What are the accelerations in the multiple-pulley problem?

[subhradeep mahata]

Homework Statement

The figure shows three blocks with their masses m,2m and 3m and two pulleys. Assuming that the pulleys and strings are massless and ideal, calculate the tension in lower string and acceleration in 3m block.

Homework Equations

The Attempt at a Solution

We have to find T. Now, let's assume that the 2m block moves downwards with acceleration a, and the m block move upwards with acceleration a as well.
Accordingly,
2mg-T=2ma and T-mg=ma
adding them, we get a=g/3
substituting it in the first equation, T=4mg/3.
Also, 2T=3m * acc
On solving, we get acc =8g/9, which is acceleration of 3m block.
But, my book says that the product of tension of the segment of the string and the corresponding acceleration of the block is constant. But my answers don't seem to match this criteria.
Neither do i know the meaning of the statement or how it has been derived, nor can I find any mistake in my method(if there is any). Kindly please address the issue.

 

[hutchphd]

assume that the 2m block moves downwards with acceleration a, and the m block move upwards with acceleration a as well.

The free pulley is not fixed in space.

 

[haruspex]

my book says that the product of tension of the segment of the string and the corresponding acceleration of the block is constant.

That would be astonishing. In the absence of other forces (as for the 3m block) the ratio would be constant, since that is the mass.
Please quote the whole passage, word for word, in case there is some misunderstanding.

 

[subhradeep mahata]

@hutchphd All right, so what are the changes to be made in the approach?

 

[subhradeep mahata]

@haruspex Actually it just says in a pink little box named "trick" that the tension times acceleration of a block is constant in case of a pulley. No other references.

 

[haruspex]

@haruspex Actually it just says in a pink little box named "trick" that the tension times acceleration of a block is constant in case of a pulley. No other references.

Then I suggest you pay no attention to it. So with zero tension, the block accelerates infinitely fast?
Sounds like a misprint.

 

[subhradeep mahata]

Then I suggest you pay no attention to it.

Ok, understood.
Now, am I correct in my procedure?

 

[haruspex]

@hutchphd All right, so what are the changes to be made in the approach?

The lengths of the strings being fixed, there is a kinematic relationship between the accelerations of the three blocks.
E.g., suppose block m moves down by x and block 2m moves down by y. How far must block 3m move?

 

[subhradeep mahata]

suppose block m moves down by x and block 2m moves down by y. How far must block 3m move?

Actually, I have serious issues in figuring out these kind of things. Any suggestions? Maybe i have difficulty in visualizing these.
Also, I kind of do not understand how the arrangement works. If m moves down, shouldn't 2m move up?

 

[haruspex]

If m moves down, shouldn't 2m move up?

Not necessarily if the pulley moves down too.

One way to figure these out is just to assume a linear relationship between the positions - in this case the horizontal position of m3 (x3 from pulley, say) and the vertical positions of the other two (x1, x2 from their pulley).
So you take the relationship to be x1+b x2+c x3=0.
Now consider moving just two blocks at a time. Hold m3 still and move m1 up a bit. How much does x2 change in relation to x1? What does that tell you about b?
Now do the same holding m2 still and seeing what happens to m3.

An alternative method is to ascribe lengths to the strings and write down equations representing that they are constant.

Bear in mind that x1 and x2 are positions relative to the pulley, so their second derivatives are accelerations relative to the pulley, not accelerations in the lab frame.

 

[ehild]

Homework Statement

The figure shows three blocks with their masses m,2m and 3m and two pulleys. Assuming that the pulleys and strings are massless and ideal, calculate the tension in lower string and acceleration in 3m block.

Homework Equations

The Attempt at a Solution

View attachment 237979
We have to find T. Now, let's assume that the 2m block moves downwards with acceleration a, and the m block move upwards with acceleration a as well.

Those two blocks have the same magnitude of acceleration, one down, the other up with respect to the hanging pulley. But the pulley also accelerates, so what are the accelerations with respect to the rest frame of reference?