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The kinetic energy of a particle moving along a circle of radius R depends on the distance covered s as K.E. = αs2, where α is a constant. Find the force acting on the particle as a function of s.

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K.E. = αs2

or, 1/2 mv2 = αs2

⇒ v2 = {2α s2}/{m}    ...(i)

Differentiating both side w.r.t. 't' we have

⇒ dv/dt = {2αs}/{m} = aT  (Tangential component of acceleration)

Now centripetal acceleration is

ac = v2/R = {2αs2}/{mR}  [using equation (i)]

Net acceleration of the particle is given by

Now, force acting on the particle is given by,

F = ma = m x {2αs}/{m} √{1 + (s2/R2)} 

= 2 α s√{1 + s2/R2}

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