A particle executes simple harmonic motion between x = − A and x = + A . If time taken by particle to go from x = 0 to 2 A is 2 s, then time taken by particle in going from x = 2 A to A is

Question

A particle executes simple harmonic motion between x = − A and x = + A . If time taken by particle to go from x = 0 to 2 A ​ is 2 s, then time taken by particle in going from x = 2 A ​ to A is

TestHub · Accepted Answer

The equation of simple harmonic motion starting from the mean position is given by $x = A sin (ω t)$

For the particle to go from $x = 0$ to $x = \frac{A}{2}$ , let the time taken be $t_{1}$ .

$\frac{A}{2} = A sin (ω t_{1})$

$sin (ω t_{1}) = \frac{1}{2} \Rightarrow ω t_{1} = \frac{π}{6}$

Given $t_{1} = 2$ s, we have $ω (2) = \frac{π}{6} \Rightarrow ω = \frac{π}{12}$ rad/s

For the particle to go from $x = 0$ to $x = A$ , let the time taken be $t_{2}$ .

$A = A sin (ω t_{2})$

$sin (ω t_{2}) = 1 \Rightarrow ω t_{2} = \frac{π}{2}$

$t_{2} = \frac{π}{2 ω} = \frac{π}{2 \times \frac{π}{12}} = 6$ s

Time taken to go from $x = \frac{A}{2}$ to $x = A$ is $t_{2} - t_{1} = 6 - 2 = 4$ s.

Answer: $4$ s

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