Power transmitted by a rotating shaft: A shaft runs at N revolutions per minute (rpm) under a steady torque T. What is the power P transmitted?

Introduction / Context:
Power in a rotating shaft equals torque times angular speed. Converting rotational speed in rpm to angular velocity in rad/s is the key step. This relation is used widely in machine design and power transmission calculations.

Given Data / Assumptions:

• Torque T is constant.
• Shaft rotational speed N in rpm.
• Neglect losses so mechanical power equals T times angular velocity.

Concept / Approach:

Angular speed ω (rad/s) = 2 π N / 60. Then power P (watts) = T * ω = T * (2 π N / 60).

Step-by-Step Solution:

Verification / Alternative check:

If N = 60 rpm (1 rps), then P = 2 π T, matching T times 2 π rad/s.

Why Other Options Are Wrong:

They have missing or extra factors of 2 or 60, or incorrect dimensional form.

Common Pitfalls:

Forgetting to convert rpm to rad/s; mixing minutes and seconds leading to a factor-of-60 error.

Final Answer:

P = 2 π N T / 60