Balancing Act

As we are learning about torque and center of gravity, we are also conducting a lab. Our goal is to find the mass of a  meter stick using only the stick itself and a 100 gram lead weight. We are not allowed to use a scale to find the mass of the meter stick. 

Step 1:

When an object is balanced, the torques on either side of its center of gravity are equal. This means that, collectively, the force and lever arm on either side of the center of gravity are the same. However, one side of the stick could have a long lever arm and little force while the other side could have a lot of force with a short lever arm. As long as the counter-clockwise torque on one side equals the clockwise torque on the other side the object will be balanced because neither rotation overpowers the other so the object will not rotate either way.

When the meter stick is placed on a table with some of it hanging off the edge, the center of gravity (for my group's stick) was at the 50.25 centimeter mark on the stick. This is the point that sits on the very edge of the table where part of the stick can hang off without causing any rotation. 
When a mass is placed on the edge of the meter stick, the center of gravity is shifted to the 29.5 centimeter mark. The distance from the stick's actual center of gravity to the edge of the table, where the new center of gravity is, is now its lever arm, instead of being from the center of gravity to the end of that side of the stick. This means the lever arm is much smaller and that the force on the stick is at the end of this lever arm, not at the end of the stick.

Step 2:

We started with what we already knew about torque and balanced objects. When an object is balanced, the torques on either side of its center of gravity are equal. Therefore, it was essential that:

counter clockwise torque = clockwise torque

Since torque = force x lever arm...

force x lever arm = force x lever arm

Not only were these equations going to be important for us to use, so was the equation for weight, since the whole purpose of the lab was to find the mass of the meter stick. It is:

weight = mass x gravity (w=mg)

We kept in mind the units of all the variables we would be working with.

Weight: Newtons (also in kilogram meters per second squared: kg m/s^2)

Mass: grams and kilograms (100 grams equals 0.1 kilogram)

Gravity: 9.8 meters per second squared (m/s^2)

Force: Newtons

Lever Arm: centimeters (but usually we work in meters for lever arm)

Torque: Newton centimeters (N cm; usually it would be Newton meters)

We chose to use our knowledge of torque for this experiment and our calculations because we knew that force and weight are both in Newtons and are equal. So if we could figure out the torque of one side, we would know both the force on that side (because we could just measure the lever arm and find the force through simple math) and the torque of the other side of the center of gravity because the two torques would be equal). Using this knowledge, we could find the mass through the equation for weight. In order to complete these processes, we took measurements. We found the length of all the lever arms and the point where the center of gravity was. We also found the mass of the weight placed on the meter stick, both in grams and kilograms.

Measurements:

Center of gravity = 50.25 cm mark

Center of gravity with the weight o the meter stick = 29.5 cm mark

Mass of weight = 100 grams (0.1 kilograms)

Lever arm to the left of new center of gravity = 29.5 cm (as shown in picture above)

Lever arm to the right of new center of gravity = 20.75 cm (as shown in picture above)

Step 3:

After we had all of our preliminary measurements taken and our plan of action was established, we began calculating. First, we found the force on the left side of the center of gravity with the weight. Here are our calculations:

w=mg

  = 0.1kg (9.8 m/s^2)

  =0.98 N = Force

Then we found the torque on this portion of the ruler...

torque = force x lever arm

            = (0.98 N)(29.5 cm)

            = 28.91 N cm

Now it was time to find the torque of the other side of the center of gravity...

counter-clockwise torque = clockwise torque

                      [ 28.91 N cm = 28.91 N cm]

               force x lever arm = force x lever arm

              (0.98 N)(29.5 cm) = F (20.75 cm)

                        28.91 N cm = F (20.75 cm)

                                1.39 N = F

Since Force = weight...

1.39 N = weight

weight = mass x gravity

1.39 N = mass (9.8 m/s^2)

0.1418 kg = mass (b/c N also equal kg m/s^2 so when divided, the units cancelled out to kg)
141.8 g = mass
This was our calculated mass (141.8 grams) which we checked on the scale after the experiment concluded. The actual mass of the meter stick was 142.9 grams, so we were very close (only 1.1 grams off).