The angles theta and alpha (q&a) could be any angles, I just drew them this way. Call the tension in the left string "T1" and the other one "T2".
What forces are acting here? There are three: gravity & the two tensions
What are the magnitudes of these
forces? The force due to gravity is just mg,
to find the tensions, we must do
a little analysis...(first a sketch is a good idea)
Why are the two "X" vectors the
same length? Because the hanging mass is not moving in the
X direction; it is just hanging still.
Why aren't the two "Y" vectors
the same length? Well, if the angles q
& a were the same, the vectors would be the same length.
But in this drawing, the angle a
is
bigger, so it should be clear that its string is doing most of the work
in holding up the mass.
We know that the mass is not moving, so we know everything adds up to zero:
Sx forces
= 0
(S means
"sum")
Sy forces
= 0
Since the "X" forces are equal,
we have: T1x = T2x
Also the "Y" forces add up to mg:
T1y + T2y = mg
Using X & Y components lets us use the familiar geometric functions...
|
|
|
| sinq = T1y/T1 | sina = T2y/T2 |
| cosq = T1x/T1 | cosa = T2x/T2 |
Where did these come from?
Here are the triangles; each hypotenuse is a tension.
NOTE:
These are only sketches & may not show the vectors to scale!
By inserting the values from the
table above into the two blue equations above,
you arrive at...
T1cosq
= T2cosa
and, T1sinq
+T2sina
= mg
Then, we can isolate T1 or T2 in
the left equation and insert that value into the right equation... (to
keep this page shorter, I'll skip the insertion stuff)
and you wind up with these two equations:
mg
= T1sinq + T1cosqtana
mg = T2sina + T2cosatanq