March 25, 2015; Lab 8: Centripetal acceleration vs. angular frequency

Purpose:

To use the theoretical relationship between centripetal acceleration vs. angular frequency (ω) to determine the radius of the rotating disk and prove that the theoretical relationship works.

Experiment: 

1.  To establish a relationship between "a" and "ω", we start from the displacement equation of an object moving in circle. After two derivatives, we obtain a centripetal acceleration equation as shown below,

-To find magnitude of a(t) from x and y coordinate, we use pythagorean theorem and get,

2.    A setup requires a disk, when an object (accelerometer) mounted on top such that the object is distance "r" away from the center of the disk. We place the object almost at the the edge of the disk (making r= the radius of the disk). By using different voltage to set the nearby wheel to rotate and in contact with the metal disk, the metal disk promptly begins to rotate afterward, at a constant acceleration. Piece of a tape hanging out of accelerometer goes through the photogate (the black c-shape). Then Loggerpro is used to record the acceleration and time interval for a certain number of rotations. 

3.   We should have the following to put into excel:

• voltage
• number or rotations
• Δt for those rotations
• recorded acceleration
• rotations converted to radians (theta)
• ω: angular speed (rad/sec)= theta/Δt

Total of 5 trials

Data Analysis:

graph of acceleration (y-axis) vs. ω^2 (x-axis)

Theoretically, a= rω^2. When we graph "a" and "ω^2", we basically get the slope (from proportional fit) as a value or disk radius "r". 

We also measure the radius of the disk and it's between 0.138m and 0.14m.

Experimental r= 0.1383m. Hence, the experimental value is in the range of the actual value.

Conclusion:

Possible error or uncertainty is the fact that acceleration can only stay relatively constant and that external force such as friction during rotation got in the way. Our proportional fit gives r^2 of 0.9997, which can mean that there's 0.03% error in our data. Nonetheless, the experimental "r" is in the range of the actual "r" and we conclude that experimental data agree with equation: a= rω^2.