question_answer

A rod of length \[l\] falls on two metal pads of same height from a height h. The coefficients of restitution of the metal pads are \[{{e}_{1}}\]and \[{{e}_{2}}({{e}_{1}}>{{e}_{2}})\]. The angular velocity of the rod after it recoils is

A)  \[\frac{{{e}_{1}}}{{{e}_{2}}}l\sqrt{2gh}\]                      

B)  \[\frac{{{e}_{1}}+1}{{{e}_{2}}+1}\sqrt{2gh}\]

C)  \[\frac{{{e}_{1}}+{{e}_{2}}}{l}\sqrt{2gh}\]       

D)  \[\frac{{{e}_{1}}+1}{{{e}_{2}}-1}\sqrt{2gh}\]

Correct Answer: C

Solution :

 After collision, the end A moves with a linear velocity of \[{{e}_{1}}\sqrt{2gh}\]. Whereas end B moves with a velocity of \[{{e}_{2}}\,\sqrt{2gh}\]. \[\therefore \] The relative velocity between the ends is \[V={{e}_{1}}\,\sqrt{2gh}-{{e}_{2}}\sqrt{2gh}\] The angular velocity, \[=\frac{V}{l}=\frac{{{e}_{1}}-{{e}_{2}}}{l}\,\sqrt{2gh}\]