Three spherical shells of masses `M`, `2M` and `3M` have radii `R`, `3R` and `4R` as shows in figure. Find net potential at point `P`,

Explanation

To find the net gravitational potential at point P due to the three spherical shells, we can use the formula for the gravitational potential \( V \) due to a spherical shell, which is given by:

\[ V = -\frac{G M}{r} \]

where \( G \) is the gravitational constant, \( M \) is the mass of the shell, and \( r \) is the distance from the center of the shell to the point where the potential is being calculated.

### Step-by-step Solution:

1. **Identify the masses and radii of the shells**:
- Mass of shell 1: \( M_1 = M \), Radius: \( R_1 = R \)
- Mass of shell 2: \( M_2 = 2M \), Radius: \( R_2 = 3R \)
- Mass of shell 3: \( M_3 = 3M \), Radius: \( R_3 = 4R \)

2. **Calculate the distance from point P to the center of each shell**:
- For shell 1 (mass \( M \)):
- Distance \( r_1 = 2R \) (since point P is outside the shell)
- For shell 2 (mass \( 2M \)):
- Distance \( r_2 = 3R \)
- For shell 3 (mass \( 3M \)):
- Distance \( r_3 = 4R \)

3. **Calculate the potential at point P due to each shell**:
- Potential due to shell 1:
\[ V_1 = -\frac{G M}{r_1} = -\frac{G M}{2R} \]
- Potential due to shell 2:
\[ V_2 = -\frac{G (2M)}{r_2} = -\frac{2G M}{3R} \]
- Potential due to shell 3:
\[ V_3 = -\frac{G (3M)}{r_3} = -\frac{3G M}{4R} \]

4. **Calculate the net potential at point P**:
- The net potential \( V \) at point P is the sum of the potentials due to each shell:
\[ V = V_1 + V_2 + V_3 \]
\[ V = -\frac{G M}{2R} - \frac{2G M}{3R} - \frac{3G M}{4R} \]

5. **Finding a common denominator**:
- The common denominator for \( 2R, 3R, \) and \( 4R \) is \( 12R \).
- Rewriting each term:
\[ V = -\left(\frac{6G M}{12R} + \frac{8G M}{12R} + \frac{9G M}{12R}\right) \]
\[ V = -\frac{(6G M + 8G M + 9G M)}{12R} = -\frac{23G M}{12R} \]

### Final Result:
The net gravitational potential at point P is:
\[ V = -\frac{23G M}{12R} \]