20-May-2015: Conservation of energy/conservation of angular momentum

PURPOSE

Figure 1

In this experiment, we attempted to predict a how high above the ground a system would reach after a meter stick rotating about a pivot collides with a piece of clay at rest (Figure 1). In order to do so, we employed a combination of conservation of energy and angular momentum equations.

PROCEDURES

Figure 2

Figure 3

For the first part of our calculations, we found the moment of inertia of the meter stick, which had a mass of 0.08 kg, rotating about an axis that was 0.475 m away from its center of mass (Figure 2). We used the parallel axis theorem (I = Icm + md²) and found this value to be 0.02472 kg*m². Next, we applied the conservation of energy equation to the meter stick rotating from the horizontal position to the vertical position as shown in Figure 3. We set the reference point as the meter stick's center of mass in its final position. Therefore, the meter stick had an initial gravitational potential energy, which eventually all turned into rotational kinetic energy. We set the initial gravitational potential energy (mgh) equal to the final kinetic energy (½Iω²) and solved for the final angular velocity (ω). We found this value to be 5.492 rad/s.

Figure 4

Afterwards, we applied the conservation of angular momentum equation to the meter stick-clay system before and after the collision. For the initial angular momentum, we found the product of the moment of inertia of the meter stick by itself and the angular velocity found earlier with the energy equation (ωo = 5.492 rad/s). On the other hand, the final angular momentum was equal to the product of the moment of inertia of both the meter stick and the clay, and the angular velocity right after the collision (ωf), which was unknown. We found the moment of inertia of the piece of clay by treating it as a particle and applying the equation Ic = mcd², as shown on the bottom of Figure 4. Then, we solved for ωf, which was 2.746 rad/s.

Figure 5

Figure 6

Then, for the final step of our calculations, we set up the conservation of energy equation by using the diagram shown in Figure 5. First, we established the pivot as the reference point and up as the positive y-direction. Then, we used trigonometry to figure out that the vertical distance between the pivot and the final positions of the meter stick and piece of clay were 0.475cosθ and 0.975cosθ, respectively. This resulted in the equation shown on the very top of Figure 6. Plugging in I = Isys = 0.02472 + 0.02472 = 0.04944 kg*m², ω = 2.746 rad/s, m1 = 0.08 kg (mass of meter stick), m2 = 0.026 kg (mass of clay), and g = 9.81 m/s², we were able to eventually solve for cosθ. We found this value to be 0.7001. We took the inverse cosine of this number to find the angle that the system made with the vertical, which we found to be 45.57°. We realized later that finding the angle was unnecessary because the value we really needed was cosθ.

Figure 7

From this angle, we used basic trigonometric relationships to find the vertical distance between the final position of the piece of clay and the ground, as shown in Figure 7. One thing to note is that we treated the piece of clay as a particle as we did throughout the rest of the calculations. Therefore, the dimensions of the clay piece were not considered in the derivations. From Figure 7, it can be seen that the theoretical maximum height of the stick-clay system was 0.2924 m.

Figure 8

Finally, we ran the actual experiment and found out how high above the ground the system actually reached after the collision. We did this by utilizing video analysis in Logger Pro. As it can be seen from Figure 8, We found the height to be 0.2904 m. This gave us a percent difference of 0.6840 percent. This accurate result was due to the fact that gravity was the only significant external force acting on the system throughout the entire process.

CONCLUSION

In this experiment, we broke down a collision into three different stages: motion before the collision, the collision itself, and the motion after the collision. We utilized conservation laws during these stages, which ultimately allowed us to estimate how high above the ground a meter stick and a piece of clay would rotate after a collision occurred between the two objects.

As mentioned earlier, the percent error was very low in this experiment because the only external force that had a considerable impact on the results was gravity. However, another reason the theoretical and experimental values were so close to each other was that we did not factor in the clay piece's dimensions into our calculations. If we had tracked the distance between the end of the meter stick and the ground (which is what our theoretical value actually represents) instead of the center of the clay piece, it would have been 1 to 2 cm smaller. Therefore, the percent error would have been approximately 3 to 7 percent higher. However, this is to be expected because there were other assumptions we made in our calculations, First, we did not account for the friction between the pivot and the meter stick as it was rotating. In addition, there was probably some angular momentum lost during the collision since it was not a perfectly head-on collision. This explains why the experimental value was smaller than the theoretical one.