Lesson 7 · from §74 of the 1910 text
The Lever
The lever
A lever is a stiff bar resting on a single support, the fulcrum. The parts of the bar on either side of the fulcrum are the arms.
Anyone who has played on a teeter board knows that the heavier person must sit nearer the fulcrum. If one arm is 3 feet and the other is 4 feet, a boy weighing 100 pounds on the short arm balances a boy weighing 75 pounds on the long arm, because 3\cdot 100 = 4\cdot 75.
In general, if the arms are l_1 and l_2 and the weights on them are W_1 and W_2, the lever balances when l_1W_1 = l_2W_2.
Example. A 60-pound weight hangs 5 feet from the fulcrum. What weight 3 feet from the fulcrum on the other side will balance it? 5\cdot 60 = 3W, \quad W = 100 \text{ pounds}.
When several weights sit on one side, add their products: the weights balance when the sum of (weight × arm) on one side equals the sum on the other.
Try one:
Quick check
A boy weighing 80 pounds sits 3 feet from the fulcrum of a teeter board. How heavy is the boy who balances him 4 feet from the fulcrum on the other side?
From the history of algebra
Archimedes of Syracuse (about 287–212 BC) proved the law of the lever in his book On the Equilibrium of Planes. He is said to have boasted, "Give me a place to stand and I will move the earth." The story goes that King Hiero challenged him to prove it, and Archimedes, working a set of compound pulleys, single-handedly drew a fully loaded ship onto the shore.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Use l_1W_1 = l_2W_2. Give numbers only; the units are filled in.
A, 4 feet from the fulcrum, balances B, who is 6 feet from it. A weighs 96 pounds. Find the weight of B.
A, who weighs 100 pounds, balances B, who weighs 120 pounds. B is 80 inches from the fulcrum. How far from it is A? (Answer in feet.)
A, who weighs 125 pounds, balances B, who weighs 100 pounds. The distance between them is 9 feet. How far is each from the fulcrum?
A and B together weigh 210 pounds. They balance when A is 3 feet 9 inches from the fulcrum, and B is 5 feet from it. Find the weight of each.
A weighs 90 pounds and is 4 feet from the fulcrum. B weighs 60 pounds and is 3 feet from the fulcrum and on the same side of it as A. C, who weighs 108 pounds, is on the opposite side of the fulcrum. How far from it must C be in order to balance both A and B?