| Problem statement Solution video https://youtu.be/5OfZvgywTwU |
DISCUSSION THREAD
Ask or answer questions here. Try it. You will learn either way.
DISCUSSION and HINTS
Although end B of the cable is moving with a constant speed, block A is not. Can you see this in the animation below?
Write down the length L of the cable in terms of the two position variables sA and sB. Since the cable does not stretch, set dL/dt = 0. (Take care in performing the derivative dL/dt.) This will give you an equation that relates the speeds of blocks A and B.
Does your answer show that vA < vB (a result that is indicated in the above animation)? In fact, your result should show that 2vA < vB: does it?
Im a bit confused as to how to interpret SB due to the way its drawn, should we be using it as such or is there something we should do to better interpret its actual length
I think you can use Pythagorean theorem to get the cable length, and then use the inextensible cable formula. Deriving that once will get the V_a that is asked.
If you are referring to how to use sB given the cable is CB, you need to do Pythagorean theorem using d, sB to fid cable length CB (leave in variables because you will be taking the derivative of it)
Are we finding the speed at the particular instant in time shown in the drawing? In that case, I think you would use SB to find the length of the cable between C and B.
Yes it's the speed at the instant shown in the drawing, so you would use the pythagorean theorem in order to find the length of the cable between C and B
Do we need to use unit vectors in our dL/dt equation? I was able to obtain an answer by using only the scalar values of va and vb.
The question asks for only the speed (a scalar). There is no need to express your answer as a vector using unit vectors.
Since block A will only move in the positive or negative x direction your answer will only have i as a unit vector. However the question asks specifically for speed so I think the result will be the magnitude of the velocity so therefore you do not need an answer in terms of unit vectors.
Would S_dot_A have to be equivalent to negative V_A since the S_A vector is is the negative x-direction?
The sign on sA_dot applies only to whether sA is increasing or decreasing:
* If sA_dot > 0, then sA is INCREASING. Since sA is defined as being positive to the left in this problem, then an increasing sA means that A would be moving to the left.
* If sA_dot < 0, then sA is DECREASING. Since sA is defined as being positive to the left in this problem, then a decreasing sA means that A would be moving to the right.
In short, the sign of sA_dot is not tied to the "x" direction; instead, it is tied to the definition of sA.
Can we assume the length against the pulleys are negligible when finding L?
Yes. For pulley D, that amount of cable wrapped around the pulley is truly a constant. For pulley C, the amount of cable wrapped around the pulley is close to constant if the radius of the pulley is small.
Yes I agree, because the angle of wrap at point C would be dependent on the location of block B, we would want to assume that length to be constant. Just noting that the radii of the pulley C is small and therefore the cable length around that pulley is negligible.
Does it matter where we choose the orientation for the direction? I assumed no but wanted to double check if this class specified standard for orientation.
I believe you do not have to re orient from what is given. You can just use the directions for Sa and Sb given on the diagram.
How do you find L?
You do not need to know L. L is constant; therefore, all you need to use is dL/dt =0.
In the animation, block B is moving at a constant speed. However, block A is not. You can see this because initially block A is not moving and then as block B goes further down, block A accelerates and moves closer to the point where the cable is wrapped around. Therefore, block A does not have constant speed, as it is continually accelerating as block B moves.
I am not sure if I am correctly creating the function for L and differentiating it. Currently I have a function L in terms of s_A, the constant d, s_B, and a constant for the pulley wrap that will become zero after differentiation. Should (dL)/(dt) be in terms of s_A_dot, s_B_dot, and s_B?
In this system, does the direction of motion of block A depend on whether block B is moving up or down?
Yes, it would. Although I believe in this system, without an external force on block A or B aside from gravity, it would be impossible for block B to be moving upwards.
In this case, the diagram arrow for v_b indicates that block B is moving downwards.
This is a statement of a problem in kinematics. Therefore, we are not provided any information about what external forces that are applied.
For whatever forces are applied, the kinematics relations rely only on the fact that the cable remains taut during the motion.
Does the sign of the answer matter in this problem, since it is asking for speed, not velocity? I thought s_dot_A was the velocity but not the speed.
You are correct - your answer is a magnitude, which does not have a sign associated with it.
To be precise, sA_dot is neither a velocity nor a speed. sA_dot = ds_A/dt. The "sign" on this term indicates whether sA is increasing or decreasing.
Do we leave our answer in terms of i?
Actually not. The problem asked for speed, which is the magnitude of the velocity vector.
Im confused about the speed of block A.
Since it is a speed, I assume it will be just a scalar value?
You are correct. If you are asked for speed, provide a scalar value since speed is the magnitude of the velocity vector.