You’ve probably played on a see-saw before, or used a wheelbarrow to move dirt or tools for gardening. Maybe you’ve used tweezers to get a splinter out of your finger. While the see-saw, the wheelbarrow and the tweezers may not seem to have much in common, they are actually somewhat alike: they are all levers. A lever is simply a rigid stick or beam, which pivots around a stationary point called a fulcrum.
For any lever, there is also a point of input force, and a point of output force against some resistance. When input force is applied to a lever (like when you pick up the handles of a wheelbarrow), the resulting output force (the force acting against the resistance, which lifts the tools in the wheelbarrow) is amplified. In other words, the lever of the wheelbarrow helps you move your garden tools because it allows you to put in a small force, and turn it into a large force to lift the tools. One of the first to demonstrate the lever’s usefulness for moving heavy objects was Archimedes; according to Pappus of Alexandria, who wrote Synagoge: Book VIII in 340 AD, Archimedes once said, “Give me the place to stand, and I shall move the Earth.” (wikiquote.org)
Levers can be put in one of three classes:
Class 1 levers have the fulcrum in the middle of the lever, force is applied to one end of the lever, and the object to be moved (the resistance) is on the other end. The see-saw is an example of a class 1 lever.
Levers can be put in one of three classes:
Class 1 levers have the fulcrum in the middle of the lever, force is applied to one end of the lever, and the object to be moved (the resistance) is on the other end. The see-saw is an example of a class 1 lever.
Class 2 levers have the fulcrum at one end and the input force on the other end, with the resistance in the middle. The wheelbarrow is an example of a class 2 lever.
Class 3 levers have the fulcrum at one end and the resistance on the other end, with the input force in the middle. Tweezers are an example of a class 3 lever.
To understand how a lever helps us move things, we need to understand how the lever changes the forces that are exerted on it. Using a see-saw as an example, say you weigh 50 pounds, and your friend weighs 100 pounds. If the fulcrum of the see-saw is in the middle, your friend will be sitting on the ground and you will be up in the air! This is because your friend is exerting twice as much force on his or her end of the see-saw as you.
To understand how to cope with this, we need to understand the law of the lever. This law states that the output force on the resistance (FB) divided by the force applied on the lever (FA) equals the distance of the input force from the fulcrum (a) divided by the distance of the output from the fulcrum (b), or:
This value is also called mechanical advantage. For more information, please click here.
In order for the two of you to play on the see-saw together, you will need a mechanical advantage! We know from the law of the lever that the product of the force and the distance from the fulcrum must be equal for you both. So:
In order for the two of you to play on the see-saw together, you will need a mechanical advantage! We know from the law of the lever that the product of the force and the distance from the fulcrum must be equal for you both. So:
We can also see that this means the distance from the fulcrum to your seat must be twice the distance of the fulcrum to your friend’s seat:
So if we move the see-saw fulcrum so that the distance from your seat is twice as much as your friend’s, you will balance, and be able to play nicely on the see-saw!
The same idea applies to the wheelbarrow: if the object you are trying to move is twice as heavy as the weight you are able to lift, you will need to position this object close enough to the wheels such that it is more than twice as close to the fulcrum as you are.
TRY IT!!
NOTE: You will need to do these experiments at a playground, or another location where you can find a see-saw. You will also need to know your exact weight in pounds, as well as your friends’ weights.
Here’s what you will need:
1. A see-saw from your local playground or park. IT MUST BE ADJUSTABLE (In other words, you need to be able to move the fulcrum)
2. A friend that is lighter or the same weight as you
3. A friend that is heavier than you (this could be a parent or teacher)
4. A tape measure at least as long as the see-saw
5. A notebook to take notes and write down numbers
Here’s what you need to do:
1. Measure your see-saw from end to end. Write down the total length in feet.
2. Divide your weight by your friends’ weights. This means you will have two numbers, your weight divided by your smaller friend’s weight, and your weight divided by your bigger friend’s weight. These will be numbers W1 and W2. Write these numbers down!
3. The number ‘W1’ is likely to be close to 1, since you and your smaller friend are likely about the same size. If your smaller friend is lighter than you, that number will be greater than 1. If the number is 1, take the length of your see-saw and divide it in half, and place the fulcrum this distance from both you and your friend’s seat (so if your see-saw is 6 feet long, place the fulcrum 3 feet from your seat and 3 feet from your friend’s seat).
4. Both you and your friend now sit on the see-saw on either end, and if you are really the same weight, you should balance! If you don’t balance, try measuring the see-saw again, or check your weights again!
5. If the number W1 is greater than 1, to find how far you and your friend should be sitting from the fulcrum we need to do some algebra! Don’t worry if you haven’t learned algebra yet, there is a very easy way to figure out the answer. We are going to go through the long way too, just for good measure!
Let’s say you weigh 75 pounds, and your smaller friend weighs 50 pounds. This means your number W1 is 1.5, because 75/50 = 1.5.
We know the length of the see-saw (say 6 feet), and we know that in order to balance on the see-saw together your friend needs to be 1.5 times further from the fulcrum as you are (see the discussion above on how levers do work if you need to review this). So to find how far from the fulcrum you and your friend need to be, we set it up like this:
x = how far you need to be from the fulcrum
1.5x = how far your friend needs to be from the fulcrum (this means he/she needs to be 1.5 times as far as you).
If we add these two distances, we know they must be 6 feet, since that’s how long our see-saw is:
x + 1.5x = 6
We can add x and 1.5x to get 2.5x
2.5x = 6
Now to find x we divide 6 by 2.5
x = 6/2.5
This gives us 2.4 feet, or about 2 feet and 5 inches
x = 2.4 feet
So you must be 2 feet and 5 inches from the fulcrum. This means your friend must be 3 feet and 7 inches from the fulcrum, because:
1.5x = 2.4 x 1.5 = 3.6 feet = 3 feet 7 inches
Whew! That’s a lot of math! Now here’s the easy way: Whatever W1 is, just add 1, and divide the total length of the see-saw by that number. That’s how far you need to be from the fulcrum. Subtract that number from 6, that’s how far your friend needs to be. It’s that easy!
6. Now move the fulcrum of the see-saw so it is 2 feet and 5 inches from your end. Sit on the see-saw with your smaller friend, and you should balance! If not, double check the length of your see-saw, and your weights!
7. Now we’ll do the same for you and your bigger friend. Take your number W2 (which should be less than 1), add 1, and divide 6 by this number. This is how far you should sit from the fulcrum.
Here’s an example: Say you weigh 75 pounds, and your larger friend weighs 150 pounds. This means your W2 number is 0.5:
W2 = 75/150 = 0.5
We add 1 to this number, and divide 6 by this final number:
0.5 + 1 = 1.5
6/1.5 = 4
This is how far you should be sitting from the fulcrum of the see-saw. This means your friend should be sitting 2 feet from the fulcrum, because 6-4 = 2.
8. Now move the fulcrum of the see-saw so it is 4 feet from your end of the see-saw. Sit on the see-saw with your bigger friend, and you should balance! If not, double check the length of your see-saw, and your weights!
NOTE: You may not balance perfectly if the see-saw has some extra weight on one end after you move the fulcrum, like a metal piece that holds the see-saw in place. If this is the case, you may need to adjust slightly to perfect the balance!
Be sure to write down all your observations: Did you and your friends balance? Why or why not? Probably the numbers were not as easy as in our examples, so if you found it hard to balance this may be the culprit also!
MAKE UP YOUR OWN EXPERIMENT!
You could do a similar set of experiments with a wheelbarrow and a heavy weight. Calculate how close to the wheels of the wheelbarrow the weight would need to be for you to lift it using the same math technique used above. As always, be sure to write down all your numbers and observations!
References for more information:
J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, Theory of Machines and Mechanisms, Oxford University Press, New York.
NOTE: You will need to do these experiments at a playground, or another location where you can find a see-saw. You will also need to know your exact weight in pounds, as well as your friends’ weights.
Here’s what you will need:
1. A see-saw from your local playground or park. IT MUST BE ADJUSTABLE (In other words, you need to be able to move the fulcrum)
2. A friend that is lighter or the same weight as you
3. A friend that is heavier than you (this could be a parent or teacher)
4. A tape measure at least as long as the see-saw
5. A notebook to take notes and write down numbers
Here’s what you need to do:
1. Measure your see-saw from end to end. Write down the total length in feet.
2. Divide your weight by your friends’ weights. This means you will have two numbers, your weight divided by your smaller friend’s weight, and your weight divided by your bigger friend’s weight. These will be numbers W1 and W2. Write these numbers down!
3. The number ‘W1’ is likely to be close to 1, since you and your smaller friend are likely about the same size. If your smaller friend is lighter than you, that number will be greater than 1. If the number is 1, take the length of your see-saw and divide it in half, and place the fulcrum this distance from both you and your friend’s seat (so if your see-saw is 6 feet long, place the fulcrum 3 feet from your seat and 3 feet from your friend’s seat).
4. Both you and your friend now sit on the see-saw on either end, and if you are really the same weight, you should balance! If you don’t balance, try measuring the see-saw again, or check your weights again!
5. If the number W1 is greater than 1, to find how far you and your friend should be sitting from the fulcrum we need to do some algebra! Don’t worry if you haven’t learned algebra yet, there is a very easy way to figure out the answer. We are going to go through the long way too, just for good measure!
Let’s say you weigh 75 pounds, and your smaller friend weighs 50 pounds. This means your number W1 is 1.5, because 75/50 = 1.5.
We know the length of the see-saw (say 6 feet), and we know that in order to balance on the see-saw together your friend needs to be 1.5 times further from the fulcrum as you are (see the discussion above on how levers do work if you need to review this). So to find how far from the fulcrum you and your friend need to be, we set it up like this:
x = how far you need to be from the fulcrum
1.5x = how far your friend needs to be from the fulcrum (this means he/she needs to be 1.5 times as far as you).
If we add these two distances, we know they must be 6 feet, since that’s how long our see-saw is:
x + 1.5x = 6
We can add x and 1.5x to get 2.5x
2.5x = 6
Now to find x we divide 6 by 2.5
x = 6/2.5
This gives us 2.4 feet, or about 2 feet and 5 inches
x = 2.4 feet
So you must be 2 feet and 5 inches from the fulcrum. This means your friend must be 3 feet and 7 inches from the fulcrum, because:
1.5x = 2.4 x 1.5 = 3.6 feet = 3 feet 7 inches
Whew! That’s a lot of math! Now here’s the easy way: Whatever W1 is, just add 1, and divide the total length of the see-saw by that number. That’s how far you need to be from the fulcrum. Subtract that number from 6, that’s how far your friend needs to be. It’s that easy!
6. Now move the fulcrum of the see-saw so it is 2 feet and 5 inches from your end. Sit on the see-saw with your smaller friend, and you should balance! If not, double check the length of your see-saw, and your weights!
7. Now we’ll do the same for you and your bigger friend. Take your number W2 (which should be less than 1), add 1, and divide 6 by this number. This is how far you should sit from the fulcrum.
Here’s an example: Say you weigh 75 pounds, and your larger friend weighs 150 pounds. This means your W2 number is 0.5:
W2 = 75/150 = 0.5
We add 1 to this number, and divide 6 by this final number:
0.5 + 1 = 1.5
6/1.5 = 4
This is how far you should be sitting from the fulcrum of the see-saw. This means your friend should be sitting 2 feet from the fulcrum, because 6-4 = 2.
8. Now move the fulcrum of the see-saw so it is 4 feet from your end of the see-saw. Sit on the see-saw with your bigger friend, and you should balance! If not, double check the length of your see-saw, and your weights!
NOTE: You may not balance perfectly if the see-saw has some extra weight on one end after you move the fulcrum, like a metal piece that holds the see-saw in place. If this is the case, you may need to adjust slightly to perfect the balance!
Be sure to write down all your observations: Did you and your friends balance? Why or why not? Probably the numbers were not as easy as in our examples, so if you found it hard to balance this may be the culprit also!
MAKE UP YOUR OWN EXPERIMENT!
You could do a similar set of experiments with a wheelbarrow and a heavy weight. Calculate how close to the wheels of the wheelbarrow the weight would need to be for you to lift it using the same math technique used above. As always, be sure to write down all your numbers and observations!
References for more information:
J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, Theory of Machines and Mechanisms, Oxford University Press, New York.
