Question

The amplitude of a simple harmonic oscillator will be doubled by:a) doubling only the initial speedb) doubling the initial displacement and halving the initial speedc) doubling both the initial displacement and the initial speedd) doubling the initial speed and halving the initial displacemente) doubling only the initial displacement

Answer 1

Answer:

When both initial speed and initial displacement is doubled then amplitude will be doubled.

Explanation:

Given that :- Amplitude of simple harmonic Oscillator  is doubled.

So,

     Formula of Simple harmonic oscillator is  $X=A\sin\ (2\pi ft +\phi)$  ...........(1)

                                                            Where X = Position in (m,cm,km.....)

                                                                        A = Amplitude  in (m,cm,km.....)

                                                                        F = Frequency in (Hz)

                                                                        T = Time in (sec.)

                                                                        Ф = Phase in (rad)

  For initial displacement taking t=0 we get,

                          Initial displacement = $A\sin(\phi)$    .................(2)            

Now taking equation (1) and differentiating it w.r.t to (t) we get

                                $\frac{dx}{dt} = 2\pi fA\cos\ (2\pi ft+\phi)$

                                 $V= 2\pi fA\cos\ (2\pi ft+\phi)$

taking t=0 for initial speed then we get,

                                Initial speed = $2\pi fA\cos\phi$    ...............(3)

observing equation (2) & (3) that the initial displacement and initial speed depends on the Amplitude of the Oscillator.

Hence,

when both initial speed and displacement is doubled then amplitude will be doubled.