20-May-2015: Conservation of Energy and Angular Momentum Lab

The goal of this lab was to predict the height at which a simple pendulum would rise after colliding with a stationary piece of clay.

Setup:
To set up this lab, we used a rotary motion sensor as a pivot for a meter stick to act as a pendulum on. Once the meter stick was attached to this pivot, we setup a small piece of clay on the ground directly in path the meter stick would be falling. To ensure the collision would be inelastic, we wrapped the clay and meter stick with tape, sticky side out. Ideally, they would stick together after the collision (see Fig. 1). We then measure the mass of the meter stick as well as the clay.

Fig. 1Setting up the experiment.

Once we had the masses, we found the position of the pivot point for the simple pendulum. This was located roughly 0.0085 m away from the top end of the meter stick. Once we had all this basic information, we were ready to begin finding our prediction.

Calculating the Height:
To find the height at which the clay and meter stick system would rise, we split the collision into three components. First was the initial fall of the meter stick after being held horizontally. Second was the change in angular velocity after the collision. Third was the final height the system would reach after the collision.

PART 1: Initial Angular Velocity
To find the angular velocity of the meter stick directly before the collision, we used conservation of energy. Initially, as the meter stick was position in a completely horizontal manner, we only had gravitational potential energy. If we called the center of mass of the meter stick in the vertical direction our zero point for GPE, then our final energy would be rotational kinetic energy. (see Fig. 2)

Fig. 2
To do this calculation, we set our zero at the vertical center of mass of the meter stick.

Directly before the collision, we found our angular velocity to be 5.445 rad/s.

PART 2: Final Angular Velocity
To find the angular velocity of the system directly after the collision, we used conservation of angular momentum. However, to find the moment of inertia in the final stage of the collision, we had to include the moment of inertia of the clay as a point mass. (see Fig. 3)

Fig. 3
We used conservation of angular momentum in this section and included the moment of inertia of the clay.

Directly after the collision, we fond our angular velocity to be 3.858 rad/s.

PART 3: Finding the Height
To find the final height of the clay, we again used conservation of energy. To make the problem simpler, we called the pivot point of the meter stick our zero for GPE. This in turn would make our GPE values negative. In the final energy section of the equation, we multiplied our heights relative to the cosine of the angle at which the meter stick would make with the vertical. (see Fig. 4)

Fig. 4
To make this problem do-able, we tried to find the angle at which the pendulum swung rather than the actual height of the clay.

We found the theoretical angle of the pendulum to be 63.6 degrees. To find the height, we assumed that the clay was directly at the end of the meter stick. Thus, the height it rose would be the difference between the length of the clay to the pivot point, and the cosine of the length of the clay to the pivot point. In final, we predicted the height of the meter stick to be 0.5514 meters.

PART 4: Comparing to the Actual Height:
Using video analysis, we recorded the collision of the meter stick and clay. Setting up a horizontal and vertical axis, we placed a point directly at the maximum height of the clay. (see Fig. 5). We found this to be 0.4842 m.

Fig. 5
Using LoggerPro, we were able to find the actual height the clay rose.

We found our percent error to be 13.87 % at the conclusion of the lab. (see Fig. 6)

Fig. 6
Percent Error Calculation.

Error and Conclusion:
There were a few sources of error in this lab. For one, we assumed that the meter stick was completely uniform and it's center of mass was located directly at the 50 cm mark. Having degraded over time, this may not have been entirely true. Also, we chose to ignore the fact that energy was lost in other forms during the collision, such as in sound or friction. Thus, it makes sense that our predicted value for height is a little bit over the actual value.

Overall, the lab was a success and we were able to prove that energy and angular moment was conserved in this inelastic rotational collision.