What is the acceleration of block C in a system of connected blocks and pulleys?

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Homework Help Overview

The discussion revolves around a problem involving three blocks (A, B, and C) connected by pulleys, where the objective is to determine the acceleration of block C after the system is released from rest. The blocks have specified masses and coefficients of friction, and the problem is set within the context of classical mechanics.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the relationship between the motions of the blocks and how to express their accelerations in relation to one another. There are attempts to draw free body diagrams and identify forces acting on each block. Questions arise regarding the effects of friction and the constraints imposed by the rope's length.

Discussion Status

The discussion is ongoing, with participants exploring various interpretations of the problem. Some guidance has been offered regarding the relationships between the blocks' accelerations and the forces involved. However, there is no explicit consensus on the correct approach or solution yet.

Contextual Notes

Participants note the complexity introduced by static versus kinetic friction and the implications of the blocks' potential motion. There is uncertainty regarding the correct application of frictional forces when block A is assumed to be motionless.

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Homework Statement


Three blocks A,B and C, whose masses are 9 kg, 9 Kg and 18 kg respectively as shown in the figure, are released from rest. The co-efficient of friction between the block A and the horizontal surface is 0.5 and The co-efficient of friction between the block B and the horizontal surface is 0.5. Find the acceleration of the block C just after the release. [Take g=10 m/s^2]. All strings and pulleys are ideal.


Homework Equations





The Attempt at a Solution


I can't figure out how the motion of block B will affect the motion of the other two blocks. I am unable to form any equation here.

Any help is appreciated. Thanks!
 

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Start by drawing a diagram showing all the forces on all the blocks. Use symbols, not numbers, in order to see what is going on (plugging in numbers early obscures the physics).

Block C is the key: without it, nothing will happen. When C is added, it will (maybe, depending on its weight and the other forces) pull B to the right.
 
Pranav-Arora said:

Homework Statement


Three blocks A,B and C, whose masses are 9 kg, 9 Kg and 18 kg respectively as shown in the figure, are released from rest. The co-efficient of friction between the block A and the horizontal surface is 0.5 and The co-efficient of friction between the block B and the horizontal surface is 0.5. Find the acceleration of the block C just after the release. [Take g=10 m/s^2]. All strings and pulleys are ideal.

Homework Equations


The Attempt at a Solution


I can't figure out how the motion of block B will affect the motion of the other two blocks. I am unable to form any equation here.

Any help is appreciated. Thanks!
Just take the accelerations of the blocks as aA, aB, and aC, and figure out how they are related. The length of the rope is constant.

ehild
 
tms said:
Start by drawing a diagram showing all the forces on all the blocks. Use symbols, not numbers, in order to see what is going on (plugging in numbers early obscures the physics).

Block C is the key: without it, nothing will happen. When C is added, it will (maybe, depending on its weight and the other forces) pull B to the right.

I have drawn the free body diagrams (see attachment) But still, I am not sure what to do next?
 

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ehild said:
Just take the accelerations of the blocks as aA, aB, and aC, and figure out how they are related. The length of the rope is constant.

ehild

I have seen some examples in the past where this way of solving problems is used but I have never been able to use this way anywhere.
 
So far, so good. Now you see if you can figure out what some of the forces are in terms of the others, and you use F = ma.
 
tms said:
So far, so good. Now you see if you can figure out what some of the forces are in terms of the others, and you use F = ma.

2T-μmg=maB
T-μmg=maA
Mg-T=MaC

I have four variables and three equations. I am stuck here. :confused:
 
How are the accelerations related?
Think: if A moves to the right by ΔxA and B moves to the right by ΔxB how much the horizontal part of the rope becomes shorter? And the vertical path longer?


ehild
 
ehild said:
How are the accelerations related?
Think: if A moves to the right by ΔxA and B moves to the right by ΔxB how much the horizontal part of the rope becomes shorter? And the vertical path longer?


ehild

Does it become shorter by ΔxA+2ΔxB?
 
  • #10
Pranav-Arora said:
Does it become shorter by ΔxA+2ΔxB?

Yes, for the horizontal part. And what is the downward displacement of C then?

ehild
 
  • #11
ehild said:
Yes, for the horizontal part. And what is the downward displacement of C then?

ehild

Wouldn't that be again equal to ΔxA+2ΔxB? :confused:
 
  • #12
Yes, it is. And what about accelerations? They are the second time derivatives of displacements.

ehild
 
  • #13
ehild said:
Yes, it is. And what about accelerations? They are the second time derivatives of displacements.

ehild

aC=aA+2aB

Do I need to solve these four equations?
 
  • #14
Yes. Express all accelerations from the other three and substitute into aC=aA+2aB. You can use the numerical data it is easier.

ehild
 
  • #15
ehild said:
Yes. Express all accelerations from the other three and substitute into aC=aA+2aB. You can use the numerical data it is easier.

ehild

Equation (i): T-45=9aA
Equation (ii): 2T-45=9aB
Equation (iii): 180-T=18aC

Multiplying (ii) with 2 and adding it with (i):
5T-135=9aC ...(iv)

Multiplying (iii) with 5 and adding it with (iv)
5(180-27)=99aC
Solving this equation doesn't give me the right answer. :(
 
  • #16
Find all a-s and T. Show the results.

ehild
 
  • #17
ehild said:
Find all a-s and T. Show the results.

ehild

I only need aC and my last equation doesn't give the right answer.
 
  • #18
Find aA. If it is negative, the initial assumption that it moves was
not valid.

ehild
 
  • #19
ehild said:
Find aA. If it is negative, the initial assumption that it moves was
not valid.

ehild

Yep, aA comes out to be negative. :)

What next? Assume that A moves left?
 
  • #20
Such problems with ropes and friction, are tricky. You need to take into account that a rope can not push, and the friction can prevent some blocks from motion. You need to determine all accelerations and tensions to be sure that neither of them is negative.

ehild
 
  • #21
Pranav-Arora said:
Yep, aA comes out to be negative. :)

What next? Assume that A moves left?

No, A can not accelerate to the left, as the rope can not push. But it can be motionless. Assume that aA=0, and solve the equations.

ehild
 
  • #22
ehild said:
You need to take into account that a rope can not push, and the friction can prevent some blocks from motion. You need to determine all accelerations and tensions to be sure that neither of them is negative.

Here are the values I got:
aA=(-5/11) m/s^2

aB=45/11 m/s^2

aC=85/11 m/s^2

T=450/11 m/s^2
 
  • #23
ehild said:
No, A can not accelerate to the left, as the rope can not push. But it can be motionless. Assume that aA=0, and solve the equations.

ehild

Assuming aA=0, T=45.
Substituting T in equation (iii),
180-45=18aC
This time too, I don't get the right answer. :(
 
  • #24
ehild said:
What is the "right answer"?
ehild

##\frac{70}{9} m/s^2##
And what I get is ##\frac{15}{2} m/s^2##.
 
  • #25
The problem is that you can not make the force of friction equal to mu mg, if A does not move. It is static friction then, what you do not know. Ignore the equation for aA. T is not equal to 45.

ehild
 
  • #26
ehild said:
The problem is that you can not make the force of friction equal to mu mg, if A does not move. It is static friction then, what you do not know. Ignore the equation for aA. T is not equal to 45.

ehild

Thanks ehild! I have got the right answer now. :smile:

One more thing, how much time should a problem like this would require? I got this question in my test paper and was totally clueless at that time.
 
  • #27
Practice. The numbers are simple, so you can solve it in a few minutes. And that is a typical problem, with blocks and pulleys, connected by string.

You need to get the skill.

ehild
 

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