First slide

Simple hormonic motion

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_o from the black. The spring is then compressed by 2x_o and released. The time taken to strike the wall is

Easy

A

$\large \frac{1}{6}\sqrt {\frac{k}{m}}$
B

$\large \sqrt {\frac{k}{m}}$
C

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

$\large \frac{\pi }{4}\sqrt {\frac{k}{m}}$

Solution

Motion of the block is SHM where time period $\large T\, = \,2\pi \sqrt {\frac{m}{k}}$
The total time from A to C
$\large {t_{AC}}\, = \,{t_{AB}} + {t_{BC}} = \,\left( {\frac{T}{4}} \right) + {t_{BC}}$
where T = time period of oscillation of spring mass system
t_BC can be obtained from,
$\large BC\, = \,AB\sin (\frac{{2\pi }}{T}){t_{BC}}$
Putting $\large \frac{{BC}}{{AB}}\, = \,\frac{1}{2}$ we obtain $\large {t_{BC}}\, = \,\frac{T}{{12}}\,\,$
$\large \Rightarrow \,{t_{AC}}\, = \,\frac{T}{4} + \frac{T}{{12}} = \,\frac{{2\pi }}{3}\sqrt {\frac{m}{k}}$

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