6-7-2017: Lab 19: Conservation of Energy and Momentum - Angular

Lab 19: Conservation of Energy/Conservation of Angular Momentum

Joel Cook

Nina Song

Eric Chong

Lab finished May 31, 2017

In this lab, we are attempting to predict the height to which an object will rise after a collision and compare this value to the height obtained experimentally. 

Theory: 

For our system, a meter stick is attached to a pivot point and a piece of clay is placed on the floor in the path of the meter stick. The meter stick collides with the clay and the two rotate to some height together. We can predict the height by knowing that the energy will be conserved:

Kinetic energy of clay and meter stick together immediately after collision = Gravitational potential energy of the clay and stick at the highest point

1/2*I*ω^2 = m(stick)*g*Δh(center of mass of stick)+m(clay)*g*Δh(center of mass of clay)

To calculate the angular velocity of the system, we must recognize that the angular momentum for the collision will be conserved:

Angular momentum before = Angular momentum after

I*ω(initial) = I*ω(final)

Finally, to find the initial angular velocity, we will use the conservation of energy from when the meter stick is released to the moment it collides with the clay:

Gravitational potential energy = Kinetic energy immediately before collision

m(stick)*g*Δh(center of mass) = 1/2*I*ω^2

Therefore, by starting from the moment when the meter stick is released, we can finish with predicting the height of the center of mass of the clay after the collision.

Procedure:

The apparatus pictured below was used to allow the meter stick to rotate freely. The apparatus consists of a rod clamped to the the table, a pivot point clamped to the rod and a meter stick attached to the pivot at 10 cm from the end of the meter stick. 

Pins were taped to the bottom of the meter stick to ensure the clay will stick after the collision and the clay was stood up on pins to facilitate the collision and adhesion. The clay was positioned such that the meter stick will strike the clay at the bottom of its swing and the two will travel together to some final height before rotating back.

Slow motion video capture was used to record the collision. The video was analyzed in Logger Pro to find the final height of the clay and meter stick, as shown below. By advancing the video one frame at a  time, we were able to click on the highest point to which the clay travelled. The meter stick in the video was used to properly scale the video and the origin was placed at the point where the stick and clay collide.

Data/Analysis:

As the above photo shows, the clay rose to a height of 0.3245 meters. The following photo shows the calculations to determine the angular velocity of the stick just before the moment of collision. The moment of inertia of the meter stick, using the parallel axis theorem is:

I = 1/12*M*L^2 + M(4L/10)^2

The angular velocity of the stick was 5.68 radians/second.

Next, the angular velocity was used to calculate the angular velocity of the stick and clay after the collision. The moment of inertia of the clay:

I = ML^2

The angular velocity of the clay and stick after the collision was calculated as shown as 2.886 radians/second.

To find the height to which the clay travelled, we calculated the change in height of the center of mass of the clay:

The final height of the clay was calculated as 0.3138 meters. 

Compared to the height from the video analysis of 0.3245 meters yields a difference of 3.3%. 

Conclusion: 

The experiment was successful. Our prediction for height was only 3.3% different from the height obtained experimentally from one trial. There are sources of error worth noting, as well. Although slow motion video was used to capture the collision and subsequent motion, it is difficult to perfectly select the center of mass of the clay in the video. Our prediction and experimental value were approximately 1 cm different which could easily have occurred in video analysis. We used a very accurate scale to reduce the error in measuring the masses used in the experiment. Finally, the bottom of the meter stick and the clay were not touching the ground and there was friction not accounted for.