1-Jun-2015/3-Jun-2015: Simple harmonic motion

PURPOSE

The purpose of this experiment was to derive equations that modeled objects in simple harmonic motion. This allowed us to find the theoretical values for their periods, which we compared to the values we found experimentally.

PROCEDURES

Period of a Ring in Simple Harmonic Motion

Figure 1

Figure 2

For the first part of the experiment, we attempted to derive an equation for the period of a ring in simple harmonic motion. The first step we took in order to accomplish this task was set up the torque equation as shown in Figure 2 by using the diagram displayed in Figure 1. Using the moment of inertia of a ring rotating about its center of mass [I = 1/2 M(R² + r²), where R = 0.0696 m, r = 0.0579 m, and M does not matter because it ends up canceling out], we were able to find the moment of inertia of the ring when it was rotating about a pivot that was a distance Ravg = 0.06375 m (the average of the inner and outer radii of the ring) away from the ring's center of mass by using the parallel axis theorem (I = Icm + Md²). Next, we solved the equation for α and assumed that sinθ was approximately equal to θ for very small values of θ. We took these steps in order to put the equation in the general form of simple harmonic motion (α = -ω²θ). From the resulting equation, we were able to deduce that ω was equal to [(2gRavg)/(R² + r² + 2Ravg²)]^1/2 = 8.753 rad/s, which we plugged into T = 2π/ω to find the period. We found this value to be 0.7178 s,

Figure 3

Following the calculations, we set up the experiment as shown in Figure 3. We pulled the ring back at an angle and released it to let it oscillate. We captured the ring's period using a pre-programmed file in Logger Pro, which we found to be 0.72 s. This gave us a percent error of 0.306 percent, which is pretty much as accurate as an experiment can be. This was probably due to the fact that there wasn't much room for error since the only force affecting the ring's motion was gravity. The only factor that could have contributed to error in the results was the sway of the ring when we first released it from rest. However, this did not have a significant effect on the results because the ring stabilized itself after a few oscillations.

Period of a Triangle in Simple Harmonic Motion

Figure 4: Derivation of the moment of inertia's

We began the second part of the experiment by deriving the moment of inertia's of a triangle with the axis of rotation at its apex and a triangle with its axis of rotation at the center of its base. First, we found its moment of inertia with respect to its apex as shown in the top portion of Figure 4. We designated dM to be a thin bar with its center of mass a distance y away from the axis of rotation. Therefore, in order to find the moment of inertia of dM, we had to apply the parallel axis theorem (with d = y) on the moment of inertia equation of a thin bar rotating about its center of mass (I = 1/12 ML², where L = 2x in this case). We were able to find the moment of inertia of the entire triangle by integrating the resulting equation from 0 to H. We found this value to be I = 1/24 MB² + 1/2 MH². Then, we used the same dM to find the center of mass of the triangle. We did this by using the equation ycm = (∫dM*y)/(∫dM). We found the center of mass to be 2/3 H from the triangle's apex. From this value, we were able to apply the parallel axis theorem again (with d = 2/3 H) to find moment of inertia of the triangle with respect to its center of mass. We did this by turning the equation from I = Icm + md² into Icm = I - md². From this step, we were able to find the moment of inertia of a triangle rotating about its center of mass was I = 1/24 MB² + 1/18 MH². Then, using d = 1/3 H, we implemented the theorem a third time to find the moment of inertia of the triangle rotating about the center of its base. The resulting equation was I = 1/24 MB² + 1/6 MH².

Figure 5: Derivation of the period of the triangle rotating about its axis

Figure 6: Derivation of the period of the triangle rotating about the center of its base

After finding the moment of inertia of the triangle with respect to the two axes of rotation, we were able to derive equations for the period of the triangle in both orientations (Figures 5 and 6). Just as before, we first set up the torque equation and set everything equal to α. In addition, we assumed sinθ to be approximately equal to θ for very small values of θ. We did this to get the equation in the general form of systems in simple harmonic motion (α = -ω²θ). From the resulting equation, we concluded that ω was equal to the square root of (16gH)/(B² + 12H²) for the triangle rotating about its apex and the square root of (8gH)/(B² + 4H²) for the triangle rotating about the center of its base. Then, we found the period by dividing 2π by these values. The resulting equations were 2π[(B² + 12H²)/(16gH)]^1/2 and 2π[(B² + 4H²)/(8gH)]^1/2, respectively.

Figure 7: Theoretical values of the periods

After deriving the equation, we plugged in the appropriate values (B = 0.14 m, H = 0.147 m, and g = 9.81 m/s²) to find the theoretical periods. For the triangle rotating about its apex, we found the period to be 0.69081 s. On the other hand, for the triangle rotating about its base, we found the period to be 0.60238 s. The mathematical processes are displayed in Figure 7.

Figure 8: Rotating about its apex

Figure 9: Rotating about the center of its base

Following our calculations, we set up the triangle about its apex and released it from a slight angle to let it oscillate. Its period was measured using a pre-programmed set-up for a pendulum on Logger Pro. As it can be seen from Figure 8, the experimental value for the period was 0.695362 s, which was only 0.659 percent away from the expected value. Then, we set up the triangle about the center of its base and reran the experiment. The resulting data is shown in Figure 9. From Figure 9, it can be seen that the experimental period for the triangle rotating about its base was 0.607886 s. In this case, the percent error between the theoretical and experimental values was 0.551 percent. Once again, there was virtually no error in the results.

CONCLUSION

In this experiment, we found the moment of inertia of objects rotating about different points to set up torque equations. Then, we set everything equal to α to get the equations into the general form of simple harmonic motion (α = -ω²θ). This allowed us to figure out what ω was, which in turn allowed us to find the period of the motion. We then ran the actual experiment and compared the theoretical values to the experimental ones.

The theoretical and experimental values were almost identical. As mentioned before, the reason was because the only force affecting the ring's motion was gravity. Since the force of gravity is constant as long as we're close to the Earth's surface, there was not much room for error. One thing that was concerning at first was the sway of the objects when we first released them and let them oscillate. This ended up not having much of an affect on the results because the objects stabilized after a few oscillations. Another factor that I thought would affect the results was the friction in the pivot. This most likely did not alter the results significantly because the pivot had a smooth outer surface.