Balance Torques and Center of Gravity

Our purpose in this lab is to study the rotational equilibrium of a meter stick and to also determine its center of gravity with a system of masses.

The materials needed for this lab are:

• Meter stick
• 3 mass holders
• masses for the holders
• knife edge clamp
• 3 mass clamps

INTRODUCTION:
When an object is not moving, it has Fnet=0. We are looking at balancing a meter stick on a knife edge clamp with masses hanging on each side. We aren't working in straight line motion when working with this particular lab, we are looking at rotational motion. The force in rotational motion is called torque.

Torque can be calculated with the following equation:

Since torque is a force and we are balancing the meter stick, the Tnet=0.

The Force is the weight force which is perpendicular to the lever arm. The angle should be 90 degrees. The lever arm would be the distance from the center of gravity to where the mass, or force, is at. Since our value for theta is 90, sin(90)=1 so we don't have to take consideration of the angle in this case.

PROCEDURE:

To begin the lab we first set the meter stick on the knife edge clamp and set it to where the meter stick was perfectly balanced. This will be where the center of mass is. We found that to be at 49 cm on the meter stick.

We measured out two different total masses of 100 g or more with the masses, mass hanger, and mass clamps because you must take into account the mass of the hanger and the clamps. They affect the torque as well since it is added to the meter stick. Our first mass was 220.6 g and our second was 171.5 g. We set our second mass 40 cm away from the center of gravity and tried to balance the meter stick with the first mass on the other end of the meter stick. That ended up being 31.16 cm away from the center of gravity. We calculated our values to see how close to zero for a Tnet we obtained. 

We calculated our Tnet value and compared it to how close to 0 we could get with our measurements we obtained.

Next we put the two masses in different locations but on the same side. We added the third mass clamp with hanger and masses that was 270.8 g on the opposite side and moved it until the meter stick was balanced again. We also found how close we got to a Tnet of 0 by solving.

We set the same weights from the first test on the same side and added one to the left and balanced it out. We calculated our Tnet and compared it to what it should have been.

For our next step, we replaced the third mass with an unknown mass. We left masses 1 and 2 on the right side at 30 cm and 15 cm respectively from the center of gravity. We adjusted the unknown mass to balance the meter stick. Our distance from the center of gravity was 30.2 cm. We solved for what the mass should be and then measured the mass to see how close to the actual we obtained. We calculated the percent error we had.

We changed our third mass on the left side with a new unknown mass. We found it's mass by using our equations. We compared the mass we obtained with the actual mass of the unknown mass and the mass clamp.

Next, we put 200 g (including mass clamps and hangers since they aren't part of the ruler and are causing the ruler to have torque) at the 90 cm mark and find a new balance point on the meter stick. We found that point to be 78 cm, which is 12 cm away from the mass hanging. The center of gravity of the meter stick is 29 cm away. We can say that the center of gravity of the meter stick is where the weight force is located because that's the point where the mass is concentrated. The new center of gravity will be at the 78 cm mark. We calculated what the mass should be and compared it to the actual mass of the meter stick. The mass of the clamp that holds the meter stick is irrelevant because it is keeping the meter stick in equilibrium.

We found the new center of gravity with a 200 g mass hanging at the 90 cm mark on the meter stick. We solved to find what the mass of the meter stick is and compared it to the actual mass.

We now had to picture a similar scenario with the 200 g weight still at 90 cm but now we add an additional 100 g at the 30 cm mark. We had to calculate to find it's center of gravity and where it should be on the meter stick. We found it to be at 65.5 cm once we did our calculations. I then went ahead and made it the center of gravity and checked to see if that center would give us a net torque of 0 or how close it would be.

We calculated to find the center of mass of the meter stick when a 200 g mass is hanging at the 90 cm mark and a 100 g mass is hanging at the 30 cm mark. We calculated our center of mass to be at 65.5 cm on the meter stick.

CONCLUSION:

In all of our calculations we had a very small percent error throughout the entire lab. This is because there is a very small window, as far as length is concerned, that we can obtain to keep our meter stick in perfect balance. Also when we were calculating our percent error, we weren't comparing the net torques because our theoretical value was 0 and we cannot have a 0 in a denominator. Instead we compared the theoretical length of where we should have put our mass and where our masses actually were on the meter stick. We weren't able to apply the last scenario of finding a center of mass to the actual meter stick to see if 65.5 cm was actually the center of mass when 200 g is hanging at 90 cm and 100 g is hanging at 30 cm. We did use calculations to see how close to 0 we got for a torque net but weren't able to compare to any tests.

We weren't able to prove the last scenario to be true. I calculated what the Tnet would be with what we found if 65.5 cm on the meter stick is really the new center of mass with the given masses at the given lengths.