A body is moving on a circlar path of radius r with constant speed under the action of force `(K)/(r^(3))` Assuming infinity as zero potential energy reference, tick the correct alternative

Options

A. Kinetic energy of body is `(K)/(2r^(2))`
B. Potential energy of the body is `(K)/(2r^(2))`
C. Mechanical energy of the body is `-(K)/(2r^(2))`
D. The body is in a bound state

Answer

To solve the problem step by step, we will analyze the motion of a body moving in a circular path under the influence of a force \( \frac{k}{r^3} \).

### Step 1: Understand the Forces Acting on the Body
The body is moving in a circular path with radius \( r \) and constant speed. The only force acting on the body is the centripetal force, which is directed towards the center of the circular path.

### Step 2: Write the Expression for Centripetal Force
The expression for centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \]
where \( m \) is the mass of the body and \( v \) is its speed.

### Step 3: Set the Given Force Equal to the Centripetal Force
According to the problem, the force acting on the body is:
\[ F = \frac{k}{r^3} \]
Since this force acts as the centripetal force, we can set the two expressions equal to each other:
\[ \frac{mv^2}{r} = \frac{k}{r^3} \]

### Step 4: Solve for \( mv^2 \)
Rearranging the equation gives:
\[ mv^2 = \frac{k}{r^2} \]

### Step 5: Calculate the Kinetic Energy
The kinetic energy \( KE \) of the body is given by:
\[ KE = \frac{1}{2} mv^2 \]
Substituting the expression for \( mv^2 \) we found:
\[ KE = \frac{1}{2} \left(\frac{k}{r^2}\right) = \frac{k}{2r^2} \]

### Step 6: Calculate the Potential Energy
To find the potential energy \( U \), we use the relationship between force and potential energy:
\[ F = -\frac{dU}{dr} \]
Substituting the force:
\[ \frac{k}{r^3} = -\frac{dU}{dr} \]
Integrating both sides with respect to \( r \):
\[ dU = -\frac{k}{r^3} dr \]
Thus,
\[ U = -\int \frac{k}{r^3} dr = \frac{k}{2r^2} + C \]
Assuming that potential energy at infinity is zero (i.e., \( U(\infty) = 0 \)), we find that \( C = 0 \). Therefore:
\[ U = \frac{k}{2r^2} \]

### Step 7: Calculate Total Mechanical Energy
The total mechanical energy \( E \) is the sum of kinetic and potential energy:
\[ E = KE + U = \frac{k}{2r^2} + \frac{k}{2r^2} = \frac{k}{r^2} \]

### Conclusion
From the calculations:
1. Kinetic Energy \( KE = \frac{k}{2r^2} \)
2. Potential Energy \( U = \frac{k}{2r^2} \)
3. Total Mechanical Energy \( E = \frac{k}{r^2} \)