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 $x_{0}$ from the black. The spring is then compressed by 2 $x_{0}$ and released. The time taken to strike the wall is

Moderate

A

$\frac{1}{6} π \sqrt{\frac{k}{m}}$
B

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

$\frac{2 π}{3} \sqrt{\frac{m}{k}}$
D

$\frac{π}{4} \sqrt{\frac{k}{m}}$

Solution

The total time from A to C
$t_{AC} = t_{AB} + t_{BC}$
= $(\frac{T}{4}) + t_{BC}$
where T = time period of oscillation of spring mass system
$t_{BC} can be obtained from, BC = AB \sin (\frac{2 π}{T}) t_{BC}$
Putting $\frac{BC}{AB} = \frac{1}{2} we obtain t_{BC} = \frac{T}{12}$
$\Rightarrow t_{AC} = \frac{T}{4} + \frac{T}{12} = \frac{2 π}{3} \sqrt{\frac{m}{k}}$ .

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