Finding the tension in two ropes.

Question

I have a problem that says to find the tension in two ropes in the following figure.

The answers are 1830kg in the right rope and 2241kg in the left rope. I'm able to successfully solve the tension in the right rope, but I keep getting 1294kg for the left rope.

Here is how I'm doing it:

v = tension in right rope

u = tension in left rope

Equation 1: $v*cos(60) - u*cos(45) = 0$

Equation 2: $v*sin(60) + u*sin(45) - 2500kg = 0$

Then, from Equation 1, solve for u:

$u = \frac{v*cos(60)}{cos(45)}$

And plug this in to Equation 2 and solve for v:

$v = \frac{2500}{sin(60) + tan(45)*cos(60)} = 1830 kg$

Now that I have v, I use it in Equation 1 to solve u

$u = \frac{1830kg*cos(60)}{cos(45)} = 1294kg$

Any help is appreciated! Thanks!

Answer 1

Note that $30^\circ$ is the complement of $60^\circ$ $$u = \frac{1830kg\cdot\cos(30^\circ)}{\cos(45^\circ)} = 2241.283kg$$

Answer 2

The reason for your problem is that you have incorrectly chosen the sine and cosine functions. Only the fact that one angle is $45^\circ$ degrees allows you to get one of the forces right, since $\sin(45^\circ)=\cos(45^\circ)$

There are several ways to prevent this error. Here's one

Consider your first equation:

v∗cos(60)−u∗cos(45)=0

This equation correctly attempts to set the net horizontal force equal to zero.

Now ask yourself: what would happen to the horizontal component of the right force if the $60^\circ$ angle were to increase to nearly $90^\circ$? Clearly, the horizontal component would increase to a maximum. But you used the cosine function, which decreases to zero for a $90^\circ$ angle! Clearly, you are in trouble.

Similarly. in your second equation you are attempting to set the total of all vertical components to zero:

v∗sin(60)+u∗sin(45)−2500kg=0

If the $60^\circ$ angle increased to $90^\circ$, the vertical component would disappear, but your equation uses the $\sin$ function, which reaches a maximum for $90^\circ$

Doing this sort of reality check gives you the chance to reverse the two functions and get the correct result.