How To Make A Balance Scale

A balance does precisely what the name suggests: it balances two items. By using one, you can determine the mass of an object.

Let's walk through how to make a do-it-yourself (DIY) scale or balance, and see how the physics principle behind it works.

You will need the following to make your homemade mass balance scale:

•A sturdy beam, which can be picked based on what you will be weighing. If you will be weighing very heavy objects, you may need a piece of lumber to make a giant balance scale. More likely, you will want to make a small balance that can be used to weigh small objects such as paper clips or coins. For a small balance, you could use a popsicle stick as the beam.
•A fulcrum, which will support the beam at a single point in the middle (or very close to a single point). For a small popsicle scale, using a wedge of rubber, such as a thin eraser, could work. 
•Small objects of known weight to act as the means of measuring the mass of the unknown object. 

In order to understand the purpose of the small objects of known weight, we need to know how a balance or scale works.

The physical principle behind a beam balance is torque. A force applied to the beam at some distance from the fulcrum (which is called the lever arm), or the point where it is balanced, produces a torque. Torque gives rise to rotational motion if the torques are unbalanced.

A beam balance uses this principle for measuring mass or weight.

The formula for torque, τ, is

\(\tau = F\times r\)

where ​F​ is the force applied by the object, and ​r​ is the lever arm. Note that the operation is a cross product, which is a vector operation, and not multiplication. The cross product will only be non-zero if some component of the force is perpendicular to the lever arm.

It is clear that for a beam balance, the lever arm can be represented as a vector that begins at the fulcrum and points out towards the end of the beam. The force vector begins at the point where the mass is located, and it is parallel to the direction of gravity.

To check if this equation makes sense, think of opening a door. In order to open the door you have to pull perpendicular to the door. If you were to face the edge of the door and push or pull, you would not open the door. The equation for torque describes precisely that physical phenomena.

For two-dimensional problems, the formula becomes

\(\tau=Fr\sin{\theta}\)

in which case the cross product has been performed, and the sine of the angle between the directions of the force and the lever arm is θ. As the angle between the force and the lever arm approaches 0, the torque also goes to 0, which makes sense.

In order to use a balance to determine the mass of an object, the object of unknown mass should be placed on one end of the balance. This will induce a torque and the balance will rotate about the fulcrum and rest on the ground until the torque is balanced. So how can we balance the torque?

This is where the objects of known mass are needed.

We can slowly add the objects of known mass to the opposite end and begin to determine the appropriate force. When the beam is balanced, and both ends are at equal heights off the ground, the forces on both ends of the beam are balanced.

When this happens, you can add up the total mass that was needed to balance the beam, which determines the mass of the unknown object.

Remember, the lever arms on both sides of the beam should be exactly equal. If not, the forces that are needed to balance the torque will not be exactly equal, and there would be additional calculation required to determine the unknown mass.