A) \[\frac{1}{2}m{{v}^{2}}\]
B) \[\frac{1}{4}m{{v}^{2}}\]
C) \[\frac{1}{6}m{{v}^{2}}\]
D) \[\frac{1}{8}m{{v}^{2}}\]
Correct Answer: C
Solution :
Mass dm of spring at a distance x moves with velocity is \[{{v}_{x}}=\frac{x}{l}\cdot v\] |
Kinetic energy of mass dm is \[dK=\frac{1}{2}(dm)v_{x}^{2}\]\[\Rightarrow \]\[dK=\frac{1}{2}\cdot \frac{{{v}^{2}}{{x}^{2}}}{{{l}^{2}}}\cdot dm\] |
Kinetic energy of complete spring is \[K=\int{dK=\frac{1}{2}\cdot \frac{{{v}^{2}}}{{{l}^{2}}}\cdot \int\limits_{0}^{l}{{{x}^{2}}dm}}\] |
Here, \[dm=\frac{m}{l}dx\] |
\[\therefore \]\[K=\frac{1}{2}\cdot \frac{{{v}^{2}}}{{{l}^{2}}}\times \frac{m}{l}\int\limits_{0}^{l}{{{x}^{2}}dx=\frac{1}{6}m{{v}^{2}}}\] |
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