First slide

Applications of SHM

Question

One end of a spring of force constant k is fixed to a vertical wall and the other to a body of mass m resting on a smooth horizontal surface. There is another wall at a distance x₀from the body. The spring is then compressed by 3x₀ and released. The time taken to strike the wall from the instant of release is (given $\sin^{- 1} (1 / 3) = (π / 9))$

Difficult

A

$\frac{π}{6} \sqrt{\frac{m}{k}}$
B

$\frac{2 π}{3} \sqrt{\frac{m}{k}}$
C

$\frac{π}{4} \sqrt{\frac{m}{k}}$
D

$\frac{11 π}{18} \sqrt{\frac{m}{k}}$

Solution

when spring is compressed by 3x₀. Amplitude, A = 3x₀.
The time taken from extreme compressed position to mean position, t₁ = T/4.
If time taken (t₂) from mean position to x = x₀ is given by
$\begin{array}{l} x = A \sin \frac{2 π t_{2}}{T} \Rightarrow x_{0} = 3 x_{0} \sin \frac{2 π t_{2}}{T} \\ \sin \frac{2 π t_{2}}{T} = \frac{1}{3} \Rightarrow \frac{2 π t_{2}}{T} = \frac{π}{9} \Rightarrow t_{2} = \frac{T}{18} \\ t_{1} + t_{2} = \frac{T}{4} + \frac{T}{18} = \frac{11}{36} T = \frac{11}{36} 2 π \sqrt{\frac{m}{k}} = \frac{11}{18} π \sqrt{\frac{m}{k}} \end{array}$

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