The kinetic energy K of a particle moving along a circle of radius R depends upon the distance s as `K=as^2`. The force acting on the particle is

Options

A. (a) `2a(s^2)/(R)`
B. (b) `2as[1+(s^2)/(R)]^(1//2)`
C. (c) `2as`
D. (d) `2a`

Explanation

To find the force acting on a particle moving along a circle of radius R, given that its kinetic energy \( K \) depends on the distance \( s \) as \( K = as^2 \), we can follow these steps:

### Step 1: Relate Kinetic Energy to Velocity
The kinetic energy \( K \) of a particle is given by the formula:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity. Given that \( K = as^2 \), we can set the two expressions for kinetic energy equal to each other:
\[ as^2 = \frac{1}{2} mv^2 \]

### Step 2: Solve for Velocity
From the equation \( as^2 = \frac{1}{2} mv^2 \), we can solve for \( v^2 \):
\[ v^2 = \frac{2as^2}{m} \]

### Step 3: Differentiate to Find Tangential Acceleration
The tangential acceleration \( a_t \) is defined as the rate of change of velocity with respect to time. We can differentiate \( v^2 \) with respect to time \( t \):
\[ \frac{dv}{dt} = \frac{d}{dt} \left( \sqrt{\frac{2as^2}{m}} \right) \]
Using the chain rule, we have:
\[ \frac{dv}{dt} = \frac{1}{2} \left( \frac{2as^2}{m} \right)^{-1/2} \cdot \frac{d}{dt} \left( \frac{2as^2}{m} \right) \]
Since \( \frac{d}{dt}(s^2) = 2s \frac{ds}{dt} \), we can write:
\[ \frac{dv}{dt} = \frac{1}{2} \left( \frac{2as^2}{m} \right)^{-1/2} \cdot \frac{2a \cdot 2s \frac{ds}{dt}}{m} \]
This simplifies to:
\[ \frac{dv}{dt} = \frac{2a s \frac{ds}{dt}}{m \sqrt{\frac{2as^2}{m}}} \]

### Step 4: Find Tangential Force
The tangential force \( F_t \) acting on the particle is given by:
\[ F_t = m \frac{dv}{dt} \]
Substituting \( \frac{dv}{dt} \) from the previous step, we get:
\[ F_t = m \cdot \frac{2a s \frac{ds}{dt}}{m \sqrt{\frac{2as^2}{m}}} \]
This simplifies to:
\[ F_t = \frac{2as \frac{ds}{dt}}{\sqrt{\frac{2as^2}{m}}} \]

### Step 5: Find Centripetal Force
The centripetal force \( F_c \) required to keep the particle moving in a circle is given by:
\[ F_c = \frac{mv^2}{R} \]
Substituting \( v^2 \) from Step 2:
\[ F_c = \frac{m \cdot \frac{2as^2}{m}}{R} = \frac{2as^2}{R} \]

### Step 6: Calculate the Net Force
The net force \( F \) acting on the particle is the combination of tangential and centripetal forces:
\[ F = \sqrt{F_t^2 + F_c^2} \]
Substituting the expressions for \( F_t \) and \( F_c \):
\[ F = \sqrt{(F_t)^2 + \left(\frac{2as^2}{R}\right)^2} \]

### Final Expression
After simplifying, we can express the net force acting on the particle as:
\[ F = \sqrt{(2as)^2 + \left(\frac{2as^2}{R}\right)^2} \]