Assuming acceleration is constant

[cipotilla]

I have posted 2 problems, 12.126 and 12.72. I have read in the book that you can assume the acceleration remains constant when the forces applied are constant.

In problem 12.126 that is the case, the weights of the blocks and the forces P and T remain constant so I assumed that the acceleration of the blocks were also constant and I solved the problem successfully.

But in problem 12.72, I obviously cannot assume the acceleration is constant, why not? I am neglecting friction and the weights of the blocks are contant, the tension in the cord is also constant, isn't it? What is the difference between these problems?

 

[Doc Al]

But in problem 12.72, I obviously cannot assume the acceleration is constant, why not? I am neglecting friction and the weights of the blocks are contant, the tension in the cord is also constant, isn't it? What is the difference between these problems?

As block A moves, the angle that the cord makes will change. This will affect the forces involved. (And don't assume the tension in the cord remains the same as the angle of the cord changes.)

 

[cipotilla]

As the angle changes, I thought that the components of the tension force would change but the resusltant T wouldn't. Why is this incorrect?

 

[Doc Al]

Why do you think that?

Did you actually write down the equations and solve for the tension? Does it depend on theta?

 

[cipotilla]

I solved problem 12.72

I solved the problem and once again, I got the wrong answer because I have the wrong sign for the acceleration. I attached the problem and books solution as well as my solution.

 

[Doc Al]

I solved the problem and once again, I got the wrong answer because I have the wrong sign for the acceleration.

I didn't go through your entire solution (I will if need be), but I note that r is defined as the distance between pulley and block A. So a positive acceleration for r means that the block accelerates to the left! I think that's at the root of the problem.

I note that you have a_A = - a_B/cos(theta); that tells me right there that you are (or should be) measuring a_A with positive values going to the left. Clearly the constraint is such that when block B moves down, block A moves to the right. (Tricky stuff, I know. But hang in there!)

 

[cipotilla]

Ok, I solved the problem correctly, the signs make sense to me now. Going back to my original confusion as to why in P12.72 acceleration is not constant and in P12.126 it is. I've learned that in P12.126, acceleration is constant because the cables are all positioned at a constant angle, they hang vertically and remain vertical. Whereas in P12.72 that is not he case, the angle theta changes and so the tension in the cables either increases or decreases. Since the forces are not constant the the acceleration is not constant. (Its the sum of the forces that has to remain constant, right?) If in P12.126 the blocks were angled and that angle was changing with time, then we would not have a constant acceleration. Cables are tricky.