Magnetic Fields / Force / Current

[julianwitkowski]

Oops now I forgot a bracket... F = 1.4 2 (4/sin(½√3) sin(½√3) = 11.2 N (down)

 

[julianwitkowski]

See...

As I understand the distinction between rad and deg is that rad gives you a distance in terms of radians, which gives you an idea of how many times something goes around a circumference...

What else should I know?

 

[BvU]

My bad, sloppy reading (doing other things same time). Of course. And I was the one who threw in this 60 degrees in the first place !

Radians are a natural measure for angles. That way arc length (in a unit circle) = angle. And that way $\displaystyle \lim_{\theta\downarrow 0} \; \sin\theta < \theta <\tan\theta$ etc.

"What else should I know " is a good question. I don't have all the answers...

In degrees... 4,6 m would be the length of the wire if L = 4/sin(60°) in deg... I don't understand why it is ½√3 in radians, because 60° = π/3... No? This is the edge of my understanding of radians, or lack of it.

If L = 4/sin(½√3) = 5.25 m, F = 1.4 2 (4/sin(½√3) sin(½√3) = 30.82 N (down)

I thought F = i B (w/sin θ) sin θ =1.4⋅2⋅(4/sin(π/3)) ⋅ sin (π/3) = 11.2 N (down)

So I'm confused... want to spoil this?

Oh boy, I wish I could turn back time here. ${1\over 2}\sqrt 3$ is the cosine of $\pi\over 3$ (= 60 degrees). (comes from half of an equlateral triangle, sides $1, {1\over 2}, {1\over 2}\sqrt 3$ ).

Think you have it right, have to run.

 

[julianwitkowski]

Think you have it right, have to run.

Well this goes back to what simon said... and we're both technically right. Both π/3 and ½√3 are slight rotations.
We pulled 60° out of nowhere as a way to have an example.

I'm curious as to why a calculator gives the same answer for both expressions when ½√3 = 49.6° and π/3 = 60°...
I guess this goes back to small rotations of θ = 1, as which also gives the same answer.

Is this why my calculator is rounding these things to θ=1?

Maybe I don't understand why θ=1 is used when values of θ are small.
,,,Or why it would be beneficial to use radians for calculations a slight turn, or any turn less than 2π ?

 

[Simon Bridge]

Just some clarifications: $$\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$$ ... you can check this on your calculator.
If you make an equilateral trangle where the hypotenuse is 1 unit long and the adjacent side is 1/2 unit long, then the angle between these sides is pi/3 (60deg) and the length of the opposite side is the sine of the angle. In this set up you can find the length of the opposite side without using a calculator. Give it a go.

However, $\frac{\sqrt{3}}{2} = 49.6^\circ$ ... is nonsense.
You need to be more careful when you write down the maths - you've been told about this before (prev in thread) - remember, the people reading this don't know what you mean, they only know what you write. We cannot tell what you've done from what you've said.

If you input "square root of three all divided by two" however your calculator does this, it should output 0.86603 ... maybe more dp than that.
I use gnu octave so I just type in sqrt(3)/2, or what I want to compute, and press enter. I get:

octave:9> sqrt(3)/2
ans =  0.86603
octave:10> pi/3
ans =  1.0472
octave:11> sin(pi/3)
ans =  0.86603

... calculators often need to be placed in a different "mode" to do trig using radians. If you mix them up you can get confused.

Since $\frac{\sqrt{3}}{2} \approx 0.8660$ while $\frac{\pi}{3} \approx 1.0472$ ... this means that 60deg is not a small angle, and it would not normally be considered a small rotation.

Don't forget to always use radians ... when you write $\theta = 1$ you are saying the angle is one radian, which would be about 57.3 degrees. Also not a small angle.

But remember the difference between an angle and a rotation?

Also remember: what counts as small depends on the context.

 

[julianwitkowski]

If you make an equilateral trangle where the hypotenuse is 1 unit long and the adjacent side is 1/2 unit long, then the angle between these sides is pi/3 (60deg) and the length of the opposite side is the sine of the angle. In this set up you can find the length of the opposite side without using a calculator. Give it a go.

That is clever actually, cool :).

However, $\frac{\sqrt{3}}{2} = 49.6^\circ$ ... is nonsense.

What does this mean at wolfram?

Sorry about that I know.. I'm working on it. This is a mysterious adventure for me as I've never done any of this before.
I have a real good streak about learning in trial and error.