738.sagews

A 75 pound weight hangs from a ceiling with two uneven wires. The first wire hangs at an angle of 47 degrees, and the second at an angle of 30 degrees. What are the forces--also called tensions--on both wires? A crude picture displaying is below7

We canexpress these vectors in terms of \textbf{i} and \textbf{j}.

*Since the green vector goes left and up, green is \textbf{g} = -|\textbf{g}|\cos{47}\textbf{i} + |\textbf{g}|\sin{47}\textbf{j}
*The red vector goes right and up, which makes it \textbf{r} = |\textbf{r}|\cos{30}\textbf{i} + |\textbf{r}|\sin{30}\textbf{j}
*The resultant force is \textbf{g} + \textbf{r} = -\textbf{w} = 75\textbf{j}. \textbf{w} is the weight of the weight pulling down. It is negative because it's going down. They are equal, because they counterbalance each other. Also note that since we are considering the weight at the origin, it doesn't have an \textbf{i} component. It's just -75\textbf{j}
*So now, we can dump the whole thing into one big equation of:
(-|\textbf{g}|\cos{47}\textbf{i} + |\textbf{g}|\sin{47}\textbf{j}) + (|\textbf{r}|\cos{30}\textbf{i} + |\textbf{r}|\sin{30}\textbf{j}) = 75\textbf{j}
*This equation says that if we add both \textbf{g} and \textbf{r} we're going to get a vector, that's the negative of the weight vector \textbf{w}. This makes sense, because the forces counter each other.
*Still doing nothing more than rewriting the equation. We can equate the coefficients of both sides and end up with two simpler equations. See how easier vectors are to work with when you have them in standard form?
-|\textbf{g}|\cos 47 + |\textbf{r}|\cos 30 = 0
|\textbf{g}|\sin 47 + |\textbf{r}|\sin 30 = 75
*solving for |\textbf{r}| in the first equation we get:
|\textbf{r}| = \frac{|\textbf{g}|\cos 47}{\cos 30}
*putting this result into the second equation: |\textbf{g}|\sin 47 + \frac{|\textbf{g}|\cos 47}{\cos 30}\sin 30 == 75
*we can solve for |\textbf{g}| by multiplying the first term by \frac{\cos 30}{\cos 30} so we have a common denominator, so we have:
\frac{|\textbf{g}|\sin 47\cos 30}{\cos 30} + \frac{|\textbf{g}|\cos 47\sin 30}{\cos 30} == 75
*then we can divide each term in the numerator by \cos 30 (remember that \tan\theta = \frac{\sin\theta}{\cos\theta}):
|\textbf{g}|\sin 47 + |\textbf{g}|\cos 47\tan 30 = 75
*factoring out a |\textbf{g}| on the left side, and then diving that to the right side:
|\textbf{g}| = \frac{75}{\sin 47+\cos 47\tan 30}

#Here's how to find the exact values using Sage
#the norm of g is 75/(sin47+cos30tan47)
gnorm = (75)/(sin(47*(pi/180))+tan(30*(pi/180))*cos(47*(pi/180)))
#the norm of r is (gnorm*cos47)/cos30
rnorm = (gnorm*cos(47*(pi/180)))/(cos(30*(pi/180)))
#g is −∣g∣cos47i+∣g∣sin47j 
g = vector([(-gnorm)*cos(47*(pi/180)),gnorm*sin(47*(pi/180))])
#r is r=∣r∣cos30i+∣r∣sin30j 
r = vector([rnorm*cos(30*(pi/180)),rnorm*sin(30*(pi/180))])
w = vector([0,-75])
force = g+r
rarrow = arrow((0,0),r.element(),color='red')
garrow = arrow((0,0),g.element(),color='green')
warrow = arrow((0,0),(0,-75),color='yellow')
farrow = arrow((0,0,),force.element(),color='blue')
plot(rarrow+garrow+warrow+farrow).show(aspect_ratio=1)
(r+g) == -w

True

n(force.norm(),digits=5)

75.000