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) \[2\sqrt{5}\,\,m/{{s}^{2}}\]
Correct Answer: D
Solution :
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 \[m=\sqrt{{{2}^{2}}+{{4}^{2}}}=2\sqrt{5}\] |
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