Math Balance

A simple mechanical calculator powered by gravity and trigonometry!

Either when teaching a child to do basic addition and subtraction, or when explaining more complex concepts to a physics class, like center of mass and mechanical equilibrium, this tool will make education much more fun!

Hook the numbers on the left to add and on the right to subtract them from the total, for a maximum output of 20.
 

Hardware Needed:

• One M3x10 screw. That's it. Only one screw.
   

Printing and Assembly

The Number Tags and the Balance Arm must be completely solid and have a precise shape, so printing at 100% infill with a 0.2 mm layer height is required. 

If you don't have access to a multimaterial printer for printing the Number Tags, don't worry! You can use a technique to achieve multicolor prints with a single extruder, like showed in this video.

The Balance Base may be printed with traditional settings. Do a filament swap at 5.2 mm (layer 26) to highlight the numbers on the scale.

After printing, check the Pivot Pin and the hole in the Balance Arm for possible blobs or imperfections. They must be as regular as possible to guarantee a smooth operation.

A single M3x10 screw will hold the Pivot Pin from the back, keeping the Balance Arm in place. Make sure the Balance Arm is able to move freely and is not catching anywhere.

How does it work?

It's quite simple. The numbers placed on the left hook are added while the numbers placed on the right hook are subtracted from the total. The result is indicated by the circle cut-out in the scale at the base.

Here are more examples:

• 9 + 7 - 8 = 8 

• 6 + 9 +3 -1 - 8 = 9

Ok, but… how does it work?

Each number tag weighs a multiple of tag number 1. This means tag number 2 weighs twice as much as tag number 1, tag number 3 weighs three times as much as number 1, number four weighs four times and so on. 

By hooking a certain amount of Number Tags to the Balance Arm, the long indicator part will move - either to the right or to the left, depending on where the numbers are placed. Knowing the weight of the Number Tags and the weight and coordinates of the center of mass of the Balance Arm, it is possible to calculate what will be the angle that the indicator will move.

For example, lets take the operation 9 + 2 - 3 = 8

From the design, we know that the distance between the center of the hooks in the Balance Arm and the pivot point is 50 mm and the angle between the “plus” arm and the “indicator” arm is 130°. 
Inspecting the 3D model properties for the Balance Arm, we obtain the coordinates of its center of mass, indicated in the drawing below, and its volume V = 9.792 cm³.
Considering the mass of each tag is 0.6N g, where N represents its number, and the density of PLA is 1.2 g/cm³, what is the angle alpha correspondent to the number 8 on the scale?

The solution will be left as an exercise for the reader.