Two metallic balls of mass `m` are suspended by two strings of length `L`. The distance between upper ends is `l`. The angle at which the string will be inclined with vertical due to attraction is `(m lt lt M` where M is the mass of Earth)

Options

A. `tan^(-1)((Gm)/(gl^(2))`
B. `tan^(-1)((Gm)/(gL^(2))`
C. `tan^(-1)((Gm)/(gl))`
D. `tan^(-1)((Gm)/(gL))`

Answer

To solve the problem of two metallic balls of mass `m` suspended by strings of length `L`, with a distance `l` between their lower ends, we need to find the angle `θ` that the strings make with the vertical due to the gravitational attraction between the balls.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two metallic balls of mass `m` suspended by strings of length `L`.
- The distance between the lower ends of the strings is `l`.
- The balls will exert a gravitational force on each other, causing the strings to incline at an angle `θ` with the vertical.

2. **Forces Acting on the Balls**:
- Each ball experiences two forces:
- The gravitational force acting downwards, which is `mg` (where `g` is the acceleration due to gravity).
- The tension `T` in the string, which can be resolved into two components:
- Vertical component: `T cos(θ)`
- Horizontal component: `T sin(θ)`

3. **Setting Up the Equations**:
- For vertical equilibrium (the ball is not moving up or down):
\[ T \cos(θ) = mg \tag{1} \]
- For horizontal equilibrium (the balls are not moving sideways):
\[ T \sin(θ) = F \tag{2} \]
where `F` is the gravitational force of attraction between the two masses.

4. **Calculating the Gravitational Force**:
- The gravitational force `F` between the two balls can be calculated using Newton's law of gravitation:
\[ F = \frac{G m^2}{d^2} \]
where `d` is the distance between the centers of the two balls. For small angles, we can approximate `d` in terms of `L` and `θ`:
\[ d = 2L \sin(θ) \approx l \quad (\text{for small } θ) \]

5. **Dividing the Equations**:
- Dividing equation (2) by equation (1):
\[ \frac{T \sin(θ)}{T \cos(θ)} = \frac{F}{mg} \]
This simplifies to:
\[ \tan(θ) = \frac{F}{mg} \]

6. **Substituting the Gravitational Force**:
- Substituting the expression for `F`:
\[ \tan(θ) = \frac{G m^2 / l^2}{mg} \]
- Simplifying gives:
\[ \tan(θ) = \frac{G m}{g l^2} \]

7. **Finding the Angle**:
- Finally, we can find the angle `θ`:
\[ θ = \tan^{-1}\left(\frac{G m}{g l^2}\right) \]

### Final Answer:
The angle `θ` at which the strings will be inclined with the vertical due to the attraction is:
\[ θ = \tan^{-1}\left(\frac{G m}{g l^2}\right) \]