The mass density of a spherical body is given by `rho(r)=k/r` for `r le R` and `rho (r)=0`for r > R , where r is the distance from the centre. The correct graph that describes qualitatively the acceleration, a, of a test particle as a function of r is :

Options

A.
B.
C.
D.

Explanation

To solve the problem, we need to analyze the mass density of the spherical body and how it affects the acceleration of a test particle at different distances from the center of the sphere.

### Step-by-Step Solution:

1. **Understanding the Mass Density**:
The mass density is given by:
\[ \rho(r) = \frac{k}{r} \quad \text{for } r \leq R \]
\[ \rho(r) = 0 \quad \text{for } r > R \]
This indicates that the mass density decreases as we move away from the center of the sphere until we reach the radius \( R \).

2. **Finding the Mass Inside Radius \( r \)**:
To find the total mass \( M \) within a radius \( r \) (where \( r \leq R \)), we need to integrate the density over the volume:
\[ M = \int_0^r \rho(r) \, dV \]
The volume element in spherical coordinates is \( dV = 4\pi r^2 dr \). Thus,
\[ M = \int_0^r \frac{k}{r} \cdot 4\pi r^2 \, dr = 4\pi k \int_0^r r \, dr \]
Evaluating the integral:
\[ M = 4\pi k \left[\frac{r^2}{2}\right]_0^r = 2\pi k r^2 \]

3. **Calculating the Gravitational Field (Acceleration)**:
The gravitational field \( g \) (which is equal to the acceleration \( a \) of the test particle) at distance \( r \) is given by:
\[ g = \frac{GM}{r^2} \]
Substituting \( M \):
\[ g = \frac{G(2\pi k r^2)}{r^2} = 2\pi G k \]
Thus, for \( r \leq R \), the acceleration \( a \) is constant:
\[ a = 2\pi G k \]

4. **Finding the Mass Outside Radius \( R \)**:
For \( r > R \), the total mass \( M_1 \) of the sphere can be found by integrating from \( 0 \) to \( R \):
\[ M_1 = \int_0^R \frac{k}{r} \cdot 4\pi r^2 \, dr = 2\pi k R^2 \]

5. **Calculating the Gravitational Field for \( r > R \)**:
The gravitational field for \( r > R \) is:
\[ g = \frac{GM_1}{r^2} = \frac{G(2\pi k R^2)}{r^2} \]
Thus, for \( r > R \):
\[ a = \frac{2\pi G k R^2}{r^2} \]
This shows that the acceleration decreases with the square of the distance \( r \).

6. **Graphing the Acceleration**:
- For \( r < R \), the acceleration \( a \) is constant.
- For \( r > R \), the acceleration \( a \) decreases as \( \frac{1}{r^2} \).

### Conclusion:
The correct qualitative graph of acceleration \( a \) as a function of distance \( r \) will show a constant value for \( r < R \) and a decreasing curve for \( r > R \).