Question

A shot put iron ball of mass 3 kg is rolling on a frictionless horizontal surface with velocity 4 m/s. It collides on the free end of an ideal horizontal spring whose other end is fixed to a rigid support. If the spring constant is 32 N/m, the maximum compression produced in the spring will be ______.

Options

• \[\sqrt{2.5}m\]

• \[\sqrt{1.5}m\]

• \[\sqrt{7.3}m\]

• \[\sqrt{2.1}m\]

Answer 1

A shot put iron ball of mass 3 kg is rolling on a frictionless horizontal surface with velocity 4 m/s. It collides on the free end of an ideal horizontal spring whose other end is fixed to a rigid support. If the spring constant is 32 N/m, the maximum compression produced in the spring will be \[\sqrt{2.1}m\].

Explanation:

Gain in P.E. of spring = loss in K.E. of iron ball

\[\therefore\quad\frac{1}{2}\mathrm{kx}^{2}=\frac{1}{2}\mathrm{mv}^{2}+\frac{1}{2}\mathrm{I\omega}^{2}\]

    \[=\frac{1}{2}\mathrm{mv}^{2}+\frac{1}{2}\left(\frac{2}{5}\mathrm{mr}^{2}\right)\mathbf{\omega}^{2}\]

    \[=\frac{1}{2}\mathrm{mv}^2+\frac{1}{5}\mathrm{mv}^2\]               \[....(\because\mathrm{v}=\mathrm{r}\omega)\]

   \[=\frac{7}{10}\mathrm{mv}^2\]

\[\therefore\quad\mathrm{x}^{2}=\frac{14\mathrm{mv}^{2}}{10\mathrm{k}}\]

   \[=\frac{14\times3\times(4)^2}{10\times32}\]

   = 2.1

i.e., \[\mathbf{x}=\sqrt{2.1}\mathbf{m}\]