Q75P A uniform solid cylinder with ma... [FREE SOLUTION]

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University Physics with Modern Physics

Hugh D. Young, Roger A. Freedman

Physics

14th edition Edition 路 1596 pages

ISBN: 9780321973610

Chapter 1: Q75P (page 1)

A uniform solid cylinder with mass M and radius 2R rests on a horizontal tabletop. A string is attached by a yoke to a frictionless axle through the center of the cylinder so that the cylinder can rotate about the axle. The string runs over a disk-shaped pulley with mass M and radius R that is mounted on a frictionless axle through its center. A block of mass M is suspended from the free end of the string (Fig. P10.75). The string doesn鈥檛 slip over the pulley surface, and the cylinder rolls without slipping on the tabletop. Find the magnitude of the acceleration of the block after the system is released from rest.

Short Answer

Expert verified

$\frac{g}{3}$

Step by step solution

Acceleration

The rate of velocity changes with respect to time is known as acceleration.

Given Data

$M a s s = m r a d i u s = 2 R$

Determine the magnitude of the acceleration of the block

$Apply \underset{}{鈭�} F = m a 蟿 = I 伪 a n d a = r 伪鈬� 伪 = \frac{a}{R} For hanging mass, / F = m a M g - T_{1} = M a . . . . . . . . . . (1) F o r t h e p u l l y, 蟿 = I 伪 T_{1} R - T_{2} R = \frac{M R^{2}}{2} . \frac{a}{R} T_{1} - T_{2} = \frac{M a}{2} . . . . . . . . . . . . . . . . . . . . . . . . . . (2)$

For the cylinder,

$F = m a T_{2} - f = M a . . . . . . . (3)$

$蟿 = I 伪 f . 2 R = \frac{M . {(2 R)}^{2}}{2} . \frac{a}{2 R} f = \frac{M a}{2} . . . . . . . . . . . . (4) F r o m (3) a n d (4), T_{2} = \frac{3 M a}{2} . . . . . . (5)$

$Substituting T_{1} and T_{2} i n (2) by (1) in (5) (M g - M a) - (\frac{3 M a}{2}) = \frac{M a}{2} g - a - \frac{3 a}{2} = \frac{a}{2} a = \frac{g}{3}$

Hence the magnitude of the acceleration of the block is $\frac{g}{3}$

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Short Answer

Step by step solution

Acceleration

Given Data

Determine the magnitude of the acceleration of the block

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