Question

A spring of unstretched length $\ell$ has a mass m with one end fixed to a rigid support. Assuming spring to be made of a uniform wire, the kinetic energy possessed by it if it's free end is pulled with uniform velocity $\upsilon$ is :

• A. $\frac{1}{2}m \upsilon^{2}$
• B. $m \upsilon^{2}$
• C. $\frac{1}{3}m \upsilon^{2}$
• D. $\frac{1}{6}m \upsilon^{2}$

Answer 1

Answer: (D)

$\frac{1}{6}m \upsilon^{2}$

Answer 2

$KE =\int \frac{1}{2} dm \left(\frac{ v }{\ell} x \right)^{2}$
$=\frac{1}{2} \int \frac{ m }{\ell} d x \frac{ v ^{2}}{\ell^{2}} x ^{2}$
$=\frac{ mv ^{2}}{2 \ell^{2}}\left(\frac{ x ^{3}}{3}\right)_{0}^{\ell}$
$=\frac{ mv ^{2}}{2 \ell^{3}} \times \frac{\ell^{3}}{3}$
$=\frac{ mv ^{2}}{6}$