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Chapter 6: Problem 738

Two small and heavy sphere, each of mass \(\mathrm{M}\), are placed distance r apart on a horizontal surface the gravitational potential at a mid point on the line joining the center of spheres is (A) zero (B) \(-(\mathrm{GM} / \mathrm{r})\) (C) \(-[(2 \mathrm{GM}) / \mathrm{r}]\) (D) \(-[(4 \mathrm{GM}) / \mathrm{r}]\)

Short Answer

Expert verified
The short version of the answer is: The gravitational potential at the midpoint on the line joining the centers of the two spheres is given by the sum of the gravitational potentials due to each sphere at that point. Since the midpoint is equidistant from both spheres, we find the potential due to each sphere and add them: \(V = V_{1} + V_{2} = (-\frac{2GM}{r}) + (-\frac{2GM}{r}) = -(\frac{4GM}{r})\) Hence, the correct answer is (D) \(-\frac{4GM}{r}\).

Step by step solution

01

Identify Variables

Identify the important variables in the problem: mass of both spheres (M), distance between the centers of spheres (r), and the gravitational potential at the midpoint (V).
02

Understand Gravitational Potential Formula

Gravitational potential V at a given point due to a mass M is given by the formula: \(V = -\frac{GM}{r}\) where G is the gravitational constant, r is the distance from the point to the mass, and M is the mass.
03

Determine the Distance from each Sphere to the Midpoint

As we need the gravitational potential at the midpoint of the line connecting the centers of the spheres, we need to determine the distance from each sphere center to the midpoint. Since it's the midpoint, both distances will be the same: \(r_{1} = r_{2} = \frac{r}{2}\)
04

Compute Gravitational Potential due to each Sphere at the Midpoint

Using the gravitational potential formula \(V = -\frac{GM}{r}\), we'll need to find the gravitational potential at the midpoint caused by both spheres. We'll call these potentials V1 and V2. For Sphere 1: \(V_{1} = -\frac{GM}{r_{1}} = -\frac{GM}{\frac{r}{2}} = -\frac{2GM}{r}\) For Sphere 2: \(V_{2} = -\frac{GM}{r_{2}} = -\frac{GM}{\frac{r}{2}} = -\frac{2GM}{r}\)
05

Add Gravitational Potentials at the Midpoint

Now, we need to add the gravitational potentials (V1 and V2) at the midpoint caused by both spheres: Total Gravitational Potential (V) at Midpoint: \(V = V_{1} + V_{2} = (-\frac{2GM}{r}) + (-\frac{2GM}{r}) = -(\frac{4GM}{r})\)
06

Compare the Result to the Given Options

Comparing the calculated gravitational potential \( -\frac{4GM}{r} \) to the given options in the exercise, we see that it matches option (D). Therefore, the correct answer is (D) \(-\frac{4GM}{r}\).

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