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 its 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)

$KE=\int \frac{1}{2}dm{\left(\frac{v}{\ell }x\right)}^2$
$=\frac{1}{2}\int \frac{m}{\ell }dx\frac{v^2}{{\ell }^2}{x}^2$
$=\frac{m{v}^2}{2{\ell }^2}{\left(\frac{x^3}{3}\right)}_0^{\ell }$
$=\frac{m{v}^2}{2{\ell }^3}\times \frac{{\ell }^3}{3}$
$=\frac{m{v}^2}{6}$