Math Puzzle Solution: How Tall Is the Lantern?

Pulley systems are fundamental simple machines designed to re-orient effort or provide mechanical advantage. Understanding the difference between a single fixed pulley and a movable pulley system explains how mechanical advantage reduces the force needed to lift heavy objects.

In the graphic featured above, two workers are attempting to lift equal weights of 110 lb:

Header: “WHICH MAN IS USING LESS FORCE?”

Setup Analysis:

• Man 1 (Yellow Hardhat): Pulling a rope attached over a single fixed overhead pulley attached to a 110 lb weight.

• Man 2 (White Hardhat): Pulling a rope threaded through a movable pulley supporting a 110 lb weight and an overhead fixed pulley.

Visual Attribution:

• Source: Courtesy of Bright Side.

In this guide, we break down the mechanical advantage formulas to determine which man uses less effort.

Strategy Overview

+--------------------------------------------------------+
|               Pulley Mechanical Advantage              |
+---------------------------+----------------------------+
                            |
                            v
+--------------------------------------------------------+
|  Man 1 Setup              |  Man 2 Setup               |
|  - 1 Fixed Pulley         |  - 1 Movable + 1 Fixed     |
|  - MA = 1                 |  - MA = 2                  |
|  - Force Required = 110 lb|  - Force Required = 55 lb  |
+---------------------------+----------------------------+
                            |
                            v
+--------------------------------------------------------+
|  Core Objective                                        |
|  - Compare tension and required pull force to identify |
|    the worker using less exertion.                     |
+--------------------------------------------------------+

Physics & Force Calculation

To calculate the pull force required for each worker:

1. Man 1 (Single Fixed Pulley):

   • A single fixed pulley changes the direction of the force but offers no mechanical advantage ($\text{MA}=1$).

   • The tension in the single rope supporting the weight equals the entire load.

   • Force Required: $110\text{ lb}$.

2. Man 2 (Movable Pulley + Fixed Pulley System):

   • The load is directly attached to a movable pulley supported by two vertical segments of rope.

   • One end of the rope is anchored to the overhead beam, and the other goes up over a fixed pulley to Man 2.

   • Because two rope segments share the $110\text{ lb}$ load equally, each segment carries $55\text{ lb}$ of tension.

   • The system provides a mechanical advantage of 2 ($\text{MA}=2$).

   • Force Required: 2110 lb​=55 lb.

Therefore, Man 2 is using half as much force as Man 1!

Force & System Comparison Table

Physics Principles, Workplace Health, and Practical Skills

Understanding mechanical advantage helps prevent physical overexertion and promotes safer lifting practices in manual labor and home projects.

Applying physics concepts directly supports joint and spinal health by minimizing physical strain during heavy lifting. Developing mechanical awareness benefits professionals in engineering and business operations, helping teams streamline personal finance equipment costs, optimize logistics, and maintain active problem-solving capabilities into retirement.

Key Takeaways on Pulley Systems

• Fixed Pulleys Change Direction: A fixed pulley allows you to pull downward instead of lifting upward, but it does not reduce the load weight.

• Movable Pulleys Reduce Force: Each supporting rope segment attached to a movable pulley distributes equal fractions of the total weight.

• Trade-off Between Force and Distance: While Man 2 uses half the force ($55\text{ lb}$), he must pull twice as much rope distance to lift the load to the same height.

Conclusion

Due to the mechanical advantage of 2 provided by the movable pulley system, Man 2 only needs to exert $55\text{ lb}$ of force compared to Man 1’s $110\text{ lb}$. Therefore, Man 2 is using less force!

Did you select Man 2 right away? Share your thoughts in the comments below!

FAQ Section

Which man is using less force?

Man 2 is using less force. He only exerts $55\text{ lb}$ of effort, whereas Man 1 must exert $110\text{ lb}$.

What is the advantage of a movable pulley?

A movable pulley divides the weight across multiple supporting rope segments, reducing the required effort force by half for every supporting strand.