- #1

embphysics

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## Homework Statement

Consider a mass m on the end of a spring of force constant k and constrained

to move along the horizontal x axis. If we place the origin at the spring’s equilibrium

position, the potential energy is U = 1/2kx^2 At time t = 0 the mass is sitting at the

origin and is given a sudden kick to the right so that it moves out to a maximum

displacement x_max = A and then continues to oscillate about the origin.

(a) Write down the equation for conservation of energy and solve it to give the

mass’s velocity [itex]\dot{x}[/itex] in terms of the position x and the total energy E.

(b) Show that E = 1/2kA^2 and use this to eliminate E from your expression for [itex]\dot{x}[/itex]. Find the time for the mass to move from the origin out to a position x.

(c) Solve the result of part (b) to give x as a function of t and show that the mass

executes simple harmonic motion with period [itex]2 \pi \sqrt{m/k}[/itex]

## Homework Equations

## The Attempt at a Solution

All right. I am currently trying to evaluate the integral, which was instructed of me to find in part (b).

Before I proceed any further, here is a little preliminary work:

[itex]\dot{x}(x) = kA^2 - \frac{k}{m} x^2[/itex]

And I used this fact to find the integral: [itex] \dot{x} = \frac{dx}{dt} [/itex] which becomes by separation of variables [itex]dt = \frac{dx}{\dot{x}}[/itex]

Substituting in, and integrating both sides, this is the integral thus far:

[itex]\int_0^t dt' = \int_0 ^x \frac{dx}{\sqrt{kA^2 - \frac{k}{m}x^2 }} [/itex]

[itex]kA^2 = (k^{1/2}A)^2[/itex] and [itex]\frac{k}{m}x^2= (\omega x)^2[/itex], where

[itex]\omega = \sqrt{\frac{k}{m}}[/itex]. This leads to trigonometric substitution:

[itex]sin \theta = \frac{(k^{1/2}A)^2}{\sqrt{kA^2 -(\omega x)^2}} [/itex]

The problem I have is finding how dx is equal to dθ. I tried using the fact that [itex]\cos \theta = \frac{( \omega x)^2}{(k^{1/k} A)^2}[/itex], then taking the arccos of this, and then taking the derivative of theta with respect to x. This didn't really seem helpful.

How can I get rid of dx, so that I might have an integral involving only theta?