Procedure / Set Up:
The experiment was divided into three parts
- Determine the period of a physical pendulum composed of a solid ring, of mass M, outer radius R and inner radius r.
- Determine the period of a semicircular plate of radius R, oscillating about the midpoint of its base.
- Determine the period of a semicircular plate of radius R, oscillating about a point on its edge, directly above the midpoint of the base.
The apparatus consists of two ring stands fastened together to support a solid ring and photogate. At the bottom of the ring is a piece of tape that will pass through a photogate each time the ring moves back and fourth. The photogate will measure the period of our physical pendulum, which is what we are looking for.
In order to find the period of our physical pendulum, we will need to apply Newtons Second Law of torque, the parallel axis theorem, and simple harmonic motion.
First we calculated the moment of inertia about the pivot. Since the chosen pivot did not have a known moment of inertia, we needed to use the parallel axis theorem. Below are the calculations.
Once we calculated the new moment of inertia, we used Newtons Second Law of Torque and made our final equations look like α = -(ω^2)ø. Since ø is very small, the sinø in our equation simply becomes ø. We can calculate the period by taking 2π/ω. Our experimental value was T = 0.71767 s.
Lastly we ran the experiment and our theoretical period was T = 0.718100 s
We compared our two values T(experimental) = 0.71767 s and T(theoretical) = 0.718100 s and concluded a percent error of only 0.060 %. This shows that our modeled equation α = -(ω^2)ø is very accurate.
Part 2:
The same set up as in part 1, except we will be using a semicircular plate. The concepts are the same when determining the period of this physical pendulum. Find the center of mass, determine the moment of inertia, apply Newtons Second Law of torque, make the equation look like α = -(ω^2)ø. The dimensions of the pendulum will also be necessary.
It was first necessary to derive the center of mass of our semicircular plate. Below is the derivation. x com = 0 and the y com = 4R/3π
Next it was necessary to find the moment of inertia about the midpoint of its base, where the plate will be pivoted. I = 1/2 MR^2
After applying Newtons Second Law of torque, making the equation look like α = -(ω^2)ø, and measuring the radius R = 0.1097m, the period was determined to be T = 0.7215 s. This will be our experimental value.
Next we ran the the experiment and found our theoretical value to be T = 0.7126 s. We highlighted a fairly constant portion of the period vs time graph and took the statistics. The mean is our period.
We again compared our two values T(experimental) = 0.7215 s s and T(theoretical) = 0.7126 s and concluded a percent error of only 1.25 %. Again it shows how well our model works at predicting the period of a physical pendulum.
Part 3:
The same set up as in part 2, except the plate will be pivoted on its edge, directly above the midpoint of the base. We needed to determine the moment of inertia, apply Newtons Second Law of torque, make the equation look like α = -(ω^2)ø.
Since we know the moment of inertia about the flat end, we can use this value to determine the moment of inertia about the center of mass and then use that value to determine the moment of inertia about the end being pivoted. We are basically just shifting our way across the pivoted point on the flat end to the new pivot point directly above and parallel. We determined this moment of inertia I = 0.65117 MR^2.
Again we can apply Newtons Second Law of torque and make the equation look like α = -(ω^2)ø. We determined the experimental period to be T = 0.70707 s.
Lastly we ran the the experiment and found our theoretical value to be T = 0.6946 s. We highlighted a fairly constant portion of the period vs time graph and took the statistics. The mean is our period.
We then compared our two values T(experimental) = 0.70707 s and T(theoretical) = 0.6946 s and concluded a percent error of only 1.80 %.
Conclusion:
The goal for each part was to derive an expression that would allow us to determine the period for each particular physical pendulum. This was done using, moment of inertia, the parallel axis theorem, Newtons Second Law of Torque, and simple harmonic motion. In the end, it was extremely useful to make our equation for torque look like α = -(ω^2)ø. The angle of oscillation was small enough to let sinø=ø. For each part, our % error was very low, indicating a successful experiment. The error could have been from uncertainty in measurements, friction from the pivot, and possibly rounding errors.
