Harmonic motion:

A. Show that the period of oscillatory motion in a potential V (x) is

Delta (t)=(2m)^(1/2) *integral((dx)/(E-V(x))^(1/2)) between xb and xa

where xa and xb are the two turning points located on two opposite sides of the equilibrium point.

B. Assume that the potential is

V(x)= (k/2)(x−x0)^(1/2)

What is the equilibrium point and what are the two turning points if the maximum of the kinetic energy is E.

C. Find the period by evaluating the integral for the potential in B.

D. Write down the equation of motion corresponding to the potential in B. Solve this equation and use it to determine the period of oscillations. Compare your answer to B.

E. Use the solution of the equation of motion in D to find an explicit expression for the kinetic and the potential energies as a function of time. Compute the total energy and verify that it is a constant. Express the amplitude of the oscillations in terms of the total energy and k.

I have a worked solution for this question however it seems to jump large sections and does not explain anything very well so I am really struggling to understand it.

I would really appreciate it if anyone had the time to try and explain this the step by step.

Thank you!