Question

A spring mass system is shown in figure, spring is initially unstretched. A man starts pulling the block with constant force $F$. Find
(a) The amplitude and time period of motion of the block
(b) The kinetic energy of the block at mean position
(c) The energy stored in the spring when the block passes through the mean position.

Answer 1

From work energy theorem $(A-B)$

Total work done $=\Delta KE$

$\Rightarrow W{D}_S+W{D}_F=\Delta KE$

$\Rightarrow -\frac{1}{2}k{x}_0^2+f{x}_0=\frac{1}{2}m{V}_0^2$

$\Rightarrow -\frac{1}{2}k\frac{F^2}{k^2}+F\frac{F}{k}=\frac{1}{2}m{V}_0^2$

$\Rightarrow -\frac{F^2}{2K}+\frac{F^2}{k}=\frac{1}{2}m{V}_0^2$

$\Rightarrow \frac{F^2}{2K}=\frac{1}{2}m{V}_0^2\Rightarrow V_0^2=\frac{F^2}{mk}$

$V_0=dfracF\sqrt{mk}$

kinetic energy of block at mean position

$=\frac{1}{2}m{v}_0^2=\frac{F^2}{2K}$

Energy stored in the spring when the block passes through the mean position

$=\frac{1}{2}k{x}_0^2=\frac{1}{2}k\frac{F^2}{K^2}=\frac{F^2}{2K}$