The velocity v and displacement x of a particle executing simple harmonic motion are related by the equation v^2 144 + x^2 36 = 1 , where v is in m/s and x is in meters. If the time taken by the particle to move directly from x = 3 m to x = 6 m is N s , the value of N is ______.

Step-by-step solution

The given equation is: v^2 144 + x^2 36 = 1 Rearranging for v^2 , we get: v^2 144 = 1 - x^2 36 v^2 = 144 ( 36 - x^2 36 ) v^2 = 4(36 - x^2) Comparing this with the standard SHM equation v^2 = ^2(A^2 - x^2) , we find: ^2 = 4 = 2 rad/s A^2 = 36 A = 6 m The time period of oscillation is: T = 2 = 2 2 = s The particle needs to move from x = 3 m (which is A/2 ) to x = 6 m (which is A ). The time taken to travel from the mean position ( x = 0 ) to x = A/2 is t₁ = T 12 . The time taken to travel from the mean position to th