The first part of this lab was to predict using calculations the period of the ring rotating about a point halfway between its inner and outer radii. To do this, I used Newton's Second Law equation for torque. I found the total torque to be a product of the component of the weight, -mgsin(theta), times the distance from the pivot which was, r+(R-r)/2. Next I had to find the moment of inertia for a ring with its pivot moved a distance, r+(R-r)/2. After substituting in the values, I solved for alpha to get the equation, alpha=-2g*sin(theta)/3(R+r). It is known that for small theta, sin(theta)=theta. With this we can switch out sin(theta) for theta. The equation now reads, alpha=-2g*theta/3(R+r). This equation resembles the equation for radial acceleration, a=-(omega^2)*r. With this comparison, we can say that omega=(2g/3(R+r))^1/2. Work seen below.
Now that we have omega, the period can be solved for with the relation T=2pi/omega. The equation now reads T=2pi(3(R+r)/2g)^1/2. The radii were found using a pair of calipers; which turned out to be R=0.0696m and r=0.0576m. Plugging these values into the equation for the period gives T=0.876s. Work is seen below.
Now, to verify my prediction I placed the ring on a pivot and using a photo gate, found the period of its oscillation about its equilibrium position. Not only was a photo gate used, but a laptop and Logger Pro were used to take data. I placed a small piece of tape on the bottom of the ring and positioned the photo gate to pick up the tape passing through it. Below is my setup as well as the experimental period.
As you can see from the photo above, my calculated period is considerably higher than the experimental period. Calculating the percent error gives a value of 21.5% which is unusually high. Below are my calculations.
It could be that I measured the radii of my ring incorrectly making the period come out larger from my calculations.
For Part 2, we were asked to cut out a triangle and semicircle and find the calculated and experimental periods for either of them based on their orientation. I chose to use the triangle for my lab. After cutting out a triangle from cardboard, I took measurements using a pair of calipers and found that the height of my triangle was 15.3cm and the base was 15.0cm with a mass of 7g.
After taking measurements, I set about calculating a period for the triangle's motion as a pendulum hanging from its point. The calculated period came out to be 0.618s. Calculations for the period seen below along with calculation of the center of gravity for the triangle.
As can be seen from the pictures my calculated period for the triangle hanging about it's tip was quite off. A percent error calculation shows that I was 13.3% under my experimental value. Seen below.
