| Problem statement Solution video https://youtu.be/EFm9SObfGKA |
DISCUSSION THREAD
Ask your questions here. Or, answer questions of others here. Either way, you can learn.
DISCUSSION and HINTS
There are two critical issues involved in solving this problem.
- One, is starting out with good free body diagrams. Recall from our lecture discussion that the Newton/Euler equations typically prefer INDIVIDUAL FBDs. Note this in the discussion of Step 1 below.
- Second, we need to be careful on signs in establishing our kinematics relating the acceleration of the two blocks to the angular acceleration of the pulley.
Recall the following four-step plan outline in the lecture book and discussed in lecture:
Step 1: FBDs
Draw individual free body diagrams (FBDs) of the pulley and the two blocks. A single FBD of the entire system will not be useful here. Let's say that we employ a "standard" xy-coordinate system (with x to the right and y up) for all FBDs.
Step 2: Kinetics (Newton/Euler)
You will need an Euler (moment) equation of the pulley about point C. In addition, write down the Newton equations for the y-motion for each of the two blocks. Write down all three equations using the sign convention discussed above.
Step 3: Kinematics
Here is where we need to be careful with signs. Write down the rigid body kinematics equations relating the accelerations of A and B back to the pulley center C:
aA = aC + α x rA/C - ω2rA/C
aB = aC + α x rB/C - ω2rB/C
This pair of equations will provide you with the relationships among aA, aB and α. Take note of the signs involved in these equations.
Step 4: Solve
From your equations in Steps 2 and 3, solve for the angular acceleration of the pulley and the accelerations of the two blocks.
Even though gravity is not drawn for the problem, shouldn't it still be taken into account? Or else theres no forces acting on the system.
This has been corrected. Thanks for pointing out this.
Should we be able to assume that the system is released from rest?
Do we assume the system starts at rest so that omega is 0?
You may assume that the system starts out from rest, if you want.
It turns out that the answer for the problem does NOT depend on the initial motion of the system. Starting from rest is as good as any other initial condition.
Which radius should we use when calculating Ic in this case?
The problem statement says to treat the drum as being a homogeneous disk. Therefore, the mass moment of inertia about C is given by:
I_C = Mr_O^2/2
In example 5.A.11 solved in class, we had to recognize that Aa and Ab must be opposite in sign. Do we have to apply this logic in this question as well? After doing kinematics, the expressions for Aa and Ab seem to already be opposite in sign.
Given that A and B are on opposite ends of the disk, it should follow that a_A and a_B are opposite in sign. Consider the equations in Step 3, where a_C is 0 and r_A/C and r_B/C are opposite in sign.
I think that since the expressions result in the opposite signs, the logic we utilized in class is already implied in the results of these equations. In this problem, we can also recognize that a_A and a_B must be different signs as the string is of fixed length, and since the results of the kinematics were that they were opposite, I think the logic is a good way to verify the validity of the results for this problem.
I would be careful with this assumption, it isn't that they are opposite sign. In the example discussed in class, we solved for the acceleration at a point on the end of the drum that. was above center, and it was positive i, and for Block B we said this became negative j direction because it was attached in such arrangement. In this homework, A and B do not have equal and opposite accelerations, they are different distances away from the center so their radial distance is different in the equation
While logically, they must have opposite signs, as has been consistent throughout the course this semester, it is usually unwise/unnecessary to make assumptions such as this. As you mentioned, after setting up the equations, they are opposite in sign, which logically makes sense. If you had not assumed this, you would still arrive at the same result. For this course, it is usually smart to let your assumptions verify your calculations, rather than the other way around.
I was wondering how to know when to assume omega is zero and when we cannot make that assumption?
For this problem, the initial motion (velocities) does not influence the result.
The problem doesn't specify the initial conditions, so I don't think they should impact the result. I assumed it started at rest, so omega = zero, but I would imagine any assumption works.
You are correct - initial conditions do not influence the results.
Pertaining to assuming directions of motion, what FBDs govern which equations and their signs, for example in the is problem do we need to assume Aa negative and Ab positive in their individual systems if we had assumed for positive angular acceleration. Conversely if we assume both to be positive what impact does this have on the equations/what signs will change in order to balance the change to arrive at the same answer?
How does assuming different directions for the acceleration of the blocks impact the signs in the equations of motion, and why must the final solution remain consistent regardless of the initial assumptions?
If you work out your cross products you can actually assume that you acceleration vectors will be positive and then obtain a negative number for one of them when the assumption turns out to be false. The critical component here is that while the tension forces of both cables are pointed in the -j direction, the r vectors pertaining to the moment each tension force creates are of opposite sign, allowing you to obtain different signs for each acceleration.
How could one model the forces acting on the two blocks? Obviously they're both acted on by gravity, but is there a tension force in either? I assume so one must go up for the disk to rotate. I just don't know how to express that tension in terms of other quantities.
You model both blocks as having gravity and tension acting on them. They do move, so the acceleration is not zero, however you can relate the acceleration of the blocks to the angular acceleration of the disk using the equation a_a = a_b + α x ra_b - ω^2*ra_b, where point a would be where the cable connects to the disk, and point b is the center of the disk. This gives you another equation you can then use along with your other equations to ultimately solve for the accelerations.
Actually, it's looking like I can solve without knowing the tension. The hints say to include the Newton Equations for A and B, but I think I found a way to solve everything without them.
Brian: Please be careful on this. I might be misunderstanding what you are saying here, but do not see how it would be possible to solve the problem without dealing with the forces of interaction between the blocks and the pulley. It is the tension in the two cables that drive the rotation of the pulley.
Do we only need to calculate one moment of inertia for this problem? Or do we need a different one for block B since it's on a different radius?
You should only need one moment of inertia for the pulley because you will only need to calculate moment about point. This is because the most strategic point to use the Newton-Euler equation is point C, and the only object whose moment of inertia is relevant to the problem (due to rotation about C) is the pulley.
**moment about one point.