• # question_answer In figure shown, find the magnitude of acceleration of m, given that string is inextensible and mass less and the acceleration of M is $2\text{ }m/{{s}^{2}}$ towards left - A) $2\sqrt{3}\,\,m/{{s}^{2}}$ B) $3\sqrt{2}\,\,m/{{s}^{2}}$ C) $4\sqrt{2}\,\,m/{{s}^{2}}$ D) $2\sqrt{5}\,\,m/{{s}^{2}}$

 Let X be the leftward displacement of m and x and y be the leftward and downward displacement of M. Let $AB={{\ell }_{1}};$ $BC={{\ell }_{2}}:$ $CD={{\ell }_{3}}$ and $Am={{\ell }_{4}}$when M moves towards left, say by x, then $AB=({{\ell }_{1}}-x)$ $BC={{\ell }_{2}}$ $CD={{\ell }_{3}}-x$ $Am={{\ell }_{4}}+y$ Total length of string remain constant $\therefore {{\ell }_{1}}-x+{{\ell }_{2}}+{{\ell }_{3}}-x+{{\ell }_{4}}+y={{\ell }_{1}}+{{\ell }_{2}}+{{\ell }_{3}}+{{\ell }_{4}}$ $\therefore 2x=y$
 Acceleration of $M={{a}_{M}}=2m/{{s}^{2}}$ ${{a}_{x}}=2m/{{s}^{2}}$ $2x=y$ Double differentiating this equation $2{{a}_{x}}={{a}_{y}}$ ${{a}_{y}}=2{{a}_{x}}=4m/{{s}^{2}}$ ${{a}_{y}}$ is downward acceleration of $m=4m/{{s}^{2}}$ m is also moving in left direction along with M. m has acceleration in horizontal direction also horizontal acceleration of m is same as that of M i.e. $2\,m/{{s}^{2}}$ $\therefore$ Net acceleration of $m=\sqrt{{{2}^{2}}+{{4}^{2}}}=2\sqrt{5}$