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.

Lab 18 moment of inertia and frictional torque

Purpose: In this laboratory we are trying to determine the moment of inertia of the apparatus, and figure out a way to use video capture to determine the angular deceleration as it slows down. And calculate the frictional torque as it slows down.

The purpose of the apparatus is to use it to find the angular deceleration by taking a video of the apparatus spinning, and eventually use the ramp to find the time it takes the cart to travel one meter.

Procedure:
In this lab, the first thing we did was measure the diameters and lengths of the different cylinders that made up the rotating object on the apparatus. Then we found the volume of the two smaller cylinders and of the big disk to find the percent of volume the big disk was compared to the three put together. We then used that information to find the mass the large disk and the smaller cylinders because the mass would be the percent of the volume of each. We found the moment of inertia of the three parts together after finding the mass of each cylinder and the disk. Then we took a video of the apparatus rotating to a stop after giving it an initial push. Using the video analyzer we to marked an initial point on the apparatus and added another point every time the apparatus made a revolution so all our points were on the same spot and marked one revolution of the apparatus. Using the marked points we were able to make a position versus time graph of the points. Then we added a power fit line to get an equation of the position with respect to time. To get the deceleration we took the derivative twice of the position vs. time equation. Using newton's second law and angular acceleration we found the linear acceleration of the cart going down the ramp at the measured angle of the ramp. Plugging the acceleration in a kinematics equation we solved for the time it would take the cart to go a distance of one meter. Then we compared that value to the value we got by measuring the time it took the cart to go down the ramp a distance of one meter.

We measured the diameters and heights of the cylinders and the disks and used the measurements to find the volumes of each. Then we calculated the percent of volume the disk was so that we could find the mass because the mass percent is the same as the percent in volume. So we multiplied the percent to the mass of the whole that read on the three objects together and got the mass of the disk. And we got the mass of the smaller cylinders by subtracting the mass of the disk to the entire mass and divided it by two. The calculations for this are as follows:

To find the moment of inertia of the whole object we had to add the moment of inertia of the individual parts(the two cylinders and the disk). Individually the moment of inertia of a solid disk is 1/2mr^2 and since all three parts were solid disks we added this formula three times with the mass of the individual parts being m and r being the radius of each part. Our calculations were as follows:

Next we recorded a video of the apparatus spinning and coming to a stop, and plotted the points every time it made a revolution and put it on a distance versus time graph on logger pro. We power fitted the line to get an equation.

Using the equation of the line we took the derivative twice so that we could get the angular acceleration of the apparatus.

Using this calculated angular acceleration and converting it to linear acceleration using the equation alpha = a/r where a is the linear acceleration and r is the radius of the large disk. We used newton's second law to calculate the linear acceleration of the cart and the frictional torque of the disk.

Then we calculated the time it would take the cart would take to travel 1 meter using a kinematics equation:

This was our theoretical value so then we found the experimental value by letting the cart go down the ramp 1 meter and timing how long It would take. We did this three times and took the average time of the three.

Conclusion:
According to the data the experimental value for the time it would take the cart to travel 1 meter down the ramp was 7.98 seconds and the theoretical value that was calculated for the cart to travel 1 meter was  8.57 seconds. The values are pretty close but there is some error that could have came from the method that we came up to calculate the time theoretically. For example, when we were marking the point of one revolution the disk would take sometime it was hard to see the mark we made on the disk so we had to guess a little when the mark was at the point of revolution. this could have altered the acceleration a little. Also, the time we calculated experimentally of 7.98 seconds I feel was a little off we should have ran more trials to get a value that was more accurate.

Lab 17 Finding moment of inertia of a uniform triangle

Purpose: The purpose of this lab is to determine the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientations of the triangle.

The apparatus will be set up pretty much the same as the previous lab 16 except that this time there will be a triangle connected to the top.

The approach we took to accomplish our goal was by first determining the experimental value for the inertia of the system without the triangle and then the moment of inertia of the system with the triangle and subtracting the moment of inertia value with the triangle to the inertia value calculated without the triangle to get the inertia value of the triangle itself. We used the same method we used in the previous lab of angular acceleration where we measured the angular acceleration up and down, and got the average of the two to plug into the equation to find the inertia of the system. The equation was derived in the previous lab and is the following:

Where m is the mass of the hanging mass, g is the acceleration due to gravity, r is the radius of the torque pulley, and alpha is the average of the angular accelerations up and down.

Next we derived a formula for the inertia of the triangle with the axis of rotation being at one end so it is easier to integrate. We did this so that we could compare our experimental values for the moment of inertia with the theoretical values for the two orientations. We used the parallel axis theorem to get the moment of inertia from the center of mass of the triangle which is what we wanted.

To derive an equation for the moment of inertia of the triangle with the axis of rotation on one side, we had to take a horizontal piece dm of the triangle and find the moment of inertia of that piece and integrate the moment of inertia of that piece from 0 to h. Since the piece of the triangle was a rod being rotated about one end we knew the equation for the moment of inertia of such a rod to be 1/3ML^2 where M becomes the mass of the rod piece dm and L becomes the length x of the rod. so then we had to find a function to replace dm and we did so by making the proportions of the mass of the piece over the mass of the whole thing equal to the area of the piece over the area of the whole triangle and solving for dm. But then we had to change the x variable into y because we were integrating with respect to dy. So we found an equation for the line of the hypotenuse of the triangle and got x = -b/hy+b and plugged that into our equation. We integrated the equation and got our moment of inertia of the triangle to be:

Where M is the mass of the triangle and b is the length of the base.

We could then use the parallel axis theorem to find the moment of inertia at the center of mass of the triangle:

The derivation to find the moment of inertia of the center is the following:

As can be seen the moment of inertia for the center of mass is 1/18 M b^2 where M is the mass of the triangle and b is the length of the base.

The measurements that we took for this experiment were the mass of the hanging mass, the diameter of the torque pulley used, the mass of the triangle, the length of the base of the triangle, the length of the height of the triangle, and the angular acceleration up and down of the system. The reason we needed the mass of the hanging mass, the diameter of the torque pulley used and the angular accelerations up and down, was because we need it to plug into the first equation to find the moment of inertia of the system experimentally. the reason we needed the mass of the triangle, the length of the base, and the length of the height of the triangle is so that we could find the moment of inertia of the triangle theoretically. The measurements that we took were as follows:

We did the first trial to find the angular velocity up and angular velocity down and collected enough data to let the hanging mass go up and down three times. Then we took the derivatives of each slope to get the angular accelerations:

For the second trial we did the same thing except we put the triangle on the apparatus with the longer side facing vertically( not the hypotenuse). The data we collected was the following:

For the third trial we changed the orientation of the triangle so that it was perpendicular to the position it was in the second trial. The data we collected for this orientation was:

 The average angular accelerations were calculated for the system by itself, and with the triangle in both positions. The following shows the calculations:

 

Using this data the moment of inertia for the system alone, and with the triangle in different positions were calculated experimentally.

 

 

And the experimental moment of inertia of the triangle at the different positions was calculated to be:

 

The theoretical moment of inertia of the triangle at the different positions was calculated to be:

Conclusion:
According to the data the experimental values of .000436 for the triangle in the vertical position, and .000326 for the triangle in the horizontal position are a little different than the theoretical values of .000245 for the triangle in vertical position and .000565 for the triangle in the horizontal position. It is safe to say there is a small error either with the experimental data or rounding error in the calculations. The experimental data shows that the moment of inertia of the triangle in the vertical position is greater than the moment of inertia of the triangle in the horizontal position. While the theoretical data says the opposite. I think the theoretical data is correct and there must have been some error for the experimental data because the triangle in the horizontal position has a larger moment of inertia it is harder to move because its mass is over a larger diameter. For example, when you spin on a chair and hold your arms out you would go slower than if you were to have your arms in. The triangle in the horizontal position has a longer base than in the vertical position so it is harder to move in the horizontal position so the moment of inertia is greater.

Lab 16 angular acceleration

Purpose: The purpose of this lab for part 1 is to see what factors affect angular acceleration of system. For part 2 we are going to use the known torque and measured angular acceleration to determine the value for the moment of inertia.

The purpose of the apparatus used is to measure the angular velocity so that we then can get the angular acceleration.

Procedure:
First thing we did in this experiment was measure the diameters and the masses of the disks and pulleys and the mass of the hanging mass. Then we set up the apparatus on logger pro and changed the equation settings to 200 counts per rotation. We attached the air hose to the apparatus so that the top disk could move but the bottom disk wouldn't. We ran the trials and collected data on logger pro for the angular velocity versus time for each trial. We took the derivatives of the angular velocity graphs (for each part whether it was ascending or descending) to get the angular acceleration up or down. Then we took the average of the angular acceleration up and down to get the average acceleration overall for the trial. For the second part of the experiment we used the data from the first part of the experiment to determined the experimental values for the moments of inertia for the different combination of disks used in each trial.

The diameters and masses of the various disks and pulleys were as follows:

For the first trial we used the hanging mass, the small torque pulley and the top disk was the steel one. The data we got on logger pro is as follows:

The measured value that we got was in rotations per second squared so we multiplied it by 2pi to convert it to radians per second squared. We did this for the first three trials then for trials 4,5 and 6, we added an equation into logger pro to convert the angular acceleration to rad per sec squared for us.

For the second trial we doubled the hanging mass using the same small torque pulley and same top steel disk and collected the following data:

I did not take a pic of the third trial, but it is safe to assume it looks similar to the trials above. For the third trial we tripled the hanging mass and used the small torque pulley and the steel disk on top. The acceleration down for the third trial is 1.33 rad/sec^2, and the acceleration up is 1.43 rad/sec^2.

For the fourth trial we went back to the original hanging mass by itself and changed the torque pulley to the larger pulley and kept the top disk the steel disk. The data we collected was as follows:

For the fifth trial we kept the hanging mass the same the torque pulley was the large pulley, but the top disk was the aluminum disk. The data we collected for this trial is as follows:

 For the sixth trial we kept the hanging mass the same as the previous trial and kept the large torque pulley, but we changed the apparatus to allow the two disks to move and we used both steel disks on top and bottom. The data collected for this trial was as follows:

We copied the accelerations up and down from the computer and got the average for each trial and put everything on the following chart in our lab manual.

For the second part of the experiment we derived an equation in class to get the moment of inertia if the disk that is spinning using newton's second law(derivation is in the lab manual) and we got the equation:

Where m is the mass of the hanging mass, r is the radius of the torque pulley, g is acceleration due to gravity, and alpha is the average of the angular acceleration up and angular acceleration down.

Using the equation above we calculated the moment of inertia for the disk or disks combination for each trial.

Conclusion:
According to the data in the first part of the lab we can see that changing the hanging mass by making it bigger and keeping torque pulley and the top disk the same makes the average angular acceleration increase from .454 rad/sec^2 in trial 1 to .926 rad/sec^2 in trial 2 and 1.38 rad/sec^2 in trial 3. Changing the radius of the torque pulley from the small one in trial 1 to the larger one in trial 4 makes the acceleration increase. In trial 1 the average acceleration was .454 rad/sec^2 and in trial 4 the average acceleration was .8719 rad/sec^2. This makes sense because a larger radius equals a larger torque force. The effect of changing the rotating mass in trial 4  just letting the top steel disk spin the average angular acceleration is .8719 rad/sec^2, while in trial 5 making the top disk the aluminum disk and only allowing that disk to spins gives an angular acceleration of 2.4885 rad/sec^2. In trial 6 allowing both steel disk on top and steel disk on the bottom to move gives an angular acceleration of 1.033 rad/sec^2. It looks like making the top disk a lighter mass makes the average acceleration higher, but when it comes to trials 4 and 6 the two disks spinning together makes the acceleration value greater than just the top steel disk moving alone. For the second part of the experiment we calculated the moment of inertia for the different trials. The first three trials the moment of inertia was pretty much the same for all three trials being .0135,.0132, and .0133. Which makes sense because the top steel disk spinning was the same for all three trials. For the fourth trial the moment of inertia was a little larger than the first three trials being .0150 which should be so because we change the torque pulley from the smaller to the larger adding a little more mass to the rotating system. For the fifth trial the moment of inertia has the smallest value being .00522 which makes sense because mass of the spinning disk was the smallest. And for trial 6 the moment of inertia was .0127 which is the smallest value out of all the trials which makes sense because the mass was the greatest with both steel disks spinning. some sources of uncertainty that could have effected our calculations were the fact that we could have measured the wrong diameter on the large vernier caliper because sometimes it was hard to read the correct line that aligned the best. There also could have been sources of uncertainty from the machine itself because it is a very old machine and worn down from multiple use over the years.