What Is the Acceleration Constraint in This Pulley System?

[ezp0004]

Homework Statement

Hello. I need help with a problem that deals with the acceleration constraint of a system (URL below is to an image of the system):

http://s3.amazonaws.com/answer-board-image/e8ee7c74-664f-4220-a394-fc2b3d5bc269.jpeg

The questions asked in the problem are as follows:

1) Find the acceleration constraint for this system. It should be a single equation relating a1y, a2y, and a3y. Hint: yA is not a constant.

2) Find an expression for the tension in string "A".

3) Using m1 = 2.5kg, m2 = 1.5kg, and m3 = 4kg, find a1y, a2y, and a3y.

Homework Equations

I need to find equation for a1y in terms of a2y and a3y. I know that a1y should be equal to -a2y. However, I do not understand how a3y relate to these. Newton's second law (force=mass*acceleration) should be the only other equation needed to solve this problem.

The Attempt at a Solution

Other than recognizing that a1y = -a2y, I am complete stuck on this problem. I would appreciate any and all help that you may be able to provide. Thanks in advance.

 

[Andrew Mason]

Homework Statement

Hello. I need help with a problem that deals with the acceleration constraint of a system (URL below is to an image of the system):

http://s3.amazonaws.com/answer-board-image/e8ee7c74-664f-4220-a394-fc2b3d5bc269.jpeg

The questions asked in the problem are as follows:

1) Find the acceleration constraint for this system. It should be a single equation relating a1y, a2y, and a3y. Hint: yA is not a constant.

2) Find an expression for the tension in string "A".

3) Using m1 = 2.5kg, m2 = 1.5kg, and m3 = 4kg, find a1y, a2y, and a3y.

Homework Equations

I need to find equation for a1y in terms of a2y and a3y. I know that a1y should be equal to -a2y. However, I do not understand how a3y relate to these. Newton's second law (force=mass*acceleration) should be the only other equation needed to solve this problem.

The Attempt at a Solution

Other than recognizing that a1y = -a2y, I am complete stuck on this problem. I would appreciate any and all help that you may be able to provide. Thanks in advance.

I am not clear on the variables you are using. Please explain what you mean by a1y, a2y, a3y etc. That is an acceleration x distance.

Assume that m2=m3 and m1=m2+m3. What would the tension be on the ropes? Then change the masses slightly and analyse the change.

AM

 

[ezp0004]

I am not clear on the variables you are using. Please explain what you mean by a1y, a2y, a3y etc. That is an acceleration x distance.

Assume that m2=m3 and m1=m2+m3. What would the tension be on the ropes? Then change the masses slightly and analyse the change.

AM

I believe that a1y, a2y, and a3y are referring to the accelerations of blocks m1, m2, and m3 in the 'y' direction. Other than that, all variables should be represented in the figure that I linked.

Using m2=m3 and m1=m2+m3, does this meant that the tensions on the ropes for m2 and m3 are the same and that the tension for the rope connected to m1 is twice that of the tension for ropes connected to m2 and m3? I'm not really sure what you are trying to get at. Since the masses are given in the third question (m1+m2=m3 but m1 does not equal m2).

 

[Andrew Mason]

I believe that a1y, a2y, and a3y are referring to the accelerations of blocks m1, m2, and m3 in the 'y' direction. Other than that, all variables should be represented in the figure that I linked.

Using m2=m3 and m1=m2+m3, does this meant that the tensions on the ropes for m2 and m3 are the same and that the tension for the rope connected to m1 is twice that of the tension for ropes connected to m2 and m3? I'm not really sure what you are trying to get at. Since the masses are given in the third question (m1+m2=m3 but m1 does not equal m2).

If m1 = (m2+m3) and m2=m3, the system is in balance. Doing a free body diagram you can see that: 1. the tension in the rope through A is just m₂g and the tension in the rope through B is m₁g = (m2+m3)g

Now change it so that m2 and m3 differ by $\Delta m$ and analyse that. (Hint: think of m2 on each side with an added mass $\Delta m$ added to the one on the right (m3).

AM

 

[rwisz]

Hello, thank you for reaching out for help with this problem. The image and questions provided are very helpful in understanding the situation. The first thing to note is that this system is a classic example of a pulley system, where the strings and pulleys are massless and frictionless. This means that the tension in the strings is constant throughout the system.

To answer your first question, the acceleration constraint for this system can be found using Newton's second law, as you mentioned. We know that the net force acting on each mass is equal to its mass times its acceleration. In this system, there are three masses (m1, m2, and m3) and three unknown accelerations (a1y, a2y, and a3y).

We can start by looking at m1. The only force acting on m1 is the tension in string A, which we can call T. This tension is pulling m1 upwards, so it will have a positive acceleration in the y-direction (a1y). Using Newton's second law, we can write:

T = m1*a1y

Next, let's look at m2. The only force acting on m2 is the tension in string B, which is pulling m2 downwards. This means that m2 will have a negative acceleration in the y-direction (a2y). We can write this using Newton's second law as:

-T = m2*a2y

Lastly, we can look at m3. The only force acting on m3 is the tension in string C, which is pulling m3 downwards. This means that m3 will also have a negative acceleration in the y-direction (a3y). We can write this using Newton's second law as:

-T = m3*a3y

Now, we can combine these three equations to eliminate the tension (T) and solve for the acceleration constraint:

m1*a1y = -m2*a2y = -m3*a3y

This can also be written in a more simplified form as:

a1y = -a2y = -a3y

This is the acceleration constraint for this system, and it relates all three accelerations together.

Moving on to the second question, we can use this acceleration constraint to find an expression for the tension in string A. Since we know that a1y = -a2y, we can substitute this into the equation we found