Question

One end of a spring of force constant k is fixed to a vertical wall and the other to a block of mass m resting on a smooth horizontal surface. There is another wall at a distance $x_0$ from the block. The spring is then compressed by $2x_0$ and released. The time taken to strike the wall is

• A. $\frac{1}{6}\pi \sqrt{\frac{k}{m}}$
• B. $\sqrt{\frac{k}{m}}$
• C. $\frac{2\pi }{3}\sqrt{\frac{m}{k}}$
• D. $\frac{\pi }{4}\sqrt{\frac{k}{m}}$

Answer 1

The correct option is C

$\frac{2\pi }{3}\sqrt{\frac{m}{k}}$

The total time from A to C

$=t_{AC}=t_{AB}+t_{BC}=\left(\frac{T}{4}\right)+t_{BC}$

where T = time period of oscillation of spring - mass $syste{m}_{BC}$ can be obtained from, BC = AB $sin\left(\frac{2\pi }{T}\right)t_{BC}$ putting

$\frac{BC}{AB}=\frac{1}{2}$ we obtain $t_{BC}=\frac{T}{12}$

$\Rightarrow$ $t_{AC}=\frac{T}{4}+\frac{T}{12}=\frac{2\pi }{3}\sqrt{\frac{m}{k}}$