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. Let's denote this force as \( F \). According to the problem, we can express this force as:

\[ F = \frac{k}{r} \]

where \( k \) is a constant and \( r \) is the distance from the center of the force.

### Step 1: Equate the gravitational force to the centripetal force
For a particle in circular motion, the centripetal force required to keep the particle in orbit is given by:

\[ F_c = \frac{mv^2}{r} \]

where \( m \) is the mass of the particle and \( v \) is its speed. Since the attractive force \( F \) provides the necessary centripetal force, we can set these two forces equal:

\[ \frac{k}{r} = \frac{mv^2}{r} \]

### Step 2: Simplify the equation
We can cancel \( r \) from both sides of the equation (assuming \( r \neq 0 \)):

\[ k = mv^2 \]

From this equation, we can solve for the speed \( v \):

\[ v^2 = \frac{k}{m} \]

Taking the square root gives us:

\[ v = \sqrt{\frac{k}{m}} \]

### Step 3: Analyze the speed
From the expression for \( v \), we see that the speed \( v \) is independent of the radius \( r \). This means that regardless of how far the particle is from the center, its speed remains constant.

### Step 4: Determine the time period
The time period \( T \) of the circular motion can be found using the relationship between speed, distance, and time. The distance traveled in one complete orbit (circumference of the circle) is \( 2\pi r \), so we have:

\[ T = \frac{\text{Distance}}{\text{Speed}} = \frac{2\pi r}{v} \]

Substituting the expression for \( v \):

\[ T = \frac{2\pi r}{\sqrt{\frac{k}{m}}} = 2\pi r \sqrt{\frac{m}{k}} \]

### Step 5: Analyze the time period
From the expression for \( T \), we can see that the time period \( T \) is directly proportional to the radius \( r \). This means that as the radius increases, the time period also increases.

### Conclusion
Based on our analysis, we can summarize the findings:

1. The speed \( v \) is independent of the radius \( r \).
2. The time period \( T \) is directly proportional to the radius \( r \).

Thus, the correct statements are:
- Speed is independent of the radius (Option 2 is correct).
- Time period is directly proportional to the radius (Option 4 is correct).

### Final Answer
The correct options are 2 and 4.