P is a point at a distance r from the centre of a spherical shell of mass M and radius a, where `r lt a`. The gravitational potential at P is

Options

A. `-(GM)/(r)`
B. `-(GM)/(a)`
C. `-GM(r)/(a^(2))`
D. `-GM((a-r)/(a^(2)))`

Answer

To find the gravitational potential at point P, which is located at a distance r from the center of a spherical shell of mass M and radius a (where r < a), we can follow these steps:

### Step 1: Understand the properties of a spherical shell
According to the shell theorem, a uniform spherical shell of mass exerts no net gravitational force on any object located inside it. Therefore, the gravitational field inside the shell is zero.

### Step 2: Define the gravitational potential
The gravitational potential (U) at a point in space due to a mass M is given by the formula:
\[ U = -\frac{G M}{r} \]
where G is the gravitational constant, M is the mass of the shell, and r is the distance from the center of the mass to the point where we are calculating the potential.

### Step 3: Apply the shell theorem for the potential inside the shell
Since point P is inside the shell (r < a), the gravitational potential at point P is constant and equal to the potential at the surface of the shell.

### Step 4: Calculate the gravitational potential at the surface of the shell
The gravitational potential at the surface of the shell (at distance a from the center) is given by:
\[ U_{\text{surface}} = -\frac{G M}{a} \]

### Step 5: Conclude the gravitational potential at point P
Since the gravitational potential inside the shell is constant and equal to the potential at the surface, we can conclude:
\[ U_P = U_{\text{surface}} = -\frac{G M}{a} \]

Thus, the gravitational potential at point P, where r < a, is:
\[ U_P = -\frac{G M}{a} \]

### Final Answer
The gravitational potential at point P is:
\[ U_P = -\frac{G M}{a} \]

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