Lab 19 Conservation of Energy/ Conservation of Angular momentum

Purpose: The purpose of this laboratory was to come up with a method to calculate theoretically how high, a meter stick swinging down from the horizontal position colliding inelastically with a piece of clay, would rise after the collision. Then compare it to the actual value from capturing the collision on video.

The purpose of the apparatus was so that we could capture the collision on video and use logger pro to measure the height the meter stick and clay rose after the collision.

Procedure:
For this lab experiment we measured the mass of the meter stick and the mass of  the piece of clay that we used. We also noted the point on the meter stick where the pivot was at because it was not at completely at the end of the meter stick. Then using the information that was measured we calculated the theoretical height the meter stick would rise after the collision using conservation of energy and conservation of angular momentum. After we calculated the theoretical value that the meter stick and clay would rise after the collision we recorded a video of the collision and found out on logger pro how high the meter stick and clay would rise, then compared our results.

The measured values that we took before the experiment were the following:

Where M is the mass of the meter stick, m is the mass of the clay, L is the distance from the pivot to the long end of the meter stick, and d is the distance from the pivot to half the length of the meter stick.

Calculating the height that the meter stick and clay after the collision involved three steps. The first step involved conservation of energy. When we raised the meter stick to the horizontal position it had potential energy, and after we released the meter stick a moment right before it collides with the clay it has kinetic energy and potential energy as well because our potential energy was equal to zero at the pivot. And we found the omega right before the collision to use for the initial omega in the conservation of angular momentum equation.

The conservation of energy equation used was:

 

Where M is the mass of the meter stick, L is the distance of the meter stick from the pivot to its longest end, d is the distance from the pivot to the middle of the meter stick, g is acceleration due to gravity and omega final is angular velocity right before the collision.

Since we put our gravitational potential energy to be zero at the pivot point our initial gravitational potential energy was zero. At the end there was kinetic energy and negative gravitational potential energy. And our calculation for the omega at the end was:

Using the angular velocity calculated before the collision we used conservation of angular momentum to calculate the angular velocity after the collision. The equation that we used for conservation of angular momentum was:

 

 

 

Where M is the mass of the meter stick, L is the distance from the pivot to the long end of the meter stick, d is the distance from the pivot to half of the meter stick, omega final is the angular velocity of the meter stick before collision, m is the mass of the clay, R is the distance the clay is from the pivot point, and omega x is the angular velocity of the meter stick and clay after collision.

Our calculation for the angular velocity after the collision:

After finding the angular velocity of the meter stick and clay after the collision we used conservation of energy to find the angle that the meter stick rises after the collision. And using that angle we calculated the height that the meter and clay rose. We used the following equation:

Where M is the mass of the meter stick, L is the distance form the pivot to the long end of the meter stick, d is the distance from the pivot to half of the meter stick, m is the mass of the clay, R is the distance of the clay is from the pivot, omega x is the angular velocity after the collision, g is acceleration due to gravity, and theta is the angle the meter stick and clay elevate after the collision.

Our calculation for the angle theta is the following:

To get the height that the meter stick and clay rose we used the distance of the pivot to the end of the meter stick and subtracted the y component of the meter stick after it rose. We used the following equation:

The calculation to find h is the following:

Recording the actual experiment and using logger pro to find the height that the meter stick rose we get:

The calculated height that we got by taking the measurements and doing the calculations was .40m. While, the value that we got for recording the experiment was .3676m. The values that we got are pretty close, but there could have been some error maybe when we imputed the distance of 1 meter on the video the line we drew might not have been exactly at the meter. In all we learned how to used conservation of energy and conservation of angular momentum to calculated angular velocities before and after the collision and then theta and finally the height the meter rose after the collision.