May 31, 2017 Lab 19: Conservation of Energy/Conservation of Angular Momentum

Lab 19: Conservation of Energy/Conservation of Angular Momentum

Chris Ceron, Amy Chung, John Choi

Goal of Lab:
Test the conservation of energy and conservation of angular momentum by colliding a meter stick on pivot against a clay ball on the ground.

Theory/Introduction:
With the absence of external forces, angular momentum is conserved much like linear momentum. Conservation of angular momentum is labeled with an L and the components are as follows:





I is the inertia of the object, it could be a system of objects where the total inertia is the sum of all the individual inertia about the axis of rotation.

The conservation of energy states that these two components are equal:



We view our ball of clay as a point mass, making its inertia:



Where L is the distance away from the axis of rotation.

We use the conservation of energy to find the angular velocity of the system. If we assume we start the meter stick from a horizontal position our change in height for the meter stick is known.

Once we have the initial omega, we are able to solve for the final omega after the collision using the conservation of angular momentum.

Since the axis of rotation for the meter stick is not precisely at the end (for our experiment it was at the 2 cm mark on the stick), we use the parallel axis theorem to find a moment of inertia about a new axis:





now applying this to the conservation of angular momentum:



gives us a value of omega final.


This omega value can be used in our energy equations where the initial energy is rotational and transfers to gravitational potential:

The only unknown within this equation is the value for theta so we can isolate and solve for it:



cosine inverse gives us the angle for which the meter stick travels up after sticking with the clay.



using this angle, we can find the height the clay rises by isolating the distance the clay is away from the axis of rotation and the 1 - cos theta portion:




Apparatus and Procedure:

Apparatus used (meter stick has already collided with ball of clay)

We set up our apparatus using a stand attached to a pivot point and put the pin through the hole on the meter stick(2cm mark of the meter stick). Using Amy's slow motion capture on iPhone recording at 240 frames per sec, we took a video of the meter stick colliding with the clay. Using LoggerPro's video analysis program, we set a reference measurement, and found the highest point the clay reaches. Our reference point was the position of the clay before the collision.

Data and Calculations:

The point on the meter stick that the axis of rotation went through was the 2 cm mark (0.02 m).

	r distance from axis (m)	Mass (kg)
Meter stick	0.48	0.02357
Clay	0.98	0.1444

Above is the measured data required for calculation. Below is the LoggerPro capture:

The video capture as well as the height reading of the clay blob.

The height recorded was 0.4310 m.

The parallel axis theorem for the meter stick:



The conservation of energy to find omega initial:



and for omega final calculations, we need the total system inertia after collision:





Using these values, we found theta to equal:



and to find the max height of the clay using the angle above:



Conclusion:

Calculating for percent error between our experimental value and theoretical gives us:



This is a large margin of error that was the result of some sources of uncertainty. We neglected the friction at the axis of rotation. This causes the calculated height much higher than the experimental. We also made some other assumptions such as the clay being treated as a point mass and energy being conserved when it truly isn't (energy can be lost through friction experienced by the collision and at the pivot point during the collision). Uncertainty in our distance from axis of rotation measurements probably also played a small role as the measurements we came up with were rather rough estimates.