Work Done (Angular): When a torque $T$ acts through an angle $\theta$ , the work done is given by $W = T\theta$ . This is the rotational analog of linear work ( $W = Fs$ ).

Power Transmission: Power $P$ is the rate at which work is done. For a rotating system transmitting torque $T$ at angular velocity $\omega$ , the power is $P = T\omega$ .

Key Relationship: Power is zero if there is no rotation ( $\omega = 0$ ), even if torque is present. A rotating machine is required to convert torque into power.

Example Calculation

Problem: A machine rotates at 3000 rpm and consumes 5 kW of power. Calculate the torque at the shaft.

Solution:

Convert power to Watts: $P = 5 \text{ kW} \times 1000 \text{ W/kW} = 5000 \text{ W}$ .
Convert rotational speed from rpm to rev/s: $n = \frac{3000}{60} = 50 \text{ rev/s}$ .
Use the power-torque relationship: $P = 2\pi n T$ .
Solve for torque:

T = \frac{P}{2\pi n} = \frac{5000}{2\pi \times 50} \approx 15.9 \text{ Nm}

Additional Relations

The power formula can be expressed directly in terms of rotational speed in rpm ( $n_{rpm}$ ):

P = 2\pi \left(\frac{n_{rpm}}{60}\right) T \quad \Rightarrow \quad P \approx 0.105 \times n_{rpm} \times T

where $P$ is in Watts (W) and $T$ is in Newton-meters (Nm).

References

Original Source

Source: engineeringtoolbox.com