Question

7. A particle performing linear S.H.M. of period 2 pi seconds about the mean position 0 is observed to have a speed of b3 m/s, when at a distance b (metre) from 0. If the particle is moving away from 0 at that instant, find the time required by the particle, to travel a
further distance b.
[Ans: pi/3 s]​

Answer 1

Given :

The time period of the particle = 2π/ω

Speed of the particle = bω√3 m/s

To Find :

The time required by the particle to travel a distance of b m

Solution :

• The velocity of particle is given by

            $v = \omega\sqrt{A^{2} - x^{2} }$

• By substituting the values

         $b\omega\sqrt{3} = \omega\sqrt{A^{2} - b^{2} }$      

         $3b^{2} \omega^{2} = \omega ^{2}( A^{2} - b^{2} )$

         $3b^{2} = ( A^{2} - b^{2} )$

          $4b^{2} = A^{2}$

          $A = 2b$

• The time taken to travel distance b from mean position is

            $x = Asin\omega t$

            $b = 2bsin\omega t$    

            $Sin\omega t =\frac{1}{2}$

            $t = \frac{\pi}{6\omega }$

• Further time taken by the particle to reach mean position is

          $t =\frac{T}{4} -\frac{\pi}{6\omega}$

          $t =\frac{2\pi }{4\omega} -\frac{\pi}{6\omega}$

          $t = \frac{\pi }{3\omega}$

The time required by the particle is π/3ω sec