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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?