Question

Figure shows an inextensible string is wound over a cylinder of mass M1 , radius R fixed to ceiling. Other end of the string also wound over another cylinder of mass M2 and radius R. When system is released string start unwinding from both the cylinders, find acceleration of M2.


SOLUTION fast​

Answer 1

Answer:

Given :

▪ Radii of both cylinders = R

▪ Mass of A = M_1

▪ Mass of B = M_2

To Find :

▪ Linear acceleration of cylinder B.

Calculation :

▪ Let angular acceleration of cylinder A be α_1 and of cylinder B be α_2.

✴ Translative motion equation :

$\dashrightarrow\bf\:M_2g-T=M_2a$

✴ Rolling motion equation :

$\dashrightarrow\bf\:RT=I_2\alpha_2$

✴ Equation for pure rolling motion :

$\dashrightarrow\bf\:\alpha_2=\dfrac{a}{R}$

↗ By comparing all equations...

$\dashrightarrow\sf\:M_2g-\dfrac{I_2\alpha_2}{R}=M_2a\\ \\ \dashrightarrow\sf\:M_2g-{\huge{(}}\dfrac{M_2R^2}{2}\times \dfrac{a}{R}\times \dfrac{1}{R}{\huge{)}}=M_2a\\ \\ \dashrightarrow\sf\:M_2g-\dfrac{M_2a}{2}= M_2a\\ \\ \dashrightarrow\sf\:M_2g=\dfrac{3M_2a}{2}\\ \\ \dashrightarrow\underline{\boxed{\bf{\purple{a=\dfrac{2g}{3}}}}}\:\orange{\bigstar}$