Insecurity about introductory classical mechanics [closed]

Question

This is a problem I encountered, but not a homework assignment, I'm not only just looking for solutions here... This is my first time encountering questions like this, I'm sure you remember the first time you see mathematical proofs, you need some help with it...That's my situation, please don't report this post...I spent the entire Sunday reading relevant material but something just doesn't click, jumping from introductory physics straight into this is just too hard.....Appreciate any help and hint.

Consider a particle of mass $m$ that is located at $\vec{x}$ and is subject to the gravitational force exerted by two other particles. The latter is kept fixed at $\vec{x_{\pm}}=\pm\vec{a}$, and their masses are $\vec{M_{\pm}}=M/2\pm \Delta M$. The potential energy of the free particle is given by $$V(\vec{x})=-Gm \left(\frac{M_{+}}{|\vec{x}-\vec{x}_+|}+\frac{M_{-}}{|\vec{x}-\vec{x}_-|}\right).$$

1. To determine the total force, I use $F=-\nabla V$. Can I use that here? It just feels very weird to use that equation and then take the derivative with respect to $x$. Then the total torque due to the gravitational force is simply $N= r \times \dot{p}=r \times F$. Would that be enough to answer the total force question?

2. Recompute $\vec F$ and $\vec \tau$ by treating the two fixed particles as a single system of mass $M$ located at its center of mass. Compare the outcome to the exact answer obtained in (a), and discuss in which $\vec a$ region it is justified to treat the two fixed particles as a single system. - This question makes me wonder if i just my thoughts on a) is complete bullshit. Now I assume the potential energy equation given treats the two particles as two different particles, I should modify the equation $V(\vec{x})=-Gm(\frac{M_{+}}{|\vec{x}-\vec{x}_+|}+(\frac{M_{-}}{|\vec{x}-\vec{x}_-|})$ and since the center of mass $R=\frac{\Sigma m_ir_i}{\Sigma m_i}$, the equation becomes $V(\vec{x})=-GmR=2\Delta M\vec{a}$. So it is only justified to treat the two fixed particles as a single system when the distance between is $a$?

3. Expand the exact $\vec{F}$ and $\vec{\tau}$ obtained in (a) in the limit where $|\vec{a}|$ is very small. Include terms of the lowest order and first order in $|\vec{a}|$. Are they asking for a Taylor expansion here?

I would really really appreciate any help. I've been hopelessly stuck on this for a very long time. And i have no idea what im doing.

Answer 1

1. You're right, you can evaluate force taking the gradient of the potential.

2. Just evaluate the center of mass $\mathbf{x}_G = \dfrac{\sum_i M_i \mathbf{x}_i}{\sum_i M_i} = \dfrac{\sum_i M_i \mathbf{x}_i}{M^{tot}}$ and the force they ask for, taking the gradient of the potential

   $V^*(\mathbf{x}) = G m M^{tot} \dfrac{1}{|\mathbf{x} - \mathbf{x}_G|}$

3. From what you write, it looks like they're asking you to evaluate a power series in $\dfrac{\mathbf{a}}{|\mathbf{x}-\mathbf{x}_G|}$ is "small", i.e. when the distance between mass $m$ and the center of gravity is much larger than the distance between the masses $M^\pm$, something like a far-field expansion.

If you need further help, I'll edit my answer later.