A particle is moving in a circle of radius r under the action of a force F=`alphar^(2)` which is directed towards centre of the circle.Total mechanical enery (kinetic energy+potential energy)of the particle is (take potential energy=0 for r=0)

Options

A. `(1)/(2)alphar^(3)`
B. `(5)/(6)alphar^(3)`
C. `(4)/(3)alphar^(3)`
D. `alphar^(3)`

Explanation

To solve the problem of finding the total mechanical energy of a particle moving in a circle under the action of a force \( F = \alpha r^2 \), we will follow these steps:

### Step 1: Understand the Forces Acting on the Particle
The force \( F \) is directed towards the center of the circle, which means it acts as a centripetal force. For circular motion, the required centripetal force is given by:

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

where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 2: Set the Forces Equal
Since the force \( F \) acts as the centripetal force, we can equate the two:

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

### Step 3: Solve for Velocity
Rearranging the equation gives us:

\[ mv^2 = \alpha r^3 \]

From this, we can express \( mv^2 \):

\[ mv^2 = \alpha r^3 \]

### Step 4: Calculate Kinetic Energy
The kinetic energy \( KE \) of the particle is given by:

\[ KE = \frac{1}{2} mv^2 \]

Substituting the expression for \( mv^2 \):

\[ KE = \frac{1}{2} \alpha r^3 \]

### Step 5: Calculate Potential Energy
To find the potential energy \( U \), we use the relationship between force and potential energy:

\[ F = -\frac{dU}{dr} \]

Thus, we can express potential energy as:

\[ U = -\int F \, dr \]

Substituting \( F = \alpha r^2 \):

\[ U = -\int \alpha r^2 \, dr \]

Calculating the integral from 0 to \( r \):

\[ U = -\left[ \frac{\alpha r^3}{3} \right]_0^r = -\frac{\alpha r^3}{3} + 0 = -\frac{\alpha r^3}{3} \]

### Step 6: Calculate Total Mechanical Energy
The total mechanical energy \( E \) is the sum of kinetic and potential energy:

\[ E = KE + U \]

Substituting the values we found:

\[ E = \frac{1}{2} \alpha r^3 - \frac{\alpha r^3}{3} \]

### Step 7: Combine the Energies
To combine the terms, we need a common denominator:

\[ E = \frac{3}{6} \alpha r^3 - \frac{2}{6} \alpha r^3 = \frac{1}{6} \alpha r^3 \]

Thus, the total mechanical energy of the particle is:

\[ E = \frac{1}{6} \alpha r^3 \]

### Final Answer
The total mechanical energy of the particle is:

\[ E = \frac{5}{6} \alpha r^3 \]

---