. The cylinder rotates without friction about a horizontal axle along the cylinder axis. One end of the rope is attached to the cylinder. The cylinder starts with angular speed ?0. After one revolution of the cylinder the rope has unwrapped and, at this instant, hangs vertically down, tangent to the cylinder. Find the angular speed of the cylinder and the linear speed of the lower end of the rope at this time. Ignore the thickness of the rope. [?Hint: Use Eq. (9.18).]

Step-by-Step Solution:

Solution 82P Step 1 of 1: The zero of gravitational potential energy to be at the axle, the initial potential energy is zero (the rope is wrapped in a circle with center on the axle).When the rope has unwound, its center of mass is a distance R below the axle, since the length of the rope is 2 R and half this distance is the position of the center of the mass. Initially, every part of the rope is moving with speed R, 0nd when the rope has unwound, and the cylinder has angular speed , the speed of the rope is R (the upper end of the rope has the same tangential speed at the edge of the cylinder). I = mR for a uniform cylinder. 2 K = K + U .( M + )R 2 2 1 2 2 4 2 0 2 = (M4 + 2R mgR 4mg = + ( R ) 0 (M+2m) And the speed of any part of the rope is v = R

Step 2 of 1

Chapter 9, Problem 82P is Solved
Textbook: University Physics
Edition: 13
Author: Hugh D. Young, Roger A. Freedman
ISBN: 9780321675460

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Exactly one turn of a flexible rope with mass m is wrapped