Relationship between angular speed of conical pendulum and the tilt angle of its string

Relationship between angular speed of conical pendulum and the tilt angle of its string 
Introduction:
In the previous lab, we already determine the relationship between the angular acceleration and the angular velocity, and in this lab, we determined the relationship between the the angular speed of conical pendulum and the tilt angle of its string.

We are looking for relationship between angular speed and θ.
Apparatus and process: 

We need to use this stop watch to find the total time needed for 5 finished rotations, so we can find the average period(T) of each rotation.

This is the apparatus that will do the conical pendulum motion, in this picture, professor Wolf was adjusting the height of the paper until the pendulum barely touch while spinning in order to find h2.







We did 7 trials with different angular speed in total in this lab.

Hypothesis and data:
To find out the magnitude of  angular speed of the motion, we need to use the stop watch to find the period of each rotation, and convert the period(T) to angular speed(w).
w=2π/T
Then, we have to determine the angle of the string. We find the length of h1 and the length of the string l we used for the circular motion to calculate for angle θ:
We found that l=.177m, and thus θ = cos^(-1)(h1/.177)

The following is the prediction we made for the relationship between angular speed(w) and angle θ

And if the prediction is correct, the graph of w vs.  sqrt(g tanθ/r+0.62) should have a slope of 1.
The following is the data chart:

And finally, the following is the graph of w vs.  sqrt(g tanθ/r+0.62)

Conclusion:
 We have found that relationship between the angular speed and angle θ is "w∝sqrt(g tanθ/r+0.62)."
The slope of our graph is 0.9222, which is a very close number to 1, and thus our prediction was very accurate. The constant 0.2917 in the function that represent the graph exist because of the experimental error caused by the uncertainty of the measurement equipment we used. Also, the way we measure the height h2 is very inaccurate because we used only folded paper to estimate h2.

Centripetal acceleration as a function of angular speed

Centripetal acceleration as a function of of angular speed
Introduction:
In this activity that we did together in class, we learned how to find the relationship between angular velocity and angular acceleration with an experimental method. We used the motion of a turntable as our experiment  model.
Equipment needed and procedure:
We need to use a stopwatch to record the total time it took for the turntable to finish five spin, in order to find its average period for each 1 complete spin motion. The following is a picture of the turntable that we used to simulate rotational motion.

Data:
The data of this lab is collected together by all the classmates, and we find the average of each groups' result to be more accurate:

The data can then be organized into the following chart, as theoretically we were always told that centripetal acceleration= v^2/r=w^2*r, we make an column of w^2:

Then we used logger pro to analyze the data into the following acceleration vs w^2(rotational speed^2):

Conclusion:
The graph is very close to a straight line that inclines upward with a slope= 0.1493, so we can sum up that centripetal acceleration, a, is direct proportional to square of angular velocity,w^2. Our graph pretty much proved the equation that we normally used to find angular acceleration:
ac=v^2/r=w^2*r.

Determination of an unknown mass

Measuring the Density of Metal Cylinders/Determination of an unknown mass.

Measuring the Density of Metal Cylinders--Introduction to propagated error calculations

Introduction: In this lab, we learned to use vernier calipers and micrometers to measure objects' dimensions. We used the measurement we obtained with vernier calipers and micrometers to calculate the density of the object, and determine the propagated error.

Apparatus:

Vernier caliper with micrometer:
This device can measure length of object very precisely. We used this apparatus to obtain the dimensions of object in this lab.











Procedure and data:
First of all, we need to find out the mass of our three objects: copper cylinder, aluminum cylinder, and iron cylinder.

Copper cylinder's mass, 56.7 g

Aluminum cylinder's mass, 20.8 g

Iron cylinder's mass, 61.4 g

Then we used the caliper to measure dimensions of three metal cylinders. and we get the following measurement with uncertainty.

With dimensions of each metal cylinder, we can use the following equations to find volume and density of each of them.

Volume = π(d^2)*h

Density = mass/ volume

And use the following formula to find propagated error:

Result:

Thus, we conclude that the density included propagated error of our 3 metal cylinder to be:

(8.76±0.019)*10^3 kg/m^3

(2.68±0.020)*10^3 kg/m^3

(7.55±0.018)*10^3 kg/m^3

This result should be acceptable since our propagated error is very small.

Activity: Determination of an unknown mass
Introduction:
In this activity, we will try to solve for an unknown mass in a system that was built at equilibrium state. The one we tried to solve was unknown mass #8, and it was the following system:

Apparatus:

Protractor: We need the angle of the tension force exert on the unknown mass, in order to calculate the weight of unknown mass.














Data and Calculation:
Following picture is the measured data that we needed to find out the weight of the unknown mass:

The following are the calculation to find out our unknown mass and propagated error of it:
Calculation to obtain mass:

Calculation for propagated error:

Result:

With all the calculation we did above, the unknown mass with uncertainty is (1.39+-0.07)kg.