Rotational work and power

Rotational work becomes torque times angle

Torque transfers rotational energy through angular displacement just as force transfers translational energy through distance. Motor power is high when torque or angular speed is high.

Use the rotational differential

For a small rotation dθ, the point of application moves dr perpendicular to the torque-producing component, so dW = F_t r dθ = τ dθ. Integrating gives W = ∫τ dθ; for constant torque W = τΔθ. Divide by time: P = τω.

A motor supplies 4 N·m through 20 revolutions, Δθ = 40π rad. Work = 4(40π) = 503 J. At 1800 rpm, ω = 188.5 rad/s and power is P = 4(188.5) = 754 W.

Torque and power trade off

At fixed power, τ = P/ω, so gearing can multiply torque only by reducing angular speed. Losses make output power smaller than input power.

Work it: A 500 W motor runs at 2500 rpm. Find ideal shaft torque. If a pump needs 3 N·m, calculate the maximum speed before ignoring losses.