The gravitational force between two objects is proportional to `1//R` (and not as `1//R^(2)`) where `R` is separation between them, then a particle in circular orbit under such a force would have its orbital speed `v` proportional to

Options

A. `1/(R^(2))`
B. `R^(0)`
C. `R^(1)`
D. `1/R`

Explanation

To solve the problem, we need to analyze the relationship between gravitational force and orbital speed when the gravitational force is proportional to \( \frac{1}{R} \) instead of the usual \( \frac{1}{R^2} \).

### Step-by-Step Solution:

1. **Understanding the Gravitational Force**:
The problem states that the gravitational force \( F \) between two objects is proportional to \( \frac{1}{R} \), where \( R \) is the separation between the two objects. This can be expressed as:
\[ F = k \cdot \frac{M_1 M_2}{R} \]
where \( k \) is a constant of proportionality, and \( M_1 \) and \( M_2 \) are the masses of the two objects.

2. **Centripetal Force Requirement**:
For an object of mass \( M_2 \) moving in a circular orbit of radius \( R \) around mass \( M_1 \), the centripetal force required to keep it in orbit is given by:
\[ F_{centripetal} = \frac{M_2 v^2}{R} \]
where \( v \) is the orbital speed of the object.

3. **Setting the Forces Equal**:
In a stable circular orbit, the gravitational force must equal the centripetal force:
\[ k \cdot \frac{M_1 M_2}{R} = \frac{M_2 v^2}{R} \]

4. **Canceling \( M_2 \) and \( R \)**:
Since \( M_2 \) appears on both sides of the equation, we can cancel it out (assuming \( M_2 \neq 0 \)):
\[ k \cdot \frac{M_1}{R} = \frac{v^2}{R} \]
Now, we can multiply both sides by \( R \):
\[ k \cdot M_1 = v^2 \]

5. **Finding the Relationship of \( v \)**:
From the equation \( v^2 = k \cdot M_1 \), we can express \( v \) as:
\[ v = \sqrt{k \cdot M_1} \]
Here, \( k \) and \( M_1 \) are constants, meaning that \( v \) does not depend on \( R \) at all.

6. **Conclusion**:
Since \( v \) is independent of \( R \), we can say that the orbital speed \( v \) is proportional to \( R^0 \) (which is 1). Therefore, the correct answer is that the orbital speed \( v \) is proportional to \( R^0 \).

### Final Answer:
The orbital speed \( v \) is proportional to \( R^0 \).