A fulcrum is placed under a lever. 2 objects must balance on the board (distance in feet of each object....)?

Question

The distance in feet of each object from the center of the board must VARY INVERSELY with its weight in pounds. In the diagram shown, what is the distance of each animal from the center of the board?

The diagram has a picture of a cat on the left (7.5lbs) and a dog on the right (12.5lbs). The total distance of the lever is 5ft (left side to right side). That's all the information given. I've been trying this for hours, now it's 1AM and I'm giving up and passing it on.

Please, my grade depends on your help!

Answer 1

Let x ft be the distance the cat is from the fulcrum. Then (5 ft - x ft) is the distance of the dog from the fulcrum because the total length of the board is 5 ft, and x ft + (5 ft - x ft) = 5 ft.

The product of the weight of the object on one side times its distance from the fulcrum must equal the product of the weight times the distance from the fulcrum of the object on the other side. Therefore:

x ft (7.5 lb) = (5ft - x ft)(12.5 lb) which is also equivalent to
x ft / 12.5 lb = (5 ft - x ft) / 7.5 lb.

Now, just solve for x.

(x ft) 7.5 lb = 62.5 ft-lb - (x ft)12.5 lb.
(x ft) 7.5 lb + (x ft) 12.5 lb = 62.5 ft-lb
(x ft) 20.0 lb = 62.5 ft-lb
(x ft) 20.0 lb / 20 lb = 62.5 ft-lb / 20 lb
x ft = 3.125 ft

So the cat is 3.125 ft from the fulcrum. Then the dog is (5 ft - 3.125 ft) = 1.875 ft from the fulcrum.

Check by multiplying them out and setting them equal to each other. If they are equal, we have the right numbers.

(7.5)(3.125) = (12.5)(1.875) ?
23.4375 = 23.4375 !

The number of ft-lb on one side equals that on the other side, so our solution is correct.

Notice that the distance of each object from the fulcrum varies inversely with their weight. The cat weighs less, therefore it is farther away from the fulcrum than the dog. The inverse is true for the dog. The dog weighs more, so it is closer to the fulcrum than the cat.

Hope this helps save your grade. Just study and make sure you get the concept down, so that you can do it yourself. There will be no help like this come exam time.