My friend is doing M1 next week (I've already sat it) and in some insane exam paper her teacher has set her, is this beast:
A rock climber has weight 550N. Rope attached to waist belt passes through ring which is fixed at point higher up cliff. She loses foothold and starts to move in direction 20 degrees to the horizontal. Tension in rope is 560N. Calculate angle rope makes with vertical.
I should mention that in M1 we use modelling assumptions of no resistive forces, weight is vertically downwards and that g=9.8, if that makes a difference.
Have a go at it, my solution was INSANELY RIDICULOUS, and though by using an iterative trigonometric formula I managed to get it, there must me an easier way. If you want to try, the mark scheme's answer was
My solution is below, if you think I may be exaggerating how convoluted my solution was, take a look.
If you want to know how I did all the resolving etc, then you’ll find a nice picture of my working here: http://dl.dropbox.com/u/498402/mechanics-y%20part%20of%20solution.png.
For details of how I got from that equation to the one that I plopped into the equation solver, working is here: http://dl.dropbox.com/u/498402/maths-y%20part%20of%20solution.jpg.
By changing tan(20) and the fractions into decimals, I arrived at this equation which I felt inadequate to solve: [math]0.132-0.132cos^2(x)= -(1100/560)cosx+cos^2(x)+302500/313600[/math].
If we pop that equation into the equation solver http://www.numberempire.com/equationsolver.php then we get x = acos(5^(3/2)*sqrt(553465*cosx-234598)/sqrt(39869606)) .
If we use WolframAlpha to solve that, we get this: http://www.wolframalpha.com/input/?i=x+%3D+acos%285^%283%2F2%29*sqrt%28553465*cosx-234598%29%2Fsqrt%2839869606%29%29+.
If we use the bigger solution, x ~ 0.7432060359962328, and ask WolframAlpha to convert that to radians for us, then we get this: http://www.wolframalpha.com/input/?i=+0 ... to+degrees .
And strangely enough, it gives us 42.58 degrees... i.e. 42.6! So my ridiculous solution was right, but we must have missed something obvious that allowed us to do it.
I know I've missed something obvious, because there is no way that would be in a Mechanics 1 exam. If anyone has any blinding insights that my friend and I missed, please do tell.