A solid circular shaft runs at 150 rpm and transmits a steady torque of 150 kg·m. What is the horsepower (HP) transmitted?

Introduction / Context:
Power transmitted by a rotating shaft is directly related to torque and angular speed. Converting to horsepower in engineering units requires careful use of consistent unit systems.

Given Data / Assumptions:

• Torque T = 150 kg·m (kilogram-force metre).
• Speed N = 150 rpm.
• Use metric engineering relation between torque, speed, and HP.

Concept / Approach:
Power P (in HP) for torque in kg·m and speed in rpm is:P_HP = (2 * π * N * T) / 4500This follows from P (kgf·m/s) = 2π N T / 60 and 1 HP = 75 kgf·m/s.

Step-by-Step Solution:
P_HP = (2 * π * 150 * 150) / 4500.Compute numerator: 2 * π * 22500 = 45000 π.Divide by 4500 → P_HP = 10 π.

Verification / Alternative check:
Convert to watts using 1 HP ≈ 735.5 W if needed; numeric value 10π HP ≈ 31.416 HP, consistent with P = τ ω.

Why Other Options Are Wrong:

• π, π², 1/π are not consistent with the torque–speed product and HP conversion factor.

Common Pitfalls:

• Using N in rad/s without converting units, or mixing N·m with kg·m.

Final Answer:
10 π