Suppose
a mass m is supported by two ropes, as shown below. One rope extends
horizontally from the wall, and the other rope goes to the ceiling at an angle q from the vertical. Calculate the
tension in each rope.

We begin by making a free-body diagram of the mass m. This is a picture
that isolates the mass from its surroundings, and shows only those forces
acting on the mass. The forces acting on this mass are the tensions in the two
ropes, which we will call T_{1} and T_{2}, and
its weight mg, which of course points down:

We
will also need to break the T_{2} vector into x and y
components (the other two force vectors are already on either the x or y
axis). This becomes

Now,
for the total force to add up to zero, all the vectors along the y axis
and all the vectors along the x axis must add up to zero.

1.
Sum of forces in the x-direction equals zero

T_{2}
sin q -T_{1} = 0

2.
Sum of forces in the y-direction equals zero

T_{2}
cos q -mg = 0

We
now have two equations in two unknowns, T_{1} and T_{2}
(we are assuming that m and q
are known). The second equation can be solved for T_{2} to give

This
can be substituted into the first equation, which can then be solved for T_{1}: