Lab 20 Physical Pendulum lab

Purpose: The purpose of this laboratory is to derive expressions for the period of various physical pendulums and verify the predicted periods by experiment.

The purpose of the apparatus is to measure the period of the pendulum so that the measured period could be compared to the period calculated theoretically.

Procedure:
In the first part of the lab we predicted the period for a metal ring. To do this we measured the diameter of the outer ring and the diameter of the inner ring. Then we used newton's second law to write an equation where the torque is equal to the moment of inertia of the ring times the angular acceleration. We rearranged the equation to get angular acceleration by itself and were able to find the angular velocity. We then used the angular velocity to calculate the period. Then we measured the actual period of the ring as a pendulum and we compared the two results.

For the second part of the experiment we cut out a half circle out and measured the radius of the half circle. Then we found the center of mass of the half circle so that we would know the distance of the force creating the torque as the pendulum swung. Then we derived an expression for the moment of inertia for the half circle because it was not one that we knew. Then we used newton's second law to and rearranged the equation to find the angular velocity. Using the angular velocity we found the period for the half circle as a pendulum. We did this for the pivot being at the top of the half circle and for the pivot being at the bottom of the half circle.

To find the angular velocity (omega) of a pendulum we wanted to use newton's second law equation and manipulate it so that it would look like this:

When we manipulate the equation to look like the one above, whatever is in the parentheses is omega^2. So we could then find omega and plug it into the following equation to find the period T:

The calculations for the period of the ring as a pendulum were as follows:

The equation for newton's second law was torque equaled moment of inertia times angular acceleration. We then solved for the angular acceleration to get it in the form that we wanted and the sintheta turned into just theta because theta was small. We added the negative sign because the force acting on the pendulum was always opposite its direction of motion. We then found omega and plugged it into the second equation to find the period which was .717 sec. We set up our ring on the stand and measured the period with logger pro and we got the measured period to be .719190 sec.

For the second part of the experiment we cut out a half circle and found the period if we let it swing as a pendulum in two positions.

The first thing we did was find the center of mass of the half circle, because the center of mass is where the force of the weight is moving the half circle as a pendulum. So to find the center of mass found an expression for a small piece of the mass and integrated it from 0 to R and the calculation was the following:

R being the radius of the half circle.

After we found the center of mass of the half circle we had to find the moment of inertia of the half circle and we did so by taking a small piece of the half circle and finding the moment of inertia of that piece and integrating it from 0 to R. The calculation went as follows:

After finding the moment of inertia we used newton's second law to find an equation that we could manipulate into the form alpha = -(    )theta. So that we could then find omega and use it to find the period. Our calculation to find the period for the half circle being hanged by the round side was:

 

The calculation to find the period for the pivot at this point:

the value that we measured on logger pro was:

Then we derived an equation to find the period if the pivot point were on the other side of the half circle:

 

The calculation to find the pivot point at this point:

The period value that we measured with logger pro for this point was.

Conclusion:
The period calculated theoretically for the half circle with the pivot on the round part was .7699 sec while the period measured was .6147 sec the values are pretty close so they are good for the equipment that we used. While the period calculated theoretically for the half circle with the pivot on the long part of the half circle was .3933 sec and the measured was .6189 sec. the values are more off for this one most likely because the calculation might be incorrect that is the only thing I can think of but I felt like it was the way it should have been done.

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.