A particle moving under a conservative force

[jamie.j1989]

Homework Statement

From, Classical mechanics 5th edition, Tom W.B. Kibble, Frank H. Berkshire
Chapter 2, problem 30

A particle moving under a conservative force oscillates between x1₁ and x₂. Show that the period of oscillation is

τ = 2\int^{x_{2}}_{x^{1}}\sqrt{\frac{m}{2(V(x_{2})-V(x))}}dx

Homework Equations

m\ddot{x} + F(x) = 0

F(x) = -\frac{d}{dx}V(x)

The Attempt at a Solution

m\ddot{x} + F(x) = 0

→ m\ddot{x} -\frac{d}{dx}V(x) = 0

→ \int^{x_{2}}_{x_{1}}m\ddot{x}dx = V(x₂)-V(x₁)

Im not sure if I've started right and if I have I don't know how to go forward with the \ddot{x}

 

[CAF123]

Rewrite $$m\ddot{x} + \frac{dV}{dx} = 0 \,\,\text{as}\,\,\frac{d}{dt}\left(\frac{1}{2}m\dot{x}^2 + V(x)\right) = 0$$ and carry on from there.

 

[jamie.j1989]

When I try that I just end up with

m\int\dot{x}\frac{d\dot{x}}{dx}dx + V(x) = 0

by parts on the integral just sends me in a circle?

 

[CAF123]

If $$\frac{d}{dt}\left(\frac{1}{2}m\left(\frac{dx}{dt}\right)^2 + V(x) \right) = 0\,\,\,\text{then}\,\,\,\frac{1}{2}m\left(\frac{dx}{dt}\right)^2 + V(x) = \text{const}$$ Use what you know about the particle at the boundaries of its oscillations (i.e at $x_1$ and $x_2$) to obtain the constant.

Once you have this, you can separate variables to find T.