If a particle of mass m is moving in a horizontal circle of radius r with a centripetal force ` (-1//r^(2))` , the total energy is

Options

A. `-(4)/(r) `
B. `-(2)/(r) `
C. `-(1)/(r)`
D. `-(1)/(2r)`

Explanation

To solve the problem, we need to find the total energy of a particle of mass \( m \) moving in a horizontal circle of radius \( r \) with a centripetal force given by \( F_c = -\frac{1}{r^2} \).

### Step-by-Step Solution:

1. **Identify the centripetal force**:
The centripetal force acting on the particle is given as:
\[ F_c = -\frac{1}{r^2} \]
For our calculations, we will consider the magnitude of the centripetal force:
\[ F_c = \frac{1}{r^2} \]

2. **Relate centripetal force to velocity**:
The centripetal force can also be expressed in terms of the mass \( m \) and the velocity \( v \) of the particle:
\[ F_c = \frac{mv^2}{r} \]
Setting the two expressions for centripetal force equal gives:
\[ \frac{mv^2}{r} = \frac{1}{r^2} \]

3. **Solve for velocity**:
Rearranging the equation to solve for \( v^2 \):
\[ mv^2 = \frac{1}{r} \]
\[ v^2 = \frac{1}{mr} \]

4. **Calculate kinetic energy (KE)**:
The kinetic energy of the particle is given by:
\[ KE = \frac{1}{2} mv^2 \]
Substituting \( v^2 \) from the previous step:
\[ KE = \frac{1}{2} m \left(\frac{1}{mr}\right) = \frac{1}{2r} \]

5. **Calculate potential energy (PE)**:
The potential energy can be found using the relationship between force and potential energy:
\[ F = -\frac{dU}{dr} \]
Rearranging gives:
\[ dU = -F \, dr \]
Integrating from infinity to \( r \):
\[ U = -\int_{\infty}^{r} F \, dr = -\int_{\infty}^{r} -\frac{1}{r^2} \, dr \]
This gives:
\[ U = -\left[-\frac{1}{r}\right]_{\infty}^{r} = -\left(0 - \left(-\frac{1}{r}\right)\right) = -\frac{1}{r} \]

6. **Calculate total energy (E)**:
The total energy \( E \) is the sum of kinetic energy and potential energy:
\[ E = KE + U = \frac{1}{2r} - \frac{1}{r} \]
Simplifying this:
\[ E = \frac{1}{2r} - \frac{2}{2r} = -\frac{1}{2r} \]

### Final Answer:
The total energy of the particle is:
\[ E = -\frac{1}{2r} \]