Angular Velocities using Relative Velocities

In summary, the problem involves determining the angular velocities of links OB and BA in a slider-crank mechanism, given that the slider's velocity is 7.5 m/s to the left. One approach is to find the relative positions of points A and O with respect to point B, but this only works for the specific instant shown in the diagram. Another approach is to look at velocity components, where the two ends of a rigid rod must have the same velocity component along the rod. Ultimately, it is necessary to consider the angles as variables and differentiate to solve for the angular velocities. Using symbols instead of numbers can make the reasoning easier to follow.
  • #1
sakau2007
7
0

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.
 

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  • #2
it would be better if you write down the complete question, it may create confusion otherwise!
 
  • #3
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.
 
  • #4
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.
 
  • #5
sakau2007 said:
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.
 
  • #6
haruspex said:
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.
 
  • #7
sakau2007 said:
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.
 
  • #8
sakau2007 said:
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.
 
Last edited:

1. What is the definition of angular velocity?

Angular velocity is the measure of how fast an object is rotating or turning around a central point. It is usually measured in radians per second or degrees per second.

2. How is angular velocity related to relative velocities?

Angular velocity is the ratio of an object's angular displacement to the time it takes to rotate. Relative velocities refer to the velocity of an object with respect to another object or reference frame. In the case of angular velocities, it refers to the velocity of an object with respect to the central point around which it is rotating.

3. What is the formula for calculating angular velocity using relative velocities?

The formula for calculating angular velocity using relative velocities is: ω = v/r, where ω is the angular velocity, v is the tangential velocity, and r is the distance between the object and the central point.

4. How does the direction of angular velocity relate to the direction of relative velocities?

The direction of angular velocity is always perpendicular to the plane of rotation, while the direction of relative velocities depends on the direction of the tangential velocity. If the tangential velocity is in the same direction as the radius, the angular velocity is positive (counterclockwise rotation). If the tangential velocity is in the opposite direction to the radius, the angular velocity is negative (clockwise rotation).

5. Can you give an example of how to calculate angular velocity using relative velocities?

Sure! Let's say a point on a rotating wheel has a tangential velocity of 10 m/s and the distance from the center of the wheel to the point is 0.5 m. To calculate the angular velocity, we can use the formula ω = v/r = 10 m/s ÷ 0.5 m = 20 radians per second. This means that the point on the wheel is rotating at a rate of 20 radians per second around the central point.

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