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?
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AP = 2 π N T / 60
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BP = π N T / 60
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CP = 2 π N T / 30
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DP = π N T
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EP = T / (2 π N)
Answer
Correct Answer: P = 2 π N T / 60
Explanation
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:
Convert N rpm to rad/s: ω = 2 π N / 60.Power: P = T * ω = 2 π N T / 60.Units: T (N·m) * ω (rad/s) ⇒ N·m/s (W).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