The system shown in figure is released from rest at t=0. Block A hits the pulley at t=x and B hits the at t=y. If pulley and strings are ideal the find the ratio of x and y .

To solve the problem of finding the ratio of times \( x \) and \( y \) when blocks A and B hit the pulley, we need to analyze the motion of both blocks under the influence of gravity. Let's break down the scenario step by step.

Understanding the Setup

Assume we have two blocks, A and B, connected by a string that passes over a pulley. Block A is hanging vertically, while Block B is also hanging but on the opposite side of the pulley. Both blocks are released from rest at the same time, and we want to determine when each block reaches the pulley.

Key Concepts

• Acceleration due to Gravity: Both blocks will accelerate downwards due to gravity, which is approximately \( 9.81 \, \text{m/s}^2 \).
• Equations of Motion: We can use the equations of motion to determine the distance each block travels and the time taken to reach the pulley.

Analyzing Block A

Let’s denote the distance Block A travels to reach the pulley as \( d_A \). Since it starts from rest, we can use the equation:

\( d_A = \frac{1}{2} g t_A^2 \)

Here, \( t_A \) is the time taken for Block A to reach the pulley, and \( g \) is the acceleration due to gravity. Rearranging gives us:

\( t_A = \sqrt{\frac{2d_A}{g}} \)

Analyzing Block B

Now, let’s consider Block B. If we denote the distance Block B travels to reach the pulley as \( d_B \), we can use a similar equation:

\( d_B = \frac{1}{2} g t_B^2 \)

Here, \( t_B \) is the time taken for Block B to reach the pulley. Rearranging gives us:

\( t_B = \sqrt{\frac{2d_B}{g}} \)

Finding the Ratio of Times

Now, we need to find the ratio of \( t_A \) to \( t_B \). This can be expressed as:

\( \frac{t_A}{t_B} = \frac{\sqrt{\frac{2d_A}{g}}}{\sqrt{\frac{2d_B}{g}}} \)

Since \( g \) and the factor of \( 2 \) cancel out, we have:

\( \frac{t_A}{t_B} = \sqrt{\frac{d_A}{d_B}} \)

Conclusion

Thus, the ratio of the times \( x \) and \( y \) when blocks A and B hit the pulley is given by:

\( \frac{x}{y} = \sqrt{\frac{d_A}{d_B}} \)

In summary, to find the ratio of the times when each block hits the pulley, you need to know the distances each block travels. This ratio will depend on the specific distances \( d_A \) and \( d_B \) that you have in your problem setup. If both blocks are of equal mass and start from the same height, then \( d_A \) and \( d_B \) will be equal, leading to a ratio of 1. However, if they are at different heights, you will need to substitute those values into the equation to find the exact ratio.