What Is the Central Potential V(r) for a Particle's Sinusoidal Trajectory?

[stunner5000pt]

the trajectory of a particle moving in a cnetral potential V(r) is given by $r = a + b \sin(\eta \phi)$ where a b and eta are constants

Compute the potential V(r) in which the particel moves (for arbitrary a, b, and eta) and sketch $V_{eff} (r) = V(r) + \frac{L^2}{2mr^2}$ for cases a =b

sketching is not hte problem here.. i just need to find out V(r)
well i know that i need to compute the force f first and then integrate f w.r.t. r to get V(r)
find find $$u = \frac{1}{r} = \frac{1}{a + b \sin(\eta \phi)}$$
then using $$f = -\frac{L^2 u^2}{m} (u'' + u)$$
and $$u '' = \frac{2b^2 \eta^2 \cos^2 (\eta \phi)}{r^3} + \frac{b \eta^2 \sin(\eta \phi)}{r^2}$$ (phew!)

once i sub expressions for u'' and u i get
$$f(r) = - \frac{L^2}{mr^2} \left( \frac{2b^2 \eta^2 \cos^2 (\eta \phi)}{r^3} + \frac{b \eta^2 \sin(\eta \phi)}{r^2} \right)+ \frac{1}{r}$$
is that fine? Can i simplify that any more in terms of r?

are the steps correct? Any problems with the derivative? Just need to know if i can simplify any further.
Your help is greatly appreciated!

Thank you

 

[Jhero]

for your question. Your steps for finding the potential V(r) look correct. To simplify the expression further, you could combine the terms with a common factor of r and then integrate to get V(r). This would give you a simpler expression for V(r). Also, you could try substituting r = a + b sin(ηφ) into the expression for V(r) and see if you can simplify it that way.

As for any problems with the derivative, it looks like you have correctly taken the second derivative of u and substituted it into the expression for f(r). However, it might be helpful to double check your calculations just to make sure there are no errors.

In terms of sketching Veff(r), remember that it is a sum of two terms: V(r) and L^2/2mr^2. So when a = b, the first term will be a cosine function and the second term will be a constant. This should help you visualize the shape of Veff(r) and how it changes with different values of a, b, and η.

I hope this helps. Good luck with your calculations and sketching!