Power transmitted by a rotating shaft\n\nA shaft rotates at angular speed ω (in rad/s) while transmitting a torque T (in N·m). What is the mechanical power developed?

Introduction / Context:
Converting rotational motion to power is routine in machine design. Power relates torque and angular speed; getting the correct form and units prevents large sizing errors in shafts, couplings, and motors.

Given Data / Assumptions:

• Torque T in newton-metre (N·m).
• Angular speed ω in radians per second (rad/s).
• Rigid shaft; steady transmission (no transients or losses considered).

Concept / Approach:
Instantaneous mechanical power equals torque multiplied by angular velocity. The radian is dimensionless, so units reduce to watt = N·m/s. No additional 2π factor is used when ω is already in rad/s.

Step-by-Step Solution:
Definition: Power P = work rate.In rotation, infinitesimal work dW = T * dθ; dividing by dt gives dW/dt = T * dθ/dt.Since dθ/dt = ω, we get P = T * ω.Therefore, the correct power in SI units is P = T * ω watts.

Verification / Alternative check:
If speed is given in revolutions per minute (N rpm), then ω = 2πN/60 rad/s, yielding P = T * (2πN/60). This explains why a 2π factor appears only when starting from rpm rather than ω.

Why Other Options Are Wrong:
(b) double counts the 2π when ω is already in rad/s; (c) and (d) invert the relationship; (e) mixes constants applicable to rpm, not to ω.

Common Pitfalls:
Confusing rpm and rad/s; dropping unit consistency; using T in N·mm without converting to N·m.

Final Answer:
T * ω watts