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Alison Whitney November 22, 2023
Group Members: Rook Bergeron
Background: Forces are a quantifiable push or pull measurements on an object that is exerted through the presence of another object. Linear forces such as an applied force, move an object straight in the x, y, or z directions. However, when a force is applied it manner that causes an object to rotate, it is represented as torque. The concept of torque was orginally investigated by Archimedes using levers in the 200s BCE. Today, it is understood as the rate of change in an objects angular momentum. Torque is represented by the equation:
Torque = r F sin (theta)
The variable r represents the position vector that relates the pivot point of the object to the location at which the force is being applied. F represents the force itself, and theta represents the angle between the direction of the force and the position vector.
When an object is static, it is known that the sum of the forces will be equal to 0, or that the forces will balance each other out so that there is no net movement. The same principle is true for torque. When the object is defined to be static, the sum of all of the torques will be equal to 0. These two concepts aid in solving for unknown variables when approaching objects with rotational movement.
This experiment attempted to use the processing of balancing a seesaw composed of a meter stick using weights on either end. The relative masses that were added to the sides and distance from the pivot point were used to determine if torque was in fact conserved. The pivot point was then moved off center and the process was repeated. The null hypothesis for the study was the sum of the torques will not be balanced. The alternate hypothesis was that the sum of the torques will be balanced and therefore be equal to 0.
Methods:
A meter stick was balanced on a pivot point with the point of rotation on the center of mass.
A weight was added to one side of the stick.
A second weight was added to the other side of the stick at a different distance from the pivot point than the first weight.
Mass was slowly added to either weight until the meter stick was balanced again.
The mass of each weight and their respective distances from the pivot point was recorded.
The masses were then removed and the pivot point of the meter stick was altered to be different from the center of mass.
The stick was then rebalanced by adding mass to the weights again.
The mass of each weight and their distances from the pivot point was recorded.
An image of all of the materials for the experiment
An image of all of the materials for the experiment
An image of the meter stick before it the masses were added to balance it
An image of the meter stick before it the masses were added to balance it
An image of the balanced meter stick with the pivot point on the center of mass
An image of the balanced meter stick with the pivot point on the center of mass
An image of the balanced meter stick with the pivot point not on the center of mass.
An image of the balanced meter stick with the pivot point not on the center of mass.
Equations:
Sum of the Forces = F1 + F2 + F3 ... + Fx = 0
Sum of the Torques = t1 + t2 + t3 ... + tx = 0
Torque = r F sin(theta)
Fg = m * g
Percent Error = ((measured - known) / measured) * 100
Written calculation for trial with the pivot point on the center of mass
Written calculation for trial with the pivot point on the center of mass
Written calculation for the trial with the pivot point away from the center of mass
Written calculation for the trial with the pivot point away from the center of mass
Raw Data:
Pivot Point on the Center of Mass:
Mass 1 = 0.133 kg Distance 1 from the Left End of the Meter Stick = 0.225 m
Mass 2 = 0.077 kg Distance 2 from the Left End of the Meter Stick = 0.982 m
Pivot Point not on the Center of Mass:
Mass 1 = 0.091 kg Distance 1 from the Left End of the Meter Stick = 0.225 m
Mass 2 = 0.286 kg Distance 2 from the Left End of the Meter Stick = 0.982 m
Known Mass Stick = 0.073 kg Distance of Center of Mass from the Left End of the Meter Stick = 0.750 m
Distance of Pivot Point from the Left End of the Meter Stick = 0.503 m
Results:
The sum of the torques when the pivot point was on the center of mass was equal to 0.
When the sum of the torques when the pivot point was not on the center of mass was set to 0, the mass of the meter stick was calculated to be 0.075 kg.
When the measured mass of stick was compared to the known mass, the percent error was found to be 3.1%.
Discussion: The alternate hypothesis that the sum of the torques will be balanced and therefore equal to 0 was supported by the results of the experiment and the null hypothesis was able to be rejected. When the pivot point was aligned on the center of mass, the sum of the torques was equal to 0, as shown by the written calculation above, meaning that the mathematical representation of the scenario matched the balanced meter stick observed in the experiment. Using this information, when the pivot point was moved, the sum of the torques was set to 0. The mass of the meter stick was then able to be calculated as 0.075 kg, with only a 3.1% error from the known mass of 0.073 kg. This result was within the 5% margin of error, and therefore the experiment was determined to be successful at accurately finding the mass of the meter stick. Therefore, both parts of the experiment supported the alternate hypothesis and the null hypothesis was rejected. There is room for future experimentation as to how to apply this principle on a larger scale or to objects that are moving along the beam.
Errors: There was some error in the experiment as determined by the 3.1% error found in the second part of the study. This was likely a result of not having weights available at less than 1 g, and not using a level to ensure the balancing of the meter stick was exact. Future experiments should utilize a level and access the smallest weights available to improve the precision. However, the 3.1% error was still within the 5% margin, so the method used in this experiment was still determined to be accurate.
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