Apr 1st, 2015: Lab 9: Centripetal force with a motor

Purpose:

We want to find the relationship between angular speed (ω) and angle (θ), and compare the relationship to the experimental data value and calculate its propagated uncertainty.

Experiment:
setup:
_ An object (attached at the end of the string) rotates around the tripod stand that when it rotates, it makes a total radius of (R+Lsinθ) from the center as shown in the diagram below,

Diagram of apparatus. The derived equation on the right side will be explained below.

_ The tripod stand is at height "H" from the ground (from where the string's attached) and when the object rotates, it makes angle θ with the vertical axis and height "h" from the ground.
_ By using higher voltage on the motor for several trials, the motor spins at a higher angular speed (ω), makes larger angle, larger total radius, and "h" from the ground increases. 
Deriving an equation:
_ Before measuring H, R, and length of string (l), we have to come up with an equation that relates ω to θ.

Actual apparatus. Ring stand with horizontal piece of paper (to measure "h") not shown here.

Deriving the equation
_ FBD, and sum of forces:

* The radius is R+Lsinθ because the omega would have been larger if we only count in Lsinθ.
_Now we can measure H, R, and l, which will remain constant throughout the lab:

• l= 1.654+/- 0.01 m
• H= 1.97 +/- 0.01 m
• R= 0.975 +/- 0.01 m

_ For this experiment, the varied data include:

• "h". We measure this by adjusting the rind stand taped with horizontal piece of paper until the top part of paper barely hits by the rotating object when it comes around.
• "ω", we collect omega by choosing number of rotations and measure the time it takes for the object to rotate in those number of rotations. The time measured has to be continuous on the stopwatch and the start time should be when the object is approximately aligned with the ring stand.
• θ, can be calculated from each trial using cosθ= (H-h)/l ; hence θ= arccos[(H-h)/l]

Data

Omega (actual) is calculated, separately, on calculator using the derived equation then plugged into excel.

Calculate the propagated uncertainty

_ Trial 5 data:

_ Next, we use these 3 actual omegas to calculate the error with experimental omega:

_ Graph of experimental vs. actual omega:

Note that linear regression line, r^2= 0.99906.

Our graph gives r^2 = 0.99906. This represents that the experimental and actual ω are similar in 99.906% of the 6 trials collected. The other 100- 99.906 = 0.094% represents the error percentage in the 6 trials. Our propagated uncertainty calculation shows that the ω (experimental) has 0.11% error from our highest and lowest ω (actual), which is very close to the error predicted by the graph, 0.094%. This shows that our accuracy percentage of experimental data (from propagated uncertainty calculation) agree with the regression line (r^2) the graph gives.

Conclusion

_ We've observed that from the equation ω= [gtanθ/(R+Lsinθ)]^1/2  ; as θ approaches 90 degree, omega gets bigger, but the angle could never make 90 degree with the vertical. Therefore, θ is proportional to ω.

_ When we compare the experimental and actual angular speed on the graph, r^2 = 0.99906, which means there's little error in the data we collected.
_ The 0.11% that we calculated is a result categorized  as systematic error. Since different groups measure R, H, l differently, we all got data that have slightly different ω actual and experimental, hence different data distribution. Note also that if we didn't include propagated uncertainty, we would've ended with experimental ω that has 0.47% from actual ω.
-Within this amount of uncertainty and error, we conclude that the derived equation for ω and θ, agrees with our experimental data.