A) \[\frac{1}{2}\lambda \ell {{v}^{2}}+\frac{\lambda {{\ell }^{2}}g}{2}\]
B) \[\lambda \ell gv\]
C) \[\frac{1}{2}\lambda {{v}^{3}}+\frac{\lambda \ell vg}{2}\]
D) \[\lambda \ell vg+\frac{1}{2}\lambda {{v}^{3}}\]
Correct Answer: C
Solution :
[c] Initial\[KE.=0\], Initial \[P.E.=0\] When the rope is just pulled off the table, final \[\operatorname{K}.E =\frac{1}{2}(\lambda \ell ){{v}^{2}}, final P.E. = (\lambda \ell )g\ell /2\] time taken \[=t=\ell /v\] \[\operatorname{Average} power =\frac{net change in energy}{time}\] \[=\frac{\frac{1}{2}\lambda \ell {{v}^{2}}+\lambda \ell g\ell /2}{\ell /v}=\frac{1}{2}\lambda {{v}^{3}}+\frac{\lambda \ell vg}{2}\]
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