# Q. A particle of mass 1 unit is moving along the circumference of a circle of radius 3 units with a variable speed `v=6t - t^2.` Find the rate of work done on the body at `t=1.` a) 20 b) 30 c) 36...

Q. A particle of mass 1 unit is moving along the circumference of a circle of radius 3 units with a variable speed `v=6t - t^2.` Find the rate of work done on the body at `t=1.`

a) 20

b) 30

c) 36

d) zero

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### 1 Answer

The particle of mass 1 unit is is moving along the circumference of a circle of radius 3 units with a variable speed given by v = f(t) = 6t - t^2. The acceleration of the particle traveling along a circle of radius r is a = v^2/r = (6t - t^2)^2/3.

The force acting on the particle is m*a = m*(6t - t^2)^2/3

At any moment of time, the force on the particle is in a direction perpendicular to its displacement as the displacement is in a direction tangential to the circle and the force acts towards the center.

As a result the work done on the particle is 0.

**The correct answer is option D.**

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Only in the **uniform circular motion** the force is directed towards the center of the circle, and thus is not doing work. If the motion is non uniform there will be two forces acting on the mass. First force is towards the center and its **instantaneous** value is

`F_n =m*V^2/R`

This normal force is not doing any work (because it is perpendicular to the instant speed).

The second force is tangential to the trajectory and its value is

`F_t = m*a =m*dV/dt =m*(6-2t)`

**This force is doing work because is parallel to the speed.**

The work done is

`W = F_t(1)*v(1) =1*(6-2)*(6-1) =20J`

**Therefore the correct answer is a) 20 J**

**Sources:**