Video Transcript
A body of weight 𝑤 is attached to
a wall by a string of length 25 centimeters. It is held in equilibrium by the
effect of a horizontal force of magnitude 93 gram-weight that keeps the body 15
centimeters away from the wall. Determine 𝑇 and 𝑤.
In order to answer this question,
we will consider a triangle of forces in equilibrium. There are three forces acting at
point 𝐶. We have the weight of the body 𝑤,
a horizontal force of magnitude 93 gram-weight, and the tension force in the
string. Since the system is in equilibrium,
there is no resultant force. And to form a triangle of forces
with zero resultant, the magnitudes of the forces must be in the same ratio as the
lengths of the sides of the triangle.
We will therefore begin by
calculating the missing length of the triangle, side 𝐴𝐵. The Pythagorean theorem states that
𝐴 squared plus 𝐵 squared is equal to 𝐶 squared, where 𝐶 is the length of the
longest side of a right triangle and 𝐴 and 𝐵 are the lengths of the shorter
sides. Substituting the values from the
diagram, we have 𝑥 squared plus 15 squared is equal to 25 squared. This simplifies to 𝑥 squared plus
225 is equal to 625. Subtracting 225 from both sides, we
have 𝑥 squared is equal to 400. We can then square root both sides
of the equation. And since 𝑥 must be positive, we
have 𝑥 is equal to 20. Side length 𝐴𝐵 is equal to 20
centimeters.
We might also notice that the side
lengths of our triangle are in the ratio 15:20:25. This simplifies to three, four,
five. And we have a three-four-five
Pythagorean triple.
As already mentioned, the
magnitudes of the forces must be in the same ratio as the lengths of the sides of
the triangle. This means that 𝑇 over 25 is equal
to 93 over 15, which is equal to 𝑤 over 20. Dividing each of the denominators
by five, we have 𝑇 over five is equal to 93 over three, which is equal to 𝑤 over
four. We can now solve the equation 𝑇
over five is equal to 93 over three to calculate the value of 𝑇. 93 divided by three is 31. And multiplying through by five, we
have 𝑇 is equal to 31 multiplied by five, which is equal to 155. The tension in the string is
therefore equal to 155 gram-weight.
We can repeat this process to
calculate the weight of the body 𝑤. This is equal to 31 multiplied by
four. The weight of the body is therefore
equal to 124 gram-weight. And we now have values of 𝑇 and 𝑤
as required. 𝑇 is equal to 155 gram-weight, and
𝑤 is equal to 124 gram-weight.
