A uiform ring of mass M and radius R is placed directly above a uniform sphere of mass 8M and same radius R. The centre of the sphere. The gravitational atraction between the sphere and the ring is

Options

A. `(8GM^(2))/(R^(2))`
B. `(2GM^(2))/(sqrt(3)R^(2))`
C. `(3GM^(2))/(2R^(2))`
D. `(sqrt(3))/(2)(GM^(2))/(R^(2))`

Answer

To find the gravitational attraction between a uniform ring of mass \( M \) and radius \( R \) placed directly above a uniform sphere of mass \( 8M \) and the same radius \( R \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Gravitational Field Due to the Sphere**:
The gravitational field \( E \) at a distance \( h \) above the center of a uniform sphere is given by the formula:
\[ E = \frac{GM}{r^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the sphere, and \( r \) is the distance from the center of the sphere to the point where the field is being calculated.

2. **Calculate the Gravitational Field at the Position of the Ring**:
Since the ring is placed directly above the center of the sphere at a distance \( R \) (the radius of the sphere), the gravitational field at the location of the ring is:
\[ E = \frac{G(8M)}{R^2} \]
Simplifying this gives:
\[ E = \frac{8GM}{R^2} \]

3. **Calculate the Gravitational Force on the Ring**:
The gravitational force \( F \) acting on the ring due to the gravitational field created by the sphere can be calculated using the formula:
\[ F = m \cdot E \]
where \( m \) is the mass of the ring. Substituting the expression for \( E \):
\[ F = M \cdot \frac{8GM}{R^2} \]
Simplifying this gives:
\[ F = \frac{8GM^2}{R^2} \]

4. **Final Result**:
Therefore, the gravitational attraction between the sphere and the ring is:
\[ F = \frac{8GM^2}{R^2} \]

### Conclusion:
The gravitational attraction between the sphere and the ring is \( \frac{8GM^2}{R^2} \).