The kinetic energy k of a particle moving along a circle of radius R depends on the distance covered. It is given as K.E. = as² where a is a constant. The force acting on the particle is

The correct option is 2as(1+ s²/R²)⁰.⁵.

To find the total force acting on a particle of mass m moving along a circle of radius R, we must consider both the tangential force and the centripetal (radial) force acting on it.

Let's break down the solution step-by-step:

Step 1: Express the kinetic energy and find the centripetal force
The kinetic energy of the particle is given as:

$\mathrm{K.E.}=a{s}^2$

Since kinetic energy is also defined as $\mathrm{K.E.}=\frac{1}{2}m{v}^2$ (where v is the velocity of the particle), we can write:

$\frac{1}{2}m{v}^2=a{s}^2$

This gives the expression for $m{v}^2$ as:

$m{v}^2=2a{s}^2$

The centripetal force ($F_c$) acting on a particle in circular motion is:

$F_c=\frac{m{v}^2}{R}$

Substituting the value of $m{v}^2$ into this formula, we get:

$F_c=\frac{2a{s}^2}{R}$

Step 2: Find the tangential force
The tangential force ($F_t$) is related to the rate of change of kinetic energy with respect to the distance covered s.
Using the work-energy theorem, the tangential force is given by:

$F_t=\frac{d\left(\mathrm{K.E.}\right)}{ds}$

Differentiating $\mathrm{K.E.}=a{s}^2$ with respect to s, we obtain:

$F_t=\frac{d}{ds}\left(a{s}^2\right)=2as$

Step 3: Calculate the total net force
Since the centripetal force and the tangential force act perpendicular to each other, the total net force (F) is the vector sum of these two components:

$F=\sqrt{F_c^2+F_t^2}$

Substitute the values of $F_c$ and $F_t$ into the equation:

$F=\sqrt{{(\frac{2a{s}^2}{R}}^2+{\left(2as\right)}^2}$

Factor out ${\left(2as\right)}^2$ from under the square root:

$F=\sqrt{{\left(2as\right)}^2\cdot (\frac{s^2}{R^2}+1)}$

Simplifying the expression gives:

$F=2as\sqrt{1+\frac{s^2}{R^2}}$

Writing the square root as a fractional exponent of 0.5:

$F=2as{(1+\frac{s^2}{R^2})}^{0.5}$

Thus, the force acting on the particle is indeed 2as(1 + s²/R²)⁰.⁵.