September 1st, 2014- Relationship between angular speed and Theta

Purpose: The purpose of this lab was for us to find the relationship between omega and theta using a conical pendulum.
Apparatus: 

This is our conical pendulum. This apparatus has a pole attached to the top of it and a string tied to the end of the pole. It is hooked up to a motor which cause the pole to pin in a clockwise direction. 

What we did:  First we took the measurements of the  few items we could obtain using a meter stick. We found that the Height of the Conical pendulum was 211 cm and the length of the string was 165 cm. Next we took the time for the weight at the end of the string to revolve around the apparatus 10 times. Once we got the time we recorded our data. After we had to determine the height of the mass by setting up a tripod and attaching a piece of paper onto it. To get the height we would have to position the tripod close enough for the weight to hit it and elevate the tripod to the point where the mass just hits the top of the paper.

Once the mass hit the paper we got the height of the paper relative to the ground by using a meter stick and recorded our data. That completed the experiment for one velocity. Professor Wolf would then raised velocity and we would obtain the same data for that speed. We continued this process for 7 different speeds. This is what we obtained

TIME FOR 

TEN ROTATIONS------HEIGHT OF MASS AT THAT VELOCITY

37 SEC                           49CM               

30 SEC                           72CM

26SEC                            92.3CM

25 SEC                          104CM

23 SEC                          119 CM

21 SEC                           132 CM

19 SEC                           174CM

WHAT WE DID WITH THIS INFORMATION: First we had to come up with a formula to be able to use our data. We set up an image of the apparatus and had to find out a few unknowns.

H= height of the apparatus.=211cm

L=length of string.

H2= the the height of the mass while spinning= column 2 above

H-H2=to the height of the string to the top of the apparatus.

Theta= acos(H-H2)/L 

We then made a body diagram of the mass and the tension. We separated the tension vector into x and y components. We set the x component= m*(V^2/R). We then found R using the Pythagorean theorem. Finally we manipulated and solved the equation to find the relationship between omega and theta.                                BELOW ARE ALL THE STEPS MENTIONED ABOVE.

Next we opened up Logger Pro and Inserted all the information we obtained.(Below)

We Then Graphed it and put a linear Fit (Below)

Conclusion: After an extensive lab we were able to find the Relationship between Omega(W) and Theta by finding the Height of the pendulum,the length of the string, the time it takes for the mass to revolve around the pendulum, and the height for that time period. We found that the relationship is that tan(theta)=(Omega^2*R)/G.