Question

The kinetic energy k of a particle moving along a circle of radius R depends on the distance covered s as k=a$s^2$ where a is a constant. The force acting on the particle is

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

According to given problem $\frac{1}{2}$ m $v^2$ = a$s^2$ = v = s $\sqrt{\frac{2a}{m}}$
So $a_r$ = $\frac{v^2}{R}$ = $\frac{2a{s}^2}{mR}$ ..(i)
Further more as $a_t$ = $\frac{dv}{dt}$ = $\frac{dv}{ds}$.$\frac{ds}{dt}$ = v$\frac{dv}{ds}$ ..(ii)
(By chain rule)
Which in light of equation (i) i.e v= s$\sqrt{\frac{2a}{m}}$ yields

$a_t$ = [s $\sqrt{\frac{2a}{m}}$][$\sqrt{\frac{2a}{m}}$] = $\frac{2as}{m}$ ..(iii)
So that a = $\sqrt{{a_r}^2+{a_t}^2}=\sqrt{{\left[\frac{2a{s}^2}{mR}\right]}^2+[{\frac{2as}{m}}^2}]$
Hence a = $\frac{2as}{m}$$\sqrt{1+\left[{\frac{s}{R}}^2\right]})\therefore$ F = ma = 2as $\sqrt{1+{\left[\frac{s}{R}\right]}^2}$