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 **

**θ** = 2π rad

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 **

**θ** = 200 rad

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 **

**θ** = **6 π** rad

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.