Find the velocity of the hanging block if the velocities of the free ends of the rope are as indicated in the figure.

To determine the velocity of the hanging block in a pulley system, we need to analyze the relationship between the velocities of the different segments of the rope and the block itself. This involves understanding how the motion of the rope affects the motion of the block. Let's break this down step by step.

Understanding the System

In a typical pulley system, when one end of the rope is pulled, it causes the other end to move as well. If we denote the hanging block's velocity as \( v_b \) and the velocities of the free ends of the rope as \( v_1 \) and \( v_2 \), we can establish a relationship based on the geometry of the system.

Establishing Relationships

Assuming the rope is inextensible and the pulley is frictionless, the total length of the rope remains constant. If one side of the rope moves up, the other side must move down by the same amount to maintain this constant length. This leads us to a fundamental relationship:

• If the rope is pulled at one end, the block's velocity will be influenced by the velocities at the ends of the rope.
• For example, if both ends of the rope are moving, we can express the relationship as:

v_b = (v_1 + v_2) / 2

Applying the Concept

Let’s say you have a scenario where the velocities of the free ends of the rope are given as \( v_1 = 3 \, \text{m/s} \) and \( v_2 = 1 \, \text{m/s} \). Plugging these values into our equation gives:

v_b = (3 \, \text{m/s} + 1 \, \text{m/s}) / 2 = 4 \, \text{m/s} / 2 = 2 \, \text{m/s}

Visualizing the Motion

Think of it like this: if you pull one end of a rope, the other end has to move in response. If you pull one side faster than the other, the block will move at a velocity that is the average of the two. This average accounts for the fact that the rope is not stretching, and thus the motion is interconnected.

Final Thoughts

In summary, to find the velocity of the hanging block, you need to consider the velocities of the free ends of the rope and apply the relationship derived from the constraints of the system. This method can be applied to various pulley systems, making it a versatile tool in mechanics. Always remember to check the direction of the velocities as well, as they can affect the signs in your calculations.