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Old Jun 20th 2017, 01:28 PM   #1
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Dynamics of rotational motion problem

Hello, I have this problem:

A uniform solid cylinder with mass M and radius R has rolled up a string. The string runs through a pulley with negligible mass.

a)Calculate the acceleration of mass m
b)Discuss the conditions under which the cylinder slides while rotating



could any one help me to solve this problem?
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Old Jun 21st 2017, 09:14 AM   #2
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If those are solutions at the bottom of your diagram, then my calculations do not agree.

Part (a) ... assuming no slippage by the cylinder on the horizontal surface

$F_{net} = ma$

(1) $mg - T = ma$

(2) $T + f_s = Ma$

$\tau_{net} = I \alpha$

$TR - f_sR = \dfrac{1}{2}MR^2 \cdot \dfrac{a}{R}$

(3) $T - f_s = \dfrac{Ma}{2}$

sum equations (2) and (3) ...

$2T = \dfrac{3Ma}{2} \implies T = \dfrac{3Ma}{4} \implies f_s = \dfrac{Ma}{4}$

substitute for $T$ in equation (1) ...

$mg - \dfrac{3Ma}{4} = ma$

$4mg - 3Ma = 4ma \implies a = \dfrac{4mg}{3M+4m}$

Part (b) ... for no slippage, $f_s \le \mu Mg$.

Slippage will occur when $\mu < \dfrac{f_s}{Mg}$

$\mu < \dfrac{1}{Mg} \cdot \dfrac{Ma}{4} = \dfrac{a}{4g} = \dfrac{1}{4g} \cdot \dfrac{4mg}{3M+4m} = \dfrac{m}{3M+4m}$
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