# Numerical problem on rotational work | Numerical on work done by the torque

In this post, we will find the equation of the work done by Torque. (In other words, we will work with the equation of rotational work.) Then we solve a few numerical problems using this formula of rotational work.

## Equation of the work done by Torque | equation of rotational work

Here we will find out the equation of the work done by Torque. Let’s consider a scenario where we apply a force of F N to the edge of a tire to get a car moving. Then, what work do we do over s m of travel?

We will use this equation of work done: W = Fs

We can also think about this force rotationally. In the case of you applying force to the edge of a tire to get a car moving, the distance s equals the radius multiplied by the angle through which the wheel turns, s = θ r , so you get this equation:
W = Fs = F θ r
But the torque, τ , equals Fr in this case.

So we can easily write: W = Fs = F θ r = Fr θ = τ θ

work done by Torque = Torque x angular displacement
W = τ θ (equation of rotational work)

## Numerical problems on work done by the torque

1 ) If you apply a torque of 500.0 N-m to a tire and turn it through an angle of 2π radians, what work have you done?

Solution:

τ = 500.0 N-m

W = τ θ = 500 x 2π J = 3140 Joule

2 ) How much work do you do if you apply a torque of 6.0 N-m over an angle of 200 radians?

Solution:

τ = 6 N-m

W = τ θ = 6 x 200 J = 1200 Joule

3 ) You’ve done 20.0 J of work turning a steering wheel. If you’re applying 10.0 N-m of torque, what angle have you turned the steering wheel through?

Solution:

W = 20 J

τ = 10.0 N-m

W = τ θ
=> θ = W / τ = 20/10 rad = 2 rad.

4 ) How much work do you do if you apply a torque of 75 N-m through an angle of 6 π radians?

Solution:

τ = 75 N-m

W = τ θ = 75 x 6 π J = 1413 Joule

5 ) You’ve done 350 J of work turning a bicycle tire. If you’re applying 150 N-m of torque, what angle have you turned the wheel through?

Solution:

W = 350 J

τ = 150 N-m

W = τ θ
=> θ = W / τ = 350/150 rad = 2.33 rad.

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