Why Does Mass M Move at u/cosθ in a Pulley System?

[Prannoy Mehta]

Homework Statement

In the arrangement as shown in the figure below, the ends P and Q of an inextensible string move downwards with uniform speed u. Pulleys A and B are fixed. Mass M moves upward with a speed:

a. 2ucosθ
b. u/cosθ
c. 2u/cosθ
d. ucosθ

Homework Equations

None, I am not sure whether it falls under general physics or homework section. [/B]

The Attempt at a Solution

The strings move with velocity u in their respective directions , thus block will move with velocity 2ucosθ. Since each of the two strings has velocity in vertical direction as ucosθ .

But the answer says it is u/cosθ . Which seems right as when you take components of this total velocity of block M in directions of strings, it gives components as u, but those 2 components don't add up to 2ucosθ. What is the mistake here ? Why are they not adding up to ucosθ?

Thank you for all the support and help.

 

[Orodruin]

The strings are not moving along their transversal direction. They are getting shorter at a given rate. Based on trigonometry, you can find how much shorter the strings become when the mass moves up a certain distance.

 

[TSny]

The point where the strings meet moves upward at the unknown speed v. What must be the component of this velocity in a direction parallel to one of the sloping strings?

 

[Prannoy Mehta]

A small correction to the problem the final answer is u/cosθ. It's very weird.

I din't get how it not being transversal effects this relation, clearly. If I suppose manage to stop time, after recording the instantaneous speeds of both the strings(Since they are at the same angle and move with the same velocity, I will assume it to be u). They both will move with u in their respective directions, and their vertical components will add up, and result in 2ucosθ. And also, if I take the angle approaching 90. The final answer yields in the velocity of it going upward with a velocity tending to infinity, which is not true. And at 90, it should be zero in reality, which the equation I got perfectly relates whereas the answer yields it as infinity, which by any standards is wrong.

 

[Orodruin]

With your assumptions, the strings would be moving apart, which does not happen. I suggest you follow the steps proposed in my earlier post. When the strings are near horizontal, a small change in the string length will result in a large change in the height. This is why the result goes to infinity.

 

[Prannoy Mehta]

It yielded me the right answer, sir u/cosθ. Thank you.
It was just a doubt in my mind why my assumption failed over there. I understood it now.