Consider an attractive force which is central but is inversely proportional to the first power of distance. If a particle is in circular orbit, under such a force , which of the following statements are correct ?

Options

A. the speed is directly proportional to the square root of orbital radius
B. the speed isindependent of radius
C. the period is independent of radius
D. the period is directly proportional to radius.

Explanation

To solve the problem, we need to analyze the given attractive force, which is inversely proportional to the first power of distance, and how it affects a particle in a circular orbit.

### Step-by-Step Solution:

1. **Understanding the Force**:
The force \( F \) acting on the particle is given by:
\[ F = \frac{k}{r} \]
where \( k \) is a constant and \( r \) is the distance from the center.

2. **Centripetal Force Requirement**:
For an object moving in a circular orbit, the centripetal force \( F_c \) required to keep the object in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

3. **Equating Forces**:
Since the attractive force provides the necessary centripetal force, we can set the two forces equal:
\[ \frac{k}{r} = \frac{mv^2}{r} \]

4. **Simplifying the Equation**:
By multiplying both sides by \( r \) (assuming \( r \neq 0 \)), we get:
\[ k = mv^2 \]
From this, we can express \( v^2 \) as:
\[ v^2 = \frac{k}{m} \]
This shows that the speed \( v \) is constant and independent of the radius \( r \).

5. **Finding the Time Period**:
The time period \( T \) for one complete revolution in a circular orbit is given by:
\[ T = \frac{2\pi r}{v} \]
Substituting \( v \) from the previous step:
\[ T = \frac{2\pi r}{\sqrt{\frac{k}{m}}} = \frac{2\pi r \sqrt{m}}{\sqrt{k}} \]
This indicates that the time period \( T \) is directly proportional to the radius \( r \).

6. **Conclusion**:
- The speed \( v \) is constant and independent of the radius \( r \).
- The time period \( T \) is directly proportional to the radius \( r \).

### Correct Statements:
Based on the analysis:
- The speed is constant and independent of the radius (Correct).
- The time period is directly proportional to the radius (Correct).

### Final Answer:
The correct statements are (B) and (D).

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