20-May-2015: Conservation of Energy/Conservation of Angular Momentum

Purpose: Using our understanding of conservation of energy and angular momentum, we were to predict how high a pivoted stick/clay combination will rise after the stick inelastically collides with a stationary blob of clay.

Procedure: In order to set up our apparatus, we first grabbed a tall vertical rod and attached a smaller horizontal rod at the top so that we could hang a meter stick just above the table. Next, we pivot the meter stick near but not exactly at its end and wrap tape (inside out) around the bottom so it could swing and catch the stationary blob of clay at the bottom of the vertical. The clay blob was also wrapped with tap to ensure that they stuck when colliding.

lab  apparatus with video capture

Our goal was to calculate how high the stick/clay combination would rise after being released from a horizontal position by using our understanding of conservation of energy and conservation of angular momentum. Then, by setting up Logger Pro's video capture software along with a camera to record how high the combination rose in our experiment, we were to compare our theoretical result with our actual experimental result.

In order to derive our theoretical result, we had to first measure the the mass of the meter stick, the mass of the clay blob, and the length of the meter stick from the pivot to the bottom end. The idea is to break the problem down into three parts, from conservation of energy before the collision, to conservation of angular momentum during the collision, and back to conservation of energy to find the maximum height of the system.

Our measured values are below
Mass of stick - 142 g
Mass of clay - 32.0 g
Length of stick from pivot - 98.5 m

We know that the moment of inertia pivoted at the center is 1/12 ML^2, but the pivot is not exactly at the end of the stick. We had to apply the parallel axis theorem to derive our new moment of inertia for our calculations.

Part 1: Conservation of Energy

At the horizontal position, the meter stick has a potential energy and that energy transfers completely into kinetic energy at the bottom of the vertical. We want to use conservation of energy to solve for the angular velocity of the stick just before it collides with the clay blob. We set our gravitational potential energy to 0 at the top of the pivot so that it has negative potential energy at the vertical position.

Part 2: Conservation of Angular Momentum

Now that we have the angular velocity of the stick at the bottom of the rotation just before the collision, we can use conservation of angular moment to solve for the angular velocity of the stick/clay system after the collision. We also had to solve for the new moment of inertia of the system using the parallel axis theorem and adding the moment of inertia of the clay blob using its distance from the pivot.

Part 3: Conservation of Energy

Using the angular velocity of the system, we can solve for the maximum angle the system makes using conservation of energy once again. Then we can use that angle to solve for the maximum height of the system.

Below is our calculations for each part.

calculations for maximum height of the stick/clay system

Our theoretical max height of the system came out to 0.40 meters.

Because we have our theoretical value of the maximum height, we can now use our video capture to record the actual maximum height that the system reaches after swinging down from its horizontal position. We set the stick in the horizontal position, begin recording, and release the stick into motion.

video capture of maximum height of system

Above is the image of our maximum height of the system. We set a ruler at the bottom and marked a distance of .3 meters so that the video capture could scale the height of the stick and we set the origin of the axis at the point of collision. By plotting the point on the image, we were able to get the x and y position on a graph shown below.

position of maximum height of stick/clay

Results: Our experimental value came out to h = 0.3676 m. compared to our theoretical value of h = 0.4000 m, we have a percent error of 8.1 %.

Conclusion/Uncertainty: By using the conservation of energy and angular momentum theories, we were able to predict the maximum height of a swinging stick/clay system. We broke the problem down into a conservation of energy problem before the collision, a conservation of angular momentum problem during the collision, and again a conservation of energy problem after the collision. Then we used the video capture to record the actual maximum height of the system to find that our theoretical value to be 8.1% off of the experimental value. Although our percent error was slightly higher than we would have liked, we can say that our results were reasonably accurate. We can attribute some of the error to the fact that the pivot was not exactly frictionless and also the video capture method does not give us the exact value for the maximum height of the system because we can only stop the video at certain intervals and it is hard to perfectly pinpoint the center of the system at the height we decided was the max. Finally there is some propagated error in our measurements of the masses and lengths of stick and center of mass.