What is the acceleration of the blocks in a pulley constraint problem?

AI Thread Summary
The discussion revolves around calculating the acceleration of blocks in a pulley system with smooth surfaces. Participants emphasize the importance of correctly applying the constraint equations for the lengths of the strings, noting that the sum of accelerations must equal zero. The initial attempts at solving the problem lead to incorrect results, prompting a reevaluation of the equations used. It is clarified that the acceleration of block B coincides with the center of mass due to the specific mass values and string constraints in this scenario. The conversation highlights the complexity of such problems and the necessity for precise formulation of equations.
Vriska
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Homework Statement


Need to find the acceleration of the blocks, all surfaces are smooth : https://drive.google.com/file/d/0B5bF4yPQOejoazhWeEgySmtfSnc/view?usp=drivesdk

Homework Statement


Need to find the acceleration of the blocks

Homework Equations


a1 + a2 + a3 =0 (acceleration of strings lengths must sum to zero)

F = ma

The Attempt at a Solution


Acceleration of topmost string = a_a(acceleration of block a)
- a_b (is this right? I feel like this should be right but idk why)Acceleration of middle string also must be the same

And acceleration of bottom string is simply T (where T is tension in the string, assuming mass is 1)

3T = 4a_b (for block b), F - 2T = 2a_a, T = a_c.

Solving all this gives me the wrong answer : /I'm suspecting somethings wrong with the first step, is it?

Homework Equations


a1 + a2 + a3 =0 (acceleration of strings lengths must sum to zero)

F = ma

The Attempt at a Solution


Acceleration of topmost string = a_a(acceleration of block a)
- a_b (is this right? I feel like this should be right but idk why)Acceleration of middle string also must be the same

And acceleration of bottom string is simply T (where T is tension in the string, assuming mass is 1)

3T = 4a_b (for block b), F - 2T = 2a_a, T = a_c.

Solving all this gives me the wrong answer : /I'm suspecting somethings wrong with the first step, is it?
 
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If the problem is asking for the acceleration of the 3 block system as a whole, the solution is a lot simpler than you think.
 
PhanthomJay said:
If the problem is asking for the acceleration of the 3 block system as a whole, the solution is a lot simpler than you think.

What would that mean? The acceleration of the centre of mass of the system? I'm not sure whether that fact helps
 
Yes, what is the acceleration of the center of mass of the system (if that is what the problem is asking...please post the question exactly as given)
 
PhanthomJay said:
Yes, what is the acceleration of the center of mass of the system (if that is what the problem is asking...please post the question exactly as given)

It's asking the individual acceleration of the blocks unfortunately
 
Vriska said:
It's asking the individual acceleration of the blocks unfortunately
Oh. In which case, it seems like the best way to proceed is to take free body diagrams of each block and solve for the accelerations of each block with respect to the ground.
 
PhanthomJay said:
Oh. In which case, it seems like the best way to proceed is to take free body diagrams of each block and solve for the accelerations of each block with respect to the ground.

Isn't that what i did? Isnt giving me the right answer
 
Hello ,

Next time please try and upload image on PF using upload button .

1506069109341-2-jpg.jpg


Your constraint equation isn't correct . Can you express the length of the string in terms of the displacements of the three blocks .
 

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conscience said:
Hello ,

Next time please try and upload image on PF using upload button .

View attachment 211520

Your constraint equation isn't correct . Can you express the length of the string in terms of the displacements of the three blocks .

Hi

I had significant difficulty in uploading the image so i had to use drive : /. In terms of displacement you say? I'm finding it difficult but so far I've got this : change in top length of string = x_a(displacement of a) - x_b, change in middle length of string =x_a-x_b + x_c, change in bottom string in just x_c, they should all sum to zero right?
 
  • #10
Vriska said:
Hi

I had significant difficulty in uploading the image so i had to use drive : /. In terms of displacement you say? I'm finding it difficult but so far I've got this : change in top length of string = x_a(displacement of a) - x_b, change in middle length of string =x_a-x_b + x_c, change in bottom string in just x_c, they should all sum to zero right?

No .

Place your coordinate system to the right of block B . Left direction is taken positive. Let A at any instant be at distance x1 from origin , B at x2 , C at x3 .

Use a pen and a paper , think carefully and tell what is the length of the string in terms of x1,x2,x3 ?
 
  • #11
conscience said:
No .

Place your coordinate system to the right of block B . Left direction is taken positive. Let A at any instant be at distance x1 from origin , B at x2 , C at x3 .

Use a pen and a paper , think carefully and tell what is the length of the string in terms of x1,x2,x3 ?

Is it 2(x1-x2)+ x3 = l ? You'd have subtract that little bit of string attached to p1 and p2 so 2(x1-x2)+ x3 = l+2q , so i differentiate twice and I'll get constraints. Thank you
 
  • #12
Vriska said:
Is it 2(x1-x2)+ x3 = l ?

No . Aren't both blocks B and C moving :wink: ?
 
  • #13
conscience said:
No . Aren't both blocks B and C moving :wink: ?

Right, x2 - x3 then?
 
  • #14
Vriska said:
Right, x2 - x3 then?

Should that be x3 - x2 ?
 
  • #15
conscience said:
Should that be x3 - x2 ?

Yes! So now my equations are :

2(a1-a2)+a3-a2= 0,
1 - 2T =2a1, (assyming F = 1 and m = 1
3T =4a2,
T =-a3 (minus because its going opposite to coordinate system)

Does this look alright? I'm not getting the right answer. (except for block b whose accleartion is equal to that of the centre of mass, is this a coincidence?)
 
  • #16
Vriska said:
T =-a3 (minus because its going opposite to coordinate system)

What is the direction of tension on C ? In F= Ma , you put a sign on the force , not on the acceleration term :smile:
 
  • #17
conscience said:
What is the direction of tension on C ? In F= Ma , you put a sign on the force , not on the acceleration term :smile:

Ah that makes more sense. Rest of the equations are alright right?
 
  • #18
Vriska said:
I'm not getting the right answer.

What are you getting ?
 
  • #19
conscience said:
What are you getting ?

a1 =13/42, according to the book its 4/21 (at F=1, m =1)
 
  • #20
Vriska said:
a1 =13/42, according to the book its 4/21 (at F=1, m =1)

I am getting a1 = 13F/42m , a2 = F/7m , a3= -4F/21 .

May be the answer key is wrong .

What is the name of the book ?
 
  • #21
conscience said:
I am getting a1 = 13F/42m , a2 = F/7m , a3= -4F/21 .

May be the answer key is wrong .

What is the name of the book ?

Thank you for your help, I've been struggling with these constraints problems for quite sometime, now it makes sense . It looks like my books actually wrong, its a nondescript college issued workbook. Oh btw is the fact that accleartion of block b = acceleration of centre of mass a complete coincidence?

Edit : it isn't! When i change b to 5, the acceleration of cm is 1/8, as expected! Why is this happening?
 
  • #22
Vriska said:
Oh btw is the fact that accleartion of block b = acceleration of centre of mass a complete coincidence?

Mathematically they have same value in this problem.But that is due to string constraint and for the particular masses as given in the problem .

The CM of the system has a different location than that of block B , but due to string constraint and specific value of the masses as given in the problem, their accelerations are turning out to be same .Quite interesting !
 
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