Block on movable wedge pulley system

[Vibhor]

Homework Statement

What is the relationship between " acceleration of block with respect to wedge" and "acceleration of wedge " ?

Homework Equations

The Attempt at a Solution

If the wedge moves by x units , then the string gets loosened/shortened at three places . The top two are x units .The last one in red is xcosθ .The total effect is the block moves down by a distance (2 + cosθ)x .

So, if acceleration of the wedge is 'a' then acceleration of block with respect to wedge a_B,W = (2+cosθ)a .

But , this is incorrect .

Where am I getting it wrong ?

Many Thanks

 

[Yash123]

sorry the ans should accof wedge=2acc of blockwrtwedge

 

[Vibhor]

sorry the ans should accof wedge=2acc of blockwrtwedge

Could you explain your reasoning as well as point out flaw in my reasoning ?

@haruspex , @ehild please see this problem .

 

[Yash123]

If to displace the wedge by let's say x meters then the string which you marked with green will displace x similarly the blue one would move x too hence total disp in x dir would be equal = 2x this would result in the increase of length in the red string as the string is inextensible...
what you did is you assumed the the string connected to block will definitely move x cos(theta) that's not correct...

 

[TomHart]

If the wedge moves to the right a distance of x, I agree that the two horizontal rope segments decrease in length by x (as you have shown). But isn't it true that whatever length the rope decreased in those segments has to be the same as the increase in the length of the sloped segment of rope? The rope has to stay the same length, true? So if you lost a total of x+x=2x length of rope in the horizontal segments, don't you have to gain that same amount in the sloped segment?

 

[ehild]

Homework Statement

What is the relationship between " acceleration of block with respect to wedge" and "acceleration of wedge " ?

Homework Equations

The Attempt at a Solution

If the wedge moves by x units , then the string gets loosened/shortened at three places . The top two are x units .The last one in red is xcosθ .The total effect is the block moves down by a distance (2 + cosθ)x .

The horizontal part of the sting gets shorter by 2x. So the part along the wedge gets longer by 2x. If the velocity of the wedge is V, the relative velocity of the block with respect to the wedge is v_r=2V. You should consider the x component of that relative velocity. The the x component of velocity of the block with respect to the ground is the sum of V and the x component of v_r.