Video Transcript
A body is under the effect of
three forces of magnitudes 𝐅 sub one, 𝐅 sub two, and 36 newtons, acting in the
directions 𝐴𝐵, 𝐵𝐶, and 𝐶𝐴, respectively, where triangle 𝐴𝐵𝐶 is a
triangle such that 𝐴𝐵 equals four centimeters, 𝐵𝐶 is six centimeters, and
𝐴𝐶 is six centimeters. Given that the system is in
equilibrium, find 𝐅 sub one and 𝐅 sub two.
Remember, for the system of
forces to be in equilibrium, there must be a zero resultant force. This means that the magnitude
of the force in our triangle must be in the same ratio as the length of the
sides of the triangle. So let’s sketch that out. We have an isosceles triangle,
with one side of length four centimeters and two of length six centimeters. We can trace the corresponding
force diagram over the top, where the ratio of 𝐅 sub one and the length of side
𝐴𝐵 is equal to the ratio of 𝐅 sub two and side 𝐵𝐶, which in turn is equal
to the ratio of the 36-newton force and the length of the third side.
If we write the ratios out for
the two sides of equal lengths, we should be able to spot the value of the
magnitude of 𝐅 sub two. The only way for the statement
to be true is if 𝐅 sub two is 36 newtons. Similarly, let’s compare the
first ratio with the one for side 𝐴𝐵. We can calculate the multiplier
for the side lengths by dividing four by six, to get two-thirds. So, we also need to multiply 36
by two-thirds to find the value of 𝐅 sub one. That’s 24 newtons. So, 𝐅 sub one is 24 newtons
and 𝐅 sub two is 36 newtons.
In our final example, let’s
look at how to apply this process to a system involving a suspended object.
A uniform rod of length 50
centimeters and weight 143 newtons is freely suspended at its end from the
ceiling by means of two perpendicular strings attached to the same point on the
ceiling. Given that the length of one of
the strings is 30 centimeters, determine the tension in each string.
Let’s begin by sketching this
system out. Here is the rod, supported by
two pieces of string that meet at an angle of 90 degrees. We might begin by calculating
the length of the third side in this triangle. Let’s call that 𝑙
centimeters. This will be useful. Since we know the system is in
equilibrium, so we will be able to then find the forces in the system. Since we have a right triangle,
we can use the Pythagorean theorem. 50 squared equals 30 squared
plus 𝑙 squared. Subtracting 30 from both sides
and we get 𝑙 squared equals 50 squared minus 30 squared, which is 1600. Finally, if we take the square
root of both sides of this equation, we find that 𝑙 is equal to 40.
Next, we know that the forces
acting here are the weights of the rod and the tensions in the strings. Since the forces are in
equilibrium, they can be drawn acting at the same point. The weight of the rod acts
vertically downward. Then, we can represent the
tension in the 40-centimeter string as 𝑇 sub one and the tension in the other
string as 𝑇 sub two. This gives us the corresponding
force triangle.
Finally, we know that the ratio
of the side lengths of the triangle and the corresponding forces are equal. That’s 143 over 50 equals 𝑇
sub one over 30. We can solve for 𝑇 sub one by
multiplying through by 30, which gives us 85.8 newtons.
Let’s repeat this for 𝑇 sub
two. This time ,143 over 50 equals
𝑇 sub two over 40. So we find 𝑇 sub two equals
143 over 50 times 40, which is 114.4 newtons. So the tensions in the two
pieces of string are 85.8 newtons and 114.4 newtons.
