Question

The spring shown in figure (12−E5) is unstretched when a man starts pulling on the cord. The mass of the block is M. If the man exerts a constant force F, find (a) the amplitude and the time period of the motion of the block, (b) the energy stored in the spring when the block passes through the equilibrium position and (c) the kinetic energy of the block at this position.
Figure

Answer 1

Explanation:

(a) We know that amplitude is the maximum displacement of the block which can be defined as $\frac{F}{k}$ .                                                                                                                                                                                                                      The time period of the motion of the block is $\mathrm{T}=2 \pi \sqrt{\frac{\text { displacement }}{\text { acceleration }}}=2 \pi \sqrt{\mathrm{m} / \mathrm{k}}.$.

(b) The energy stored in the spring when the block passes through the equilibrium position is  

$E=\frac{1}{2} k x^{2}$.  Thus, on solving, we get $E=\frac{1}{2}\left(\frac{F^{2}}{k}\right)$, whereas, the potential energy is zero.  

(c) The kinetic energy of the block at this position can be found as $E=\frac{1}{2} k x^{2}$ and thus, $\mathrm{K.E}=\frac{1}{2}\left(\frac{\mathrm{p}^{2}}{\mathrm{k}}\right)$.