A body is executing SHM. When its displacements from the mean position are 4 ~cm and 5 ~cm it has velocity 10 ~cms ⁻¹ and 8 ~cms ⁻¹ respectively. Its periodic time t is

Step-by-step solution

The velocity v of a particle executing simple harmonic motion at a displacement x is given by v = A^2 - x^2 , where A is the amplitude and is the angular frequency. For the first case, 10 = A^2 - 4^2 100 = ^2 (A^2 - 16) . For the second case, 8 = A^2 - 5^2 64 = ^2 (A^2 - 25) . Dividing the two equations: 100 64 = A^2 - 16 A^2 - 25 25 16 = A^2 - 16 A^2 - 25 . Cross-multiplying: 25(A^2 - 25) = 16(A^2 - 16) 25A^2 - 625 = 16A^2 - 256 . 9A^2 = 369 A^2 = 41 . Substituting A^2 = 41 into the first equation: 100 = ^2 (41 -