Systems of pulleys – first system (textbook machine theory) For the first system of pulleys containing n pulleys, what is the velocity ratio (VR)?

Introduction / Context:
The velocity ratio (VR) of a machine indicates the ideal mechanical advantage in distance terms. In the first system of pulleys, several pulleys are combined so that the number of supporting segments equals the number of pulleys in the moving block.

Given Data / Assumptions:

• First system of pulleys as defined in standard machine theory.
• Number of pulleys = n.
• Ideal ropes (no slip, negligible stretch) and friction ignored for VR.

Concept / Approach:
For the first system, the moving block is supported by n rope segments. The effort end must move n units of length for the load to rise 1 unit, so VR equals the number of supporting segments, i.e., VR = n.

Step-by-Step Solution:

Verification / Alternative check:
Check limiting cases: n = 1 gives VR = 1 (a single fixed pulley changes direction only). For n = 2, VR = 2, and so on—matching textbook examples.

Why Other Options Are Wrong:

• n^2: not applicable in first system; confusion with other compound mechanisms.
• 2n or 2n − 1: correspond to other arrangements or counting mistakes.

Common Pitfalls:
Confusing the first system with the second (VR = 2^n) or third (VR = 2n) systems; always identify the reeving pattern first.

Final Answer:
n