A system is shown in the figure. A man standing on the block is pulling the rope. Velocity of the point of string in contact with the man is 2 m/s downwards. The velocity of the block will be [assume that the block does not rotate]
My approach:
Can you please tell why the above method is wrong, sir?
Call the string velocity at the man $v$ downwards, and the block velocity $u$ upwards, then it is true that $v = 2u$. But the question says the man pulls the string at 2m/s and the man is moving because he is standing on the block. So that means $v + u = 2$, not $v = 2$.
No, wait, is the string velocity relative to the man or not?
@JohnRennie Ok sir. Then $u=2/3$ is also incorrect. Also I don't know whether the speed mentioned in the question (2 m/s) is relative to the ground or the man. How did you infer that it's wrt. the man, sir?
@JohnRennie I don't know sir. I've directly quoted the question from my book. We need to conclude from the available data. Possibly in a way that will yield the correct answer - 2 m/s.
@JohnRennie Ok sir. I hope the problem is with the horizontal part of the string where I've written $2v$ instead of $v$. But I've given some thought and since the pulley itself moves upward with a velocity $v$, I added them both to get $2v$. Could you tell why is this incorrect?
So suppose we write the length of the string in terms of the position of the block $L = f(x)$. Now we differentiate wrt time to get $dL/dt = f'(x)$ and $dL/dt$ is the velocity of the string.
Now, the red dashed lines are fixed. The black dashed lines change position as the block moves. Let's write down the equation for the length of the string measured from the point where it is tied to the block to the point where it intersects the lower red line.
Then the horizontal bit between the two leftmost pulleys doesn't change. I'' going to ignore that because when we differentiate any constants will disappear.
Particle Accelerator
Particle Accelerator