question_answer

A block of mass \[{{m}_{1}}\] lies on a smooth horizontal table and is connected to another freely hanging block of mass \[{{m}_{2}}\] by a light inextensible string passing over a smooth fixed pulley situated at the edge of the table as shown in the figure. Initially the system is at rest with \[{{m}_{1}}\] at a distance d from the pulley. The time taken for \[{{m}_{1}}\] to reach the pulley is

A) \[\frac{{{m}_{2}}g}{{{m}_{1}}+{{m}_{2}}}\]

B) \[\sqrt{\frac{2d({{m}_{1}}+{{m}_{2}})}{{{m}_{2}}g}}\]

C) \[\sqrt{\frac{2{{m}_{2}}d}{({{m}_{1}}+{{m}_{2}})g}}\]        

D) \[\frac{4d({{m}_{1}}+{{m}_{2}})}{3g{{m}_{1}}}\]

Correct Answer: B

Solution :

[b]: Let a be common acceleration of the system.    The free body diagrams of two blocks are as shown in the figure.   Their equations of motion are        \[T={{m}_{1}}a\]                                 ...(i) \[{{m}_{2}}g-T={{m}_{2}}a\]                          ...(ii) From (i) and (ii), we get\[a=\frac{{{m}_{2}}g}{{{m}_{1}}+{{m}_{2}}}\] Using, \[s=ut+\frac{1}{2}a{{t}^{2}}\] \[\Rightarrow \]\[d=0\times t+\frac{1}{2}\frac{{{m}_{2}}g}{{{m}_{1}}+{{m}_{2}}}{{t}^{2}}\]           (Using (iii)) or\[t=\sqrt{\frac{2d({{m}_{1}}+{{m}_{2}})}{{{m}_{2}}}}\]