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.