Work done by couple

Before going through work done by couple, make sure to check torque and couple meaning from here.

Consider a wheel of radius ‘r’ rotating about an axis through a center due to the application of a couple of magnitude ‘F’ at points A and B.

Let’s consider a wheel covers angular displacement ‘\theta‘ such that the wheel moves from A to A as shown in the figure.

Work done in displacing from A to A is given by: W_{AA^'} = F \times AA^'

W_{AA^'} = F \times  \theta r    \qquad [ \theta = \frac{l}{r} \quad \rightarrow \quad \theta = \frac{AA^'}{r}]

Similarly, Work done in displacing from B to B is given by:

W_{BB^'} = F \times \theta r

The total work done by a couple can be calculated by adding work done by both forces.

    \[W = W_{AA^'} \quad +  \quad W_{BB^'}\]

    \[W = F \times \theta r + F \times \theta r\]

    \[ W = 2 F \theta r\]

    \[W = F \times (2r) \theta\]

    \[ \boxed{W = \tau \theta} \]

Where \tau is the torque due to a couple which is calculated by the product of either force with the distance between them.

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