phys4AS15hppham: Conservation of energy and angular momentum

Objective:

Apply principles of conservation of energy and angular momentum to theoretically calculate the final height of a system (clay+stick) after the stick swings and collide inelastically with the clay.

Physical problem & Approach:

Basically, we have a meter stick held from a horizontal position and a clay is on the floor. We let go of the stick, it swings and collides inelastically with the clay, then they both continue swings to the final position before swinging back down. The goal of this lab is to find the final height that the system reaches before swinging back. Here is how we approach to solve this problem:

Step 1: First of all, we apply the conservation of energy before collision. We choose to put gravitational potential energy equal 0 at the center of mass of the stick when the stick is vertical. This means that when the stick is held horizontally, it has a gravitational potential energy. Then when it swings until it is vertical, the gravitational potential energy is transferred into kinetic energy. We apply this concept to figure out the angular velocity of the stick right before it reaches to the bottom of swing.
Step 2: Then, we apply the principle of conservation of angular momentum during collision to find out the angular velocity of the system which is the combination of the stick and clay. The moment of inertia of the stick itself and that of the whole system are found in order to calculate the angular velocity after collision.

Step 3: Lastly, we apply the concept of conservation of energy again to calculate the final position of the system before it swings back down. We choose to put gravitational potential energy equal 0 at the pivot point (near the top of the stick). Given that, we have kinetic energy and gravitational potential energy for both meter stick and clay at start, then kinetic energy is completely transferred into gravitational potential energy when it reaches to the final height. With the value of angular velocity from second step, we can find the angle between its vertical position and its diagonal position. Using the angle to calculate the final height. Here is the diagram.

Calculating theoretical height of the object:

We first measure dimensions of objects we need for calculation. The meter stick is one meter long, the mass is measured by using a balance. Likewise, the mass of the clay is obtained. The distance from the edge to the pivot point is also measured. Below is the data we obtained.

Then applying the concept of conservation of energy explained in step 1 in our approach to find the angular velocity of the meter stick when it is vertical. Here is the work.

As we have the angular velocity of the meter stick, we then apply the concept of conservation of angular momentum to find the angular velocity of the system after the collision. Here is the work.

Lastly, following step 3 in our approach, apply conservation of energy to find the final position of the system.

WOW! We have a theoretical value of final position. Now what we need to do is to actually perform an experiment to check how our theoretical value compares to experimental value. The closer these values are, the more accurate we get for both our calculation and experiment.

Set-up:

Let a metal rod stand on the floor holding by three legs. Secure a horizontal rod on the vertical rod using a clamp.
Mount the rotational sensor on the horizontal rod, then nails the meter stick to the rotational sensor to get the pivot point. The other end of the meter stick is wiped with a tape. A clay stands on the floor holding by three paper clips (we use them as legs). A tape is also used to wrap around the clay in order for the inelastic collision to happen between the meter stick and clay.
Set up a camera far enough to capture the whole experiment. The camera doesn't necessarily see the initial horizontal position of the meter stick; however, it has to capture the final position of the whole system after collision. Connect the camera to Logger Pro.

Hold the meter stick at its horizontal position, then let it swing down. The meter stick collides with the clay, then both continues to swing to the final position. The video is captured and analyzed in Logger Pro.

Data Collection and Analysis:

The video is captured, and we start to analyze it by dotting position of the clay after collision until it reaches its final height.

The origin is placed at the point of collision where the clay stands, the scale is set equal the distance from the the pivot point to the clay. Here is the picture of how we analyze the video.

Discussion:

From an actual experiment, we get an experimental value of final position is 0.504m, while our theoretical value is 0.554m. The percent error we calculated is 9.03% which is pretty good given so many uncertainties happen in this experiment. Moreover, both values in this experiment are small, thus a small change in both values leads to easy change in percent error. Therefore, it's not surprised to obtain a percent error equal 9.03%.
There are some sources of uncertainties happening in this experiment:

Uncertainty in measuring dimensions of the mass of meter stick and clay.
Uncertainty in dotting points when analyzing the video.
Friction in rotational sensor.
A small amount of energy is transferred into sound and heat, not completely transferred from gravitational potential energy to kinetic energy and vice versa.
Rounding number when calculating theoretical value of final position also affects the percent error.
The meter stick is assumed perfectly straight, but it is actually not. It bends a little bit at one edge.
Uncertainty in setting a scale when analyzing the video.

Conclusion:

In this experiment, we apply the principle of conservation of energy and angular momentum to figure out the final position of the system (stick+clay) reaches before it swings back down. The conservation of energy is applied before and after collision, while the conservation of angular momentum is applied during collision. The concept of moment of inertia and parallel axis theorem are also used during calculation.
The theoretical value of final position is found to be 0.554m, while the experimental value is 0.554m. The percent error is calculated to be 9.03%, which is pretty good given errors during experiment. Our good percent error shows that the method we used to calculate the theoretical value is correct. It also refers that our set-up and procedure is "on point" in finding the experimental value.

phys4AS15hppham