Question

The kinetic energy K of a particle moving along a circle of radius R depends on the distance covered s as K = as², where a is a constant. The force acting on the particle is ______.

Options

• 2a`"s"^2/"R"`

• `2"as"(1+"s"^2/"R"^2)^(1//2)`

• 2as

• 2a`"R"^2/"s"`

Answer 1

The kinetic energy K of a particle moving along a circle of radius R depends on the distance covered s as K = as², where a is a constant. The force acting on the particle is `bbunderline(2"as"(1+"s"^2/"R"^2)^(1//2))`.

Explanation:

According to given problem

`1/2"mv"^2="as"^2` ⇒ v = `"s"sqrt(2"a")/"m"`

So a_R = `"v"^2/"R"=(2"as"^2)/"mR"`       ...(i)

Further more as a_t = `"dv"/"dt"="dv"/"ds"."ds"/"dt"="v""dv"/"ds"`        ...(ii)

(By chain rule)

Which in light of equation (i) i.e. v = s`sqrt((2"a")/"m")` yields

a_t = [s`sqrt((2"a")/"m")`] [`sqrt((2"a")/"m")`] = `(2"as")/"m"`         ...(iii)

So that a = `sqrt("a"_"R"^2+"a"_"t"^2)=sqrt([(2"as"^2)/"mR"]^2+[(2"as")/"m"]^2)` 

Hence a = `(2"as")/"m"sqrt(1+["s"/"R"]^2)`

∴ F = ma = `2"as"sqrt1+["s"/"R"]^2`