Projection of Resultant Force: Why Do We Need to Add B-Component?

[flgdx]

Homework Statement

What's the projection of the resultant force onto an said axis?

Homework Equations

Sine Law, Cosine Law

The Attempt at a Solution

For problem 2/20, the projection onto the b axis was found by multiplying the resultant force by the cosine of 30 degrees. Why does this method not work for problem 2/4? Why do we need to add in the b-component of the resultant force? The questions are uploaded as pictures along with the solutions. Thank you!

 

[BvU]

Hello fl,

Nice pictures. I can imagine you don't want to copy all the typesetting and the pictures. But I can't really imagine what it is that you could want from a possible helper !

[edit] Ahhh, the problem was in the solution... Goes to show how well posts are being read by helpful helpers who just got out of bed . Fortunately haru has had his lunch already !

 

[haruspex]

For problem 2/20, the projection onto the b axis was found

No, it is the projection onto the a axis. The references to P_b in the equations are wrong.
Also, in 2/20, R is not shown as being a resultant of other forces; it is the only force.

Why do we need to add in the b-component of the resultant force?

It is not adding a b component of R. The calculation of the projection onto the b axis here does not use R at all. It goes back to the two constituent forces and adds their b components.
This should equal the b component of R. To check that, you would need to find the angle R makes to the b axis. It seems to be a bit less than 30 degrees.

 

[flgdx]

No, it is the projection onto the a axis. The references to P_b in the equations are wrong.
Also, in 2/20, R is not shown as being a resultant of other forces; it is the only force.

It is not adding a b component of R. The calculation of the projection onto the b axis here does not use R at all. It goes back to the two constituent forces and adds their b components.
This should equal the b component of R. To check that, you would need to find the angle R makes to the b axis. It seems to be a bit less than 30 degrees.

Yes sorry! I meant to say the a axis on problem 2/20. Also for 2/4, I was talking about the b-component of the 100N Force. Sorry I made this post before going to bed hence the number of mistakes. Let me clear up my question since it sounds a bit confusing. So for problem 2/20, the projection was found using simple trigonometry, without the additional component while the case wasn't the same for 2/4. For 2/4, we needed to solve for the b-component of the 100N Force, 100*cos(50), and add that to F2 which is in line with the b axis. So what I don't get is why is the approach to each question different? Aren't they basically the same question with just different angles and values? Why can't we find the projection for 2/4 using the same method? Why doesn't 80*cos(20) give us the projection onto the b axis?

 

[haruspex]

Why doesn't 80*cos(20) give us the projection onto the b axis?

The 80N is F₂. That acts along the b axis, so its projection onto the b axis is still 80N.
80cos20 would give its projection onto the horizontal axis.

 

[haruspex]

Aren't they basically the same question with just different angles and values?

It is the same approach.
In 2/20 you have one known force at a known angle to the desired axis. In 2/4 you have two forces at known angles to the desired axis. In each case you can describe the approach as ΣF_icos(θ_i), where θ_i is the angle F_i makes to the projection axis.

 

[flgdx]

The 80N is F₂. That acts along the b axis, so its projection onto the b axis is still 80N.
80cos20 would give its projection onto the horizontal axis.

Oh sorry, I meant why does Rcos(20) not give us the projection for 2/4?

 

[haruspex]

Oh sorry, I meant why does Rcos(20) not give us the projection for 2/4?

Because the angle between R and the b axis is not 20°. As I posted, it is a little under 30°.

 

[flgdx]

Oh okay. I see where my fault lies now. It's because I assumed that R is completely horizontal with an angle of 0 degrees in reference to the dotted lines in reference to the first picture in problem 2/4. Thank you so much!

 

[flgdx]

oh and one last question pls. Why does cosine law work for 2/4 when solving for R but when I try to use sine law I get a different value?