- #1

eprparadox

- 138

- 2

## Homework Statement

The amplitude of any oscillator can be doubled by:

A. doubling only the initial displacement

B. doubling only the initial speed

C. doubling the initial displacement and halving the initial speed

D. doubling the initial speed and halving the initial displacement

E. doubling both the initial displacement and the initial speed

## Homework Equations

## x(t) = A \sin(\omega t + \delta) ##

## The Attempt at a Solution

The answer is suppose to be E. But I have no intuition for that.

Here's what I did quantitatively. I started with a simple harmonic oscillator as [tex] x(t) = A \sin(\omega t + \delta) [/tex]

Our initial conditions are ## x(0) = x_0 ## and ## \dot{x}(0) = v(0) = v_0 ##.

Plugging these initial conditions in, we get:

## x_0 = A\sin(\delta) ## and

## v_0 = A \omega \cos(\delta) ##

We can subtract these two equations to get

[tex] x_0 - v_0 = A \sin(\delta) - A\omega\cos(\delta) [/tex]

Solving for ## A ##, we get

[tex] A = \frac{x_0 - v_0}{\sin(\delta) - \omega \cos(\delta)} [/tex]

So if we want to double ## A ##, then we need to double the right side of the above equation and that amounts to doubling both the ## x_0 ## value and the initial speed. Two questions for the PF crew:

1. Does the above look like the right thought process to getting the answer (even though I already knew what the answer was)?

2. Is there an intuition to why it is that we need to double both the initial displacement and the initial speed to double the amplitude?