Systems of pulleys – third system (textbook machine theory) For the third system of pulleys with n pulleys, what is the velocity ratio (VR) expressed in terms of n?

Introduction / Context:
Velocity ratio (VR) is a key performance parameter of simple machines. For classical systems of pulleys (first, second, and third systems), VR depends on how the ropes are reeved and how many pulleys support the moving block.

Given Data / Assumptions:

• Third system of pulleys as defined in standard machine theory texts.
• Number of pulleys = n.
• Ideal kinematics without slip; friction ignored for VR.

Concept / Approach:
In the third system, the free (moving) block is supported by 2n rope segments effectively sharing the load (for the canonical arrangement), which makes the distance moved by the effort twice the distance moved by the load for each supporting segment count n, producing VR = 2n.

Step-by-Step Solution:

Verification / Alternative check:
Compare with other systems: first system VR = n; second system VR = 2^n. The third system’s linear dependence VR = 2n contrasts with the exponential dependence of the second system.

Why Other Options Are Wrong:

• n: corresponds to the first system.
• n^2: not a standard VR in these classic arrangements.
• 2n − 1: sometimes arises in variant counting errors; not the canonical result.

Common Pitfalls:
Miscounting supporting rope segments or confusing the three classical systems’ formulas.

Final Answer:
2n