Magnitude of the Projection of force "F" on the u-axis

In summary, the conversation was about finding the magnitude of the projection of a force vector parallel to a given line using two different methods. The first method is using the equation Fu = F * cos(theta) and the second method is using the dot product of F and the unit vector along the parallel line. There was confusion about the difference between the magnitude of the projection and the magnitude of the components of the force, and the professor explained that this is due to the axes not forming a 90 degree angle. It was noted that in a special case where the axes do form a 90 degree angle, the magnitude of the projection and the magnitude of the component along the U-axis can be solved using the same equation.
  • #1
mhrob24
53
9
Homework Statement
Find the magnitude of the force PROJECTED on the U and V axes
Relevant Equations
Fx = F * cos(theta) OR dot product of F and Ufu (unit vector along u-axis)
So I was watching a YouTube video preparing for a quiz on Wednesday, and I saw something that I would like clarification on. I'm pretty sure I understand what is being explained, but I just want to confirm.

1632173733274.png
The figure above is associated with the problem at hand. So I understand that to get the magnitude of the projection of a force vector that is parallel to some line, you can either use:

1. Fu = F * cos(theta)

OR

2. dot product of F and the unit vector along the parallel line (in this case, Uu)

So for this problem, using the first method, the two equations would be:

Fu = F * cos(45) AND Fv = F * cos(15)

I totally get that...what I am slightly confused about is what the professor says later in the video. He says:

"Note that the magnitude of the PROJECTION is NOT equal to the magnitudes of the COMPONENTS of this force. For that, you would need to use the parallelogram law and the law of sines".

Now, he's saying this because the axes here do not form a 90 deg angle? because for example, if the figure was like this:

1632174291203.png


and the question asked us to find the magnitude of the COMPONENTS of the force F along the u and v axes, then the equation to find Fu would still be the same : F * cos(45)...BUT, the equation to find Fv would NOT be the same as before. It would now be: F * sin(45).So this is just a special case where it just so happens that the magnitude of the PROJECTION and the magnitude of the COMPONENT along the U-axis can be solved using the same equation : F * cos(45) ?FYI, here is the video:

 
Last edited:
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  • #2
Projection is a linear function. The projection of a vector onto another is equal to the sum of the projections of its components in any orthogonal axis system.
 

Related to Magnitude of the Projection of force "F" on the u-axis

1. What is the definition of "Magnitude of the Projection of force "F" on the u-axis"?

The magnitude of the projection of force "F" on the u-axis is the length of the component of the force vector that lies in the same direction as the u-axis.

2. How is the magnitude of the projection of force "F" on the u-axis calculated?

The magnitude of the projection of force "F" on the u-axis can be calculated using the dot product formula: |F| cosθ, where |F| is the magnitude of the force vector and θ is the angle between the force vector and the u-axis.

3. What is the significance of the magnitude of the projection of force "F" on the u-axis?

The magnitude of the projection of force "F" on the u-axis represents the amount of force acting in the direction of the u-axis. This can be useful in analyzing the overall effect of a force on an object or system.

4. How does the angle between the force vector and the u-axis affect the magnitude of the projection of force "F" on the u-axis?

The magnitude of the projection of force "F" on the u-axis is directly proportional to the cosine of the angle between the force vector and the u-axis. This means that as the angle increases, the magnitude of the projection decreases and vice versa.

5. Can the magnitude of the projection of force "F" on the u-axis be negative?

Yes, the magnitude of the projection of force "F" on the u-axis can be negative. This occurs when the angle between the force vector and the u-axis is greater than 90 degrees, indicating that the force is acting in the opposite direction of the u-axis.

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