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

[Vriska]

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?

 

[PhanthomJay]

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.

 

[Vriska]

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

 

[PhanthomJay]

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)

 

[Vriska]

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

 

[PhanthomJay]

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.

 

[Vriska]

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

 

[conscience]

Hello ,

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

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

 

[Vriska]

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?

 

[conscience]

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 ?

 

[Vriska]

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

 

[conscience]

Is it 2(x1-x2)+ x3 = l ?

No . Aren't both blocks B and C moving ?

 

[Vriska]

No . Aren't both blocks B and C moving ?

Right, x2 - x3 then?

 

[conscience]

Right, x2 - x3 then?

Should that be x3 - x2 ?

 

[Vriska]

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?)

 

[conscience]

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

 

[Vriska]

What is the direction of tension on C ? In F= Ma , you put a sign on the force , not on the acceleration term

Ah that makes more sense. Rest of the equations are alright right?

 

[conscience]

I'm not getting the right answer.

What are you getting ?

 

[Vriska]

What are you getting ?

a1 =13/42, according to the book its 4/21 (at F=1, m =1)

 

[conscience]

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 ?

 

[Vriska]

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?

 

[conscience]

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 !