5-20-15: Conservation of Energy/ Conservation of Angular Momentum Lab

Lab 19: Conservation of Energy/Conservation of Angular Momentum

Purpose:
The purpose of this lab was to use the principles of conservation of energy and angular momentum to determine the max height of a swinging mass system.

Apparatus:

 

 

The apparatus used for this experiment consisted of a metal stand and a meter stick pivoted near one of its ends on a rotational sensor. Tape was wrapped around the other end of the meter stick. A piece of clay was also wrapped in tape and placed on a stand made of paper clips. The clay was strategically placed so that it would collide with the meter stick at the bottom of the swing. Both the clay and the meter stick were wrapped with tape so that the clay would stick to the meter stick and essentially create an inelastic collision.

 

A camera was also used during the experiment to record the collision and to determine the final height of the system. Furthermore, LoggerPro was used to analyze the video.

 

Procedure:

First, we recorded the mass of the clay and meter stick and measured the distance from the meter stick's center mass to the pivot point.

• Mass of clay = 19.4 g
• Mass of meter stick = 85.6 g
• Distance from CM to pivot = 0.487 m

Next, we setup the apparatus as seen above and positioned the camera so that we could see the end of the meter stick throughout the whole swing.

 

Now for the actual experiment. The meter stick pivoted near one end was released from a horizontal position. Right when it reached the bottom of the swing, the meter stick collided inelastically with a piece of clay. Then, the meter stick and clay rotated together to a final position. Video was recorded of this process. The video was used to determine the max height of the system.

 

Furthermore, we determined the max height of the system through use of the principle of energy and the principle of angular momentum.

 

Finally, we compared the theoretical max height to the experimental value.

 

 

Data:

 

Theoretical Value



As stated before, the theoretical height was found through use of the principle of energy and the principle of angular momentum.

First, we determined the meter stick's new inertia since it was not being pivoted from its center. This was done by the parallel axis theorem.

• Distance from CM to pivot = 0.487 m
• Mass of meter stick = 85.6 g

I(stick)= (1/12)ML^2 + M(.487)^2
I(stick)= 0.0274 kg*m^2

Next, we calculated the inertia of the clay. We treated this inertia as a point mass.

• Distance of clay from pivot= 0.9987 m
• Mass of clay= 19.4 g

I(clay)= 0.0194 *(0.9987)^2 = 0.0193 kg*m^2
Finally, we calculated the Inertia of the system.

• I(system)= I(stick)+I(clay)= 0.04675 kg*m^2

Now, with a calculated inertia for the stick and an inertia for the system, we can determine the max height of the rotating system.

First, we calculated the angular speed at the bottom of the swing. This was achieved based on the principle of conservation of energy. Next, we calculated the angular speed of the system after the inelastic collision. This was done through the principle of conservation of angular momentum.

In the picture above, you can see our calculated values of inertia as well as the angular speed of the stick before and after the collision.
Notice that after the collision in angular momentum, we used the inertia of the system and not the stick.

• Angular Speed at the bottom of the swing = 5.46 rad/s
• Angular Speed after the collision = 3.2 rad/s

Next, we symbolically solved for the max height of the system.

As you can see in order to find the max height of the system, we must find theta.

 

Fortunately, this was achieved through the principle of conservation of energy.

First, we established the pivot point as our GPE=zero mark. Then, we setup equations for the change in the GPE of the stick and the GPE of the clay. Finally, we took the sum of GPE of clay and meter stick and set it equal to the KE of the system. From here, we were able to solve for theta. Finally, we plugged theta into out equation for h and determined the max height of the system.



For our experiment, theta= 53.06 degrees.
Therefore, our theoretical max height of the system is 0.399 m

Experimental Value

In this portion of the lab, we captured video of the rotating system and analyzed the footage on LoggerPro.

In the picture above you can see how we tracked the movement of the clay after the collision up until the max height. Each blue point represents the clay's location at a given time interval. In order for LoggerPro to give us an appropriate max height of the system, we set a scale in the video using the meter stick. This was very easy since we already knew the length of the meter stick. Next, we established the origin at the bottom of the swing. From here, LoggerPro was able to plot the clay's movement and plot it on a graph. We adjusted the graph, so that it only gave us movement in the y-direction.

As you can see above, we used the Examine feature on LoggerPro to determine the max height of the system.

• Experimental Max Height = 0.387 m

Uncertainty/Error: The sources of uncertainty/error for this experiment are the following:

• Uncertainty in the measurements of the masses of the clay and meter stick.
• Uncertainty in the measurement of the distance between the pivot and the clay and the distance between the pivot and the meter stick center of mass.
• Uncertainty in plotting the points during the analysis of the video.
• Friction present during the experiment

Conclusion:

When comparing our theoretical value (0.399m) to our experimental value (0.387m), we can see that they are fairly close. In fact, there is only a 3% error. This helps us conclude that the sources of error or uncertainties present during the experiment are negligible to the point that it does not affect the outcome of the experiment. Furthermore, it was expected for the experimental value to be less than the theoretical value. Anything higher would raise some red flags, since the theoretical value is seen as the limit for the experiment. Lastly, we proved that the method of determining the max height of swinging system through the conservation of energy and the conservation of angular momentum is correct.