Brake power measurement with a rope brake: Assume a rope brake dynamometer around a drum. Using effective radius Re = (D + d) / 2, the brake power (B.P.) is ________ (use W = dead load, S = spring balance reading, N = rpm).

Introduction / Context:
Brake power (B.P.) is the usable power delivered at the engine shaft. A rope brake dynamometer is a simple device to measure torque via a dead weight and a spring balance around a brake drum. Getting the sign and effective radius right is key to accurate power calculation.

Given Data / Assumptions:

• Rope brake around a drum of diameter D with rope diameter d.
• Dead load W (N) hangs on one side; spring balance reads S (N) on the other.
• Effective radius Re = (D + d) / 2 (in metres).
• Shaft speed N (rpm).

Concept / Approach:
Torque T equals net circumferential force times effective radius. The net force causing braking is (W − S). Power equals torque times angular speed. Angular speed in rad/s is 2 * pi * N / 60. Combine to obtain B.P.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional check: N is rpm, so 2 * pi * N / 60 gives s^-1; multiplying by torque (N·m) yields watts, confirming consistency.

Why Other Options Are Wrong:

• Using (W + S) overestimates torque; the opposing forces act in opposite directions so net is W − S.
• Dividing by Re instead of multiplying inverts torque dimensionally.
• Both incorrect sign and inverse radius compound the error.

Common Pitfalls:
Forgetting rope thickness in effective radius or misreading the spring balance direction. Also, ensure steady state (no acceleration) so that net torque equals measured braking torque.

Final Answer:
B.P. = (2 * pi * N / 60) * (W - S) * Re