Lab 9: April 03, 2017: Centripetal Force with Motor

By -
Lab 9: Centripetal Acceleration with Motor
Amy, Chris, and John
April 03, 2017

Today's lab was about establishing the relationship between the angle a string forms with a tripod and the angular velocity an object has. 

Theory: We were given that there exists a relationship between angular speed and centripetal force. 
We know that the centripetal force of the object we are spinning on this lab is: 
F=ma(centripetal acceleration) which also equals mrw^2 (where w is angular speed).
As shown, if the centripetal force increases, the x component of the tension also increases which changes the angle theta that the string makes with the vertical.

If we call height from the ground to the top of string, H, and the height from ground to the mass, h. We can find the vertical length of the string at a particular angular speed, expressed as H-h. This forms a right triangle which lets us solve for the angle theta (L is the length of the string):
Because of this we can also calculate for the radius component of the centripetal force for a particular angular speed. If we call the length of the rod on top R, taking the sum of R and x component of string L.

Using our sum of forces equation from above we can derive an equation for omega:
In this case omega is the value of the angular speed, theta is the angle the string forms and L is the length of the string from above. We calculated the value of omega by timing the revolutions the object swung and divided 2 pi by the time for one revolution. 

Apparatus and Procedure: 

For this lab have our motor mounted on the tripod with a rod that goes up vertically with a rod going horizontal attached to it. There is a string at the end of the horizontal rod with a mass attached to the end which revolves when the power supply to the motor is turned on.
We did five trial runs where Professor Wolf increase the power level every run. For each run we record the time for ten revolutions, and the height from the ground that the mass is spinning at. We do this by setting up a stand with a piece of paper we can change the height of. The height at which the spinning mass hits the piece of paper is the height of it above the ground.

Measured Data and Graphs: 
he height from ground to top of rod, length of string and the radius from center to the point on the rod where the string was tied is given below:
HeightLengthRadius 
1.82 ± 0.001 m1.59 ± 0.001m0.75 ± 0.001m
data for our five trial runs with different angular speeds:
height (m)time for 10 rev (s)
10.3633.41
20.65527.37
30.8724.22
41.13219.92
51.3615.65
our height measurements here also have the same uncertainty value of +/- 0.001m.

Calculated Data: 

θ (degrees)ω (using angle) (rad/s)ω (using period) (rad/s)
123.31.7495052161.880630143
242.92.2293449632.295646806
353.32.5481880442.594213587
464.43.0603653493.154209492
573.13.7684441834.014814893

Methods for calculations of values were outlined in the Theory/Introduction. One difference is we recorded the time for 10 revolutions, our calculations were done by taking the time recorded and dividing by 10 to get the time for 1 revolution to fit the equation.

Plotting these two omega values:

Conclusion:
For the values we obtained in both data tables should be the same since we are calculating for the same object.  As for every experiment we have uncertainties introduced, mainly in our calculated data.  For the slope we obtained in our final graph, the value of 1.045 is very close to our expected value of 1 and the uncertainty in this measurement given by LoggerPro is reasonably low (0.01039 or 0.994%). While there is uncertainty from meter stick measurements, I believe these are almost negligible compared to our main source of error, the measurement of time for 10 revolutions. This was by far the most error prone measurement since we were eyeballing the mass spinning 10 times and recording a time when it passed a point we considered as the origin. There exists many different sources of error here (reaction time, miscounts, other human error) and it contributed to the discrepancy in values obtained for omega.
Overall, our end result produced respectable results (our final graph correlation is a relatively high value, 0.9970). We can conclude the mathematical models outlined in the Theory and Introduction for calculation of omega using the angle theta shows a relationship between the two quantities

Comments

Post a Comment

Popular posts from this blog

Lab 6: March 3, 2017: Propagated Uncertainty

By -
Image
Lab 6: Measuring the Density of Metal Cylinders  Amy, Chris, and John  March 3,2017 Today's lab purpose was to measure the density of two metal cylinders and calculate the propagated error in each one of our density measurements by using a Vernier caliper and a scale.  Theory:  In this lab we had to figure out the density of two unknown metal cylinders and calculate the propagated uncertainty for each. For this particular lab there are factors where our calculations may have been slightly incorrect propagating a further error in our data. When measuring the density of each metal we used a Vernier caliper that outputs measurement readings in centimeters and is more precise up to two decimal places.  In order to calculate propagated uncertainty in our lab we used the propagated uncertainty of density we took partial derivatives and generated this formula:  Apparatus and Procedure:  For this lab we used the Vernier caliper and a sc...
Read more

Lab 8: March 29, 2017: Centripetal Acceleration vs. Angular Frequency

By -
Image
Lab 8: Centripetal Acceleration vs. Angular Frequency Amy, Chris, and John March 29, 2017 Today's lab was about finding the relationship between centripetal force and angular speed, with multiple varying scenarios. Then graph our findings and find a graph with the best linear fit that shows us the best correlation.  Theory:   We know for an object moving in uniform circular motion, the direction of acceleration is towards the center. The force that corresponds with this acceleration is called the centripetal force and in  equation, centripetal force  is  the ma. It can be interpreted as the net force. If we had different forces that contribute to the circular motion, the sum of those forces would be equal to your centripetal force. In our lab, looking at the setup from the side of the disk, the free body diagram of the mass looks something like:     We derived the equations in order to change our variables one at the time and figu...
Read more

Lab 2: March 1, 2017: Free Fall Lab

By -
Image
Lab 2: Determination of g and some statistics for analyzing data Amy, Chris, and John March 1, 2017 Today's lab purpose was to examine the validity of the statement:  In the absence of all other external forces except gravity, a falling object will accelerate at 9.8 m/ s^2 .Calculate values of g by group and compare them to other group's findings of g and calculate the standard deviation as well as errors of uncertainty.  Theory:  For this particular lab we had to figure out a way of measuring g, looking at the distance between sparks on a long tape created by the spark generator as an object was in free fall. By measuring the distances of the position as the object was in free fall we could gather a time lapse of how fast the object was falling. We graphed the findings from our times and distances, from the tape with markings, and calculated through a velocity vs. time graph how fast the object was accelerating down or g though the slope of the data w...
Read more