29-Sep.14: The Relationship between Angular Speed and the Angle of a string

PURPOSE:
To derive a relationship between angular speed (ω) and the angle of the string above the vertical (θ) for a particular rotating apparatus.



The apparatus was a tall tripod, standing 211 centimeters high (h), with an attached motor which spun at a constant angular velocity.
At the top of the tripod was a rod, positioned perpendicular to the axis of rotation - we also attached a 165 centimeter string (L) to one end of the rod. At the bottom of the attached string, we tied a small mass. The radius of rotation (d) at the top of our apparatus was 61.5 centimeters.



 


We then measured, as a class, the amount of time it would take for 10 rotations to be completed.
After each measurement we took the average of the times.
After agreeing on a time, we measured the height (h2) of the hanging (now swinging at an angle) mass by slowly raising a piece of paper, which was attached to a sturdy pole, beneath the mass' swing-zone. We waited until the mass just barely tapped the paper, then quickly locked the paper-pole system into place and retrieved it for further measuring. We measured the height of the piece of paper (the very top of the paper where the mass came into contact with the paper) - this represents how high the swinging mass is above the ground.



We completed this process a total of eight times.
Below is a list of the times taken for 10 rotations paired with the height of the swinging mass for that run.

1) 37 sec. and 49 cm.
2) 30 sec. and 72 cm.
3) 26 sec. and 92 cm.
4) 25 sec. and 104.5 cm.
5) 23 sec. and 119 cm.
6) 21 sec. and 132 cm.
7) 19 sec. and 147 cm.
8) 17 sec. and 157 cm.

From the raw data, we recognized that with a smaller period (faster rotational velocity) there was an increased height of the mass.

We then found the period of each run by dividing the recorded times by ten (number of rotations).

T1) 3.7 sec.
T2) 3.0 sec.
T3) 2.6 sec.
T4) 2.5 sec.
T5) 2.3 sec.
T6) 2.1 sec.
T7) 1.9 sec.
T8) 1.7 sec.

Next, we calculated a value for theta (in radians) for each of the periods using the inverse cosine function.

Deriving a formula for the relationship between angular speed and the angle:

Using this derived expression we calculated values for omega for each of the experiments performed.

We then measured values for omega using the classic model,

Using our values for the first run, we found the values for the rest of the experiments using excel.

Using the ordered pairs (omega calculated, omega measured), we graphed our data to show how accurate our measurements were - a perfect measurement would cause a slope of 1 on the graph.

Our slope: 0.8740

A Great Day for Physics.

(taken:9/29/14 7:01am)