The pulley in (Figure 1) represents different pulleys that are attached – Free 85A

The pulley in (Figure 1) represents different pulleys that are attached with outer radius and inner radius indicated in the table. The horizontal rope is pulled to the right at a constant linear speed that is the same in each case, and none of the two separate ropes slips in its contact with the pulley.Rank these scenarios on the basis of the linear speed of the block.

Answer

Ranking Pulley Systems by the Linear Speed of the Block

Pulley systems with varying inner and outer radii introduce interesting dynamics in rotational and translational motion. In this analysis, we compare different pulley configurations to determine which setup allows the block to move faster based on constant linear input speed of a rope pulled horizontally.

📌 What Remains Constant Across Scenarios?

• The horizontal rope is pulled at a constant linear speed in all cases.
• No slipping occurs between the ropes and pulleys.
• Each pulley has an outer radius (R) and inner radius (r) (where the block rope is wound).

🔍 Understanding the Setup

The pulley consists of two sections:

• The outer rim is connected to the horizontal rope (input).
• The inner groove is connected to a hanging block via a rope (output).

As the outer rope is pulled, it causes the pulley to rotate. The inner rope causes the block to rise or fall based on the rotational motion.

🧠 Key Concept: Tangential Speed and Radius

The angular velocity ω of the pulley is governed by the outer rope:

v_outer = R × ω → ω = v_outer / R

This same angular velocity drives the inner rope:

v_block = r × ω = r × (v_outer / R)

Thus, the linear speed of the block is:

v_block = (r / R) × v_outer

📈 Interpretation of the Formula

The speed at which the block moves depends directly on the ratio of the inner radius to the outer radius:

• Greater r/R ratio → faster block movement
• Smaller r/R ratio → slower block movement

🔢 Ranking Method

To rank different pulleys by block speed, compute or compare:

r / R for each pulley

The higher the r/R, the higher the linear speed of the block.

📌 Final Ranking (Hypothetical Example)

🎯 Conclusion

To determine how fast a block moves in pulley systems with inner and outer radii, focus on the ratio r / R. The larger the ratio, the greater the block’s linear speed. This analysis helps in understanding compound pulley systems in physics and engineering applications.