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Byju's Answer

Standard XII

Physics

Speed of Efflux

In figure the...

Question

In figure the cylinder of mass $10 k g$ and radius $10 c m$ has a tape wrapped round it. The pulley weighs $100 N$ and has a radius $5 c m$ . When the system is released, the $5 k g$ mass comes down and the cylinder rolls without slipping. Calculate the acceleration and velocity of the mass as a function of time.

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Solution

$M = 10 k g$ , $R_{1} = 10 c m$ and $M_{2} g = 100$
For 5 kg block
$5 g - T_{2} = 5 a_{2}$ ....(1)
for pulley $R_{2} = 5 c m$
$(T_{2} - T_{1}) R_{2} = I α_{2}$
$(T_{2} - T_{1}) R_{2} = \frac{M_{2} R_{2}^{2}}{2} α_{2}$ ....(2)
, where $M_{2}$ =mass of pulley
For cylinder
$T_{1} + f = M_{1} a_{1}$ ...(3)
torque about the cylinder
$(T_{1} - f) R_{1} = \frac{M_{1} R_{1}^{2}}{2} α_{1}$ ...(4)
$a_{1} = R_{1} α_{1}$ ....(5)
Acceleration of rope at the top cylinder and over the pulley
$a_{1} + R_{1} α_{1} = R_{2} α_{2}$ ...(6)
Also the acceleration of rope at the top of puller is equal to acceleration of cylinder
$R_{2} α_{2} = a_{2}$ ...(7)
on solving all the equations
$a_{2} = \frac{4 g}{11}$
and velocity $v_{2} = a_{2} t = \frac{4 g t}{11}$

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In figure the cylinder of mass 10kg and radius 10cm has a tape wrapped round it. The pulley weighs 100N and has a radius 5cm. When the system is released, the 5kg mass comes down and the cylinder rolls without slipping. Calculate the acceleration and velocity of the mass as a function of time.