Finding the Angle of Resultant Force in a Plane

[RStars]

Homework Statement

I am having quite an issue solving this resultant force question:

The following forces act on a point in the same plane: 40N direction due north; 10N direction due east, 40N direction 50° south of west, 20N direction northwest and 15N direction 20° south of east. Calculate the value and direction of the resultant force.

I have drawn up the question to look like this:
http://img98.imageshack.us/img98/5219/forcesresultant.jpg

I have figured out the components of all forces other than at point E. How would I go about calculating the angle between E and the x axis?

Homework Equations

The Attempt at a Solution

So far I have:
A = 40j
B=10i
C= 15 cos 20i-15sin20j
D=-40cos50i-40sin50j
E=-20cos(x)i+20sin(x)j

How would I go about calculating the angle? I am thinking it might be 90-50=40° but not 100% sure.Thanks in advanced.

 

[NascentOxygen]

Hi RStars! It's halfway turn through a right angle, so is 45 degrees.

 

[RStars]

Hey, how do you infer that from the question? Is Northwest always dead on in the middle?

 

[grzz]

Yes. Northwest is at 45deg to the North or, which is the same thing, 45 to the west.

 

[RStars]

Ok thanks a lot that's exactly what I was after.

 

[NascentOxygen]

Ok thanks a lot that's exactly what I was after.

Better prepare yourself for the day you strike a question involving directions such as North North West, or maybe West North West. They can be abbreviated NNW and WNW. What angle do you reckon each of these would involve, RStars?

 

[RStars]

I am not too sure, would NNW be 22.5 degrees north from NW? and WNW be 22.5 degrees west of NW? Also I have another question about the OP. So far my working out is:

A=40j
B=10i
C=15cos(20)i-15sin(20)j
D=-40cos(50)i-40sin(50)j
E=-20cos(45)i+20sin(45)j

i=10+15cos(20)-40cos(50)-20cos(45) = -15.758
j=40-15sin(20)-40sin(50)+20sin(45) = -9.914

Resultant = 18.617

angle of resultant = tan(angle)=j/i
angle = 32.1756 degree
Not sure if I did that bit right?

Also need to figure out the direction of the resultant. Since both i and j are negative I am guessing it is in the 3rd quadrant. So would It make sense to say the resultant is 32.18 degrees south of west? Or is there an error somewhere in my method? Thanks for all the help so far.

 

[NascentOxygen]

I am not too sure, would NNW be 22.5 degrees north from NW? and WNW be 22.5 degrees west of NW? ✔

i=10+15cos(20)-40cos(50)-20cos(45) = -15.758
j=40-15sin(20)-40sin(50)+20sin(45) = -9.914

Resultant = 18.617

You could say magnitude of resultant = ...
or, |resultant| = ...

angle of resultant = tan(angle)=j/i ⟵ Careful! You mustn't equate things which are not equal
angle = 32.1756 degree ✔
Not sure if I did that bit right?

Also need to figure out the direction of the resultant. Since both i and j are negative I am guessing it is in the 3rd quadrant. So would It make sense to say the resultant is 32.18 degrees south of west? Or is there an error somewhere in my method?

Your method looks right, without checking your calculator work. A sketch to remind yourself what you are looking at is invaluable, though I think you probably are using one.

 

[RStars]

Looking back on my calculations I have found out that I have made an error in the j calculation. It should be 18.37. However when this is changed I end up with tan(angle)=18.37/-15.76

This gives me an angle of -49.37 (Is this correct? )

Also since it is now x-axis negative and y positive it would be in the second quadrant. -49.37 degrees north or west?

Not to sure what to make of this to be honest, any help would be appreciated.

 

[NascentOxygen]

Looking back on my calculations I have found out that I have made an error in the j calculation. It should be 18.37. However when this is changed I end up with tan(angle)=18.37/-15.76

This gives me an angle of -49.37 (Is this correct? )

It is if your component calculations are right.

Also since it is now x-axis negative and y positive it would be in the second quadrant. -49.37 degrees north or west?

Look at the i and j components. They will indicate whether it is more towards the north, or more towards the west.

But if you had the components as algebraic expressions and not numbers, you wouldn't be able to assess by eye their relative sizes to judge whether the angle was more north of NW, or more west of NW. So you'd draw your angle in the 4th quadrant (the answer your calculator gives you here), then draw an angle in the 2nd quadrant making an identical angle with the x-axis (but in its negative direction) as does the one in the 4th quadrant.