Angular Velocities using Relative Velocities

[sakau2007]

Homework Statement

I have attached a picture of the diagram of the problem. I am supposed to determine the angular velocities of links OB and BA if the velocity of the slider at the instant shown is 7.5 m/s to the left.

Homework Equations

Va = Vb + Vab (Velocity of A is equal to the Velocity of B plus the Velocity of A relative to B)
There may be more equations I need. I am not sure.

The Attempt at a Solution

First, I'm attempting to get the relative positions of A with respect to B and O with respect to B (this is how my teacher solved the problem).

Doing this I get:
Rab = 2.25cos(30) i - 2.25sin(30) j = 1.95i - 1.13j (I feel pretty certain this is correct).
Rob = -1cos(55) - sin(55) = -0.57i - 0.82j (My teacher has positive numbers for this which confuses me because it seems to me that the position of O relative to B on both the X and Y axes is further left and down so it should be negative. Why is my teacher not putting negative signs here?

Va = Vb + Vab
-7.5 i m/s = ?

And I am stuck here as well. Not sure how to proceed. Thanks for the help.

 

[NihalSh]

it would be better if you write down the complete question, it may create confusion otherwise!

 

[sakau2007]

The question in its entirety is:
"For the slider-crank mechanism pictured, determine the angular velocities of links OB and BA if the velocity of the slider at the instant shown is 7.5 m/s to the left."

The answers are ωab = -1.92 k rad/s and ωob = 6.52 k rad/s but I'm very confused on how to get to them.

 

[haruspex]

If you are going to do it by figuring out the relative positions, it's no use doing that only for the instant in the diagram. You need equations that give the relative positions at any angle. Then you can look at how the positions change as the angle changes.
Another approach is to look at velocity components. E.g. the two ends of a rigid rod must have the same velocity component along the rod.

 

[NihalSh]

The question in its entirety is:
"For the slider-crank mechanism pictured, determine the angular velocities of links OB and BA if the velocity of the slider at the instant shown is 7.5 m/s to the left."

The answers are ωab = -1.92 k rad/s and ωob = 6.52 k rad/s but I'm very confused on how to get to them.

the question gives the information that the length of the link won't change. So, try and relate "vertical component" and "horizontal component" of length to the length of the link and horizontal distance to the slider.

You should be able to imagine, if slider moves to the left that means horizontal distance decreases hence that would mean vertical distance should increase.

If you relate x and y of point B properly, you should be able to figure it out.

 

[sakau2007]

If you are going to do it by figuring out the relative positions, it's no use doing that only for the instant in the diagram. You need equations that give the relative positions at any angle. Then you can look at how the positions change as the angle changes.
Another approach is to look at velocity components. E.g. the two ends of a rigid rod must have the same velocity component along the rod.

Would this not be a correct way to go about solving it:

-7.5 i m/s = (ωob x rob) + (ωab x rab)

Then I could use the values for rob and rab that I found in the original post. Then we would have a system of solvable equations with two unknown (ωob and ωab) set equal to their respective i- and j- components. However, I'm still confused why my teacher is getting 0.57 i + 0.82 j for the value of rob when I feel like it should be -0.57 i - 0.82 j.

 

[NihalSh]

Would this not be a correct way to go about solving it:

-7.5 i m/s = (ωob x rob) + (ωab x rab)

Then I could use the values for rob and rab that I found in the original post. Then we would have a system of solvable equations with two unknown (ωob and ωab) set equal to their respective i- and j- components. However, I'm still confused why my teacher is getting 0.57 i + 0.82 j for the value of rob when I feel like it should be -0.57 i - 0.82 j.

Like haruspex previously mentioned, its angular quantities you have to consider. And I mentioned that you can go about finding relation, by using height of B (OB and BA's height would be same) and length of OA, but the catch is to do it in terms of angular quantities.

 

[haruspex]

Would this not be a correct way to go about solving it:

-7.5 i m/s = (ωob x rob) + (ωab x rab)

Then I could use the values for rob and rab that I found in the original post. Then we would have a system of solvable equations with two unknown (ωob and ωab) set equal to their respective i- and j- components. However, I'm still confused why my teacher is getting 0.57 i + 0.82 j for the value of rob when I feel like it should be -0.57 i - 0.82 j.

As I wrote, you need to consider the angles as variable. Rewrite your previous equation with the angle as a variable and differentiate.
Also, it will be much easier for others to follow your reasoning if you avoid using the actual numbers. Invent symbols for all of the distances etc. and write your equations in terms of them.