Lab 19 - Conservation of Energy/Conservation of Angular Momentum

Purpose: To use conservation of angular momentum and angular kinetic energy to predict how high a swinging meter stick will go after colliding inelastically with a piece of clay sitting on the ground.

Apparatus: We used a clamp to hold a sensor that is normally for logger pro for measuring angle(maybe?), except we just used the sensor as an axis of rotation because it was relatively simple to attach a meter stick to. The meter stick had a hole at the 0.015m mark,  and this is where we threaded a screw through in order to make the meter stick swing about an axis. We then bent a paper clip into an asymmetrical shape in order to provide a little stand so that a piece of clay is not making contact with the ground, which reduces friction between the clay and the ground. This is important because if the clay drags on the ground after the meter stick swings down and catches it, much energy will be lost in the system to friction/thermal energy. We wrapped tape (sticky side out) around the meter stick and around a ball of clay. The tape was to make sure that this would be an inelastic collision.

Procedure:  Mass of meter stick = 0.1612kg                    Mass of clay = 0.0387kg

We lifted the meter stick to be horizontal, then let it go. It swung down and collided inelastically with a piece of clay with tape on the ground that was being held up by a paper clip, and we took a video of the whole incident from start to finish. We then uploaded to logger pro and analyzed how high the piece of clay rose in the direction of the y-axis.

Below are my calculations for the moment of inertia of the meter stick both before and after collision. There are two different moments of inertia because after collision, there is a "point mass" rotating about an axis. This point mass is the piece of clay that sticks onto the ruler. In the calculations below, we had to use 1/12mr^2 with a parallel axis shift instead of 1/3mr^2 because the axis of rotation is not exactly at the edge of the ruler, and the parallel axis shift only works with the moment around a center of mass. We then set GPE=Rot. KE to find the value for ω at the bottom of the swing. The GPE used half the height of the meter stick because gravity "acts" on the center of mass only.

Next, just like in linear collisions, we conserved angular momentum to find the new angular velocity after impact.
Lbefore=Lafter
Iω=Iω
Lafter takes into account the moment of inertia of the clay combined with the moment for the meter stick.

The scratch marks below are where I learned that I cannot simply equate the Rot.KE back to GPE to get a value because it was way off.

What we had to do was a little more involved, but no where near impossible. Instead of calculating the new center of mass for the meter stick-clay system, we separated the two different changes of height for the two centers of mass. We said GPE(clay)+GPE(cm ruler)= 1/2Iω^2.
 Then by similar triangles, or a proportion of the change in height of the centers of mass relative to the position along the ruler they were at, to the axis of rotation, we came up with a way to eliminate one unknown variable. This made it possible to solve the equation for a single variable which was the change in height of the piece of the clay. We chose to solve for this center of mass because it was much more easier to pinpoint the position of the clay on logger pro as opposed to the center of the meter stick.

Theoretical height = 0.38246m

The picture above shows the maximum height of the clay after impact.
Max height = 0.3541-0.0001197= 0.3539803m

Percent error = 7.45%

The amount in error in this experiment is not because we did something wrong. It is because there are factors that contribute to significant error in this lab. Some of these errors include the fact that no matter how many times we tried, we could not get the collision to happen without the clay catching some friction on the ground, which uses up KE to create thermal energy. Another source of error in this lab is that mapping the position of the clay on logger pro is not very precise. This creates error because clicking a tiny bit off from the intended position on the video causes the data to be off by a magnified amount.

Conclusion: It is pretty cool to see that many of the same principles for linear calculations still apply in rotational bodies. Also, I learned that although calculations are a little more involved, they are fairly straight-forward. I have a better sense of understanding of the physical world around me and can see that in a collision, to calculate everything I need to break things up into phases. I think I have a pretty solid understanding of collisions after this lab, including collisions with rotating objects. I learned that if things stick together, momentum (whether linear or rotational) must be conserved because a "new" object with a different mass, which changes the velocity of the system. Then, energy can be "conserved" in an equation, but if you equate it back to GPE, you must find the center of mass for it to be valid.

Lab Partners:
Haokun Zhang
Christian Rivera