# Columns and the Law of Squares

There is an inverse square relationship between the length of a column and the amount of weight that it can carry. If you cut a column in half, one of the shorter pieces will carry four times as much weight as the original column. This is calculated as follows: Take the ½ (you cut the column in half) and invert it to get 2. Then square the 2 (2 x 2) to get 4. If you take a column and cut it into thirds, one of the shorter pieces will carry nine times as much weight. (Invert 1/3 to get 3 and then square the 3 to get 9). A column that is one forth (1/4) the length of another column will carries 16 times as much weight. You can only carry this so far. You can’t make a material stronger than it is. These examples assume that the only thing that changes is the length of the columns. If the column material changes in any way then the examples do not work.

An easier way to look at this may be to think of it this way. If I cut a column into 2 equal sections, the shorter section will hold four times as much weight (2 squared or 2 x 2). If I cut a column into 3 equal sections, the short section will hold nine times as much weight (3 squared or 3x3).

Here is a simple experiment to demonstrate this.

- Take a piece of 1/8" x 1/8" x 36" long balsa wood
**(do not use larger pieces and use only balsa wood)**. - Lay the stick of balsa on a gram scale. The scale will register some weight for the wood. With the balsa stick still on the scale, tare the scale (zero it). This removes the weight of the stick from the scale reading.
- Holding the balsa stick perpendicular to the scale place a finger on the top of the stick and press down until the wood starts to bend. Record the reading on the scale. The number will fluctuate so just pick a number that is in the middle of the readings.
- Record this reading - you have just calculated the "buckling" load (weight) of this particular 36" stick of wood.
- Cut the 36" stick into two 18" pieces.
- Take one of the 18" pieces and repeat steps 2,3 and 4. You have now calculated the buckling load of this 18" piece of the original 36" column. The buckling load of the 18" piece should be very close to four times the buckling load of the 36" piece.
- Take one of the 18" pieces and cut it to 12" in length. The length is now 1/3 the length of the original 36" piece. Try to predict what the buckling weight of the 12" piece will be by multiplying the buckling weight of the 36" by 9. (1/3 inverted = 3 * 3 squared = 9).
- Repeat steps 2,3 and 4 with the 12" piece and compare to your estimate.
- You can use these experiments to learn more about the properties of wood. Make up a chart and record the weight of each stick of balsa before you cut it and then the weight of each piece as you cut it into 18" and 12" pieces. How close to 1/2 the weight of the 36" stick are the two 18" pieces? How close to 1/3 the weight of the 36" stick are the three 12" pieces?

(Note: 1/8" x 1/8" balsa wood is used in this demonstration because most scales should be able to record all of the readings. Conducting the experiment with larger pieces or heavier woods could be dangerous because the amount of force required to make the pieces bend could be excessive resulting in breaking the wood and having a broken sharp point stick into a hand.)

You may wonder what good this information does, because the height of the structure has to meet the specifications of the challenge. It does not do any good to know that if you build a structure that is one half the required height that it will hold four times as much weight.

This information **is** useful because when designing and building a structure, bracing a column makes it act like two shorter columns. So, if you properly brace a 9” column in the middle you now have effectively created two 4.5" columns which can support four times as much weight.

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