7-June-2017: Physical Pendulum Lab

Physical Pendulum Lab
May Soe Moe
Lab Partners: Steven Castro, Stephanie Flores
Date: 7-June-2017

Objective: To derive expressions for the period of various physical pendulums and to verify the predicted periods by experiment

Introduction:

We did the pre-lab questions for this lab, in which we derived expressions for the moment of inertia of the semicircle rotating around its midpoint, its rim, the moment of inertia of the isosceles triangle rotating around its apex, and its midpoint. Once we get our expressions for moment of inertia of theseshapes, we derived our angular frequencies by using torque equation: T=Iα. Then, we used our previous derived equation in class, which is α=ω^2θ. We first use the torque equation and find α and put it into "α=ω^2θ" form. Whatever constant in that form is our ω^2. And we will find its square root and put into period equation: T=2π/ω. Then, we will conduct an experiment of a semicircle rotating around its midpoint, and an isosceles triangle rotating around its apex, and figure out their period by using motion sensor and Logger Pro. After that, we will compare the theoretical value and our experimental value.

Experimental Procedure:

(1) We set up to conduct the experiment as below:

Semi-circle Oscillating around its Midpoint

Isosceles Triangle Oscillating around its Apex

PhotoGate will determine the period of the triangle as the triangle oscillates.

(2) We chose our physical pendulums to be a semicircle and an isosceles triangle.

(3) Semicircle rotates and oscillates around its midpoint, and the triangle rotates around its apex.

(4) We sticked a piece of paper at the bottom of the pendulums to make sure PhotoGate can sense their motions correctly.

(5) We connected PhotoGate to our laptop and Logger Pro.

(6) After all the set up, we gave a push to the pendulums to make them start oscillating, and started to collect data in Logger Pro.

Experimental Data and Measurements:

Measurements of the Semi-circle and Triangle Pendulums

Experimental value of Period of A Semi-circle rotating around its midpoint collected by the PhotoGate

Experimental value of Period of A Triangle rotating around its Apex collected by the PhotoGate

Theoretical Calculation:

Calculating the moment of inertia of a semi-circle rotating around its midpoint

Calculating Angular Frequency 

Calculating the Moment of Inertia of an Isosceles Triangle around its Apex

Calculating Angular Frequency of the Triangle around its Apex

Calculating the period by using the derivation from above

Comparing the results

Conclusion:
Theoretical calculation part of this lab was done by finding moment of inertia around the midpoint of the semi-circle, moment of inertia around the apex of the triangle, and by using torque equation to find the angular frequencies of the oscillating semi-circle, and the triangle. Once we have the angular frequencies of the pendulums, we could calculate the period. The experimental part was done by using PhotoGate to figure out pendulums' periods. Now, when we compare the results of theoretical period and experimental period, we found that they are less than 3% error, which is really close and within acceptable range. Therefore, this lab was successful and it also proved that our derived equations of the period of oscillating triangle and semi-circle were correct.

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

Lab 19: Conservation of Energy/Conservation of Angular Momentum
May Soe Moe
Lab Partners: Ben Chen, Steven Castro, Stephanie Flores
Date: 31-May-2017

Objective: To determine how high the clay-stick combination rises after the collision, and compare the experimental results with theoretical results

Introduction: 
In this lab, we will theoretically figure out how high the clay and the stick rises up after the meter stick collides with the clay by using the conservation of angular momentum and the conservation of energy. We will use the conservation of energy first to find out the angular velocity of the stick right after the collision. Once we get the initial angular velocity, we will use the conservation of the angular momentum to find out the final angular momentum. After that, we will use the conservation of energy again to get the maximum height the clay and the meter stick reach after the collision. Experimentally, after setting up the apparatus, we will shoot a slow-motion video with an iPhone of the meter stick colliding with the clay and reaching up to the maximum height. After that, we will use Logger Pro to set up the initial point and final height of the clay and the meter stick. Logger Pro will give us the maximum height the clay and the stick reach together. We will compare what Logger Pro gives us with the theoretical calculation we did.

Experimental Procedure:
(1) We set up our apparatus as below:

Our apparatus set up

After the clay collides with the stick

(2) The pivot is at 10 cm mark, and we will consider the pivot is at 0 cm mark, the center of mass is at 40 cm mark, and the clay is at 90 cm mark.

(3) After setting up, we aim where to put the clay, so that the stick collides with the clay.

(4) We put the pins into the clay so that it would stand. The clay only sticks to the meter stick if it is standing on the pins.

(5) Once we got our aim, we set up a ring stand where the phone could capture the whole process of the stick colliding to the clay and the clay sticking to the stick.

(6) We made sure that we could see the maximum height the clay and the stick reached in the video.

(7) After recording the whole process, we inserted the video into Logger Pro.

(8) We define our x and y- axes and marked where the initial and final position of the stick. Logger Pro automatically determined the maximum height the clay and the stick reached.

What we did in Logger Pro to get the maximum height

Experimental Data: 

We got the maximum height as 0.3743, experimentally.

One thing to note in this screenshot is that our actual maximum height is -0.3743, not 0.6104. We set our x and y- coordinates wrong, therefore, it is switched. 

Theoretical Calculations:

Calculating the angular velocity at the bottom to find the angular velocity at maximum height

Calculating angular velocity at maximum height and using it to find the maximum height

Conclusion:
When we compared it to our theoretical maximum height (0.394m) and experimental maximum height (0.3743m), it has 5% difference and 5% percentage error. This is within acceptable range. The source of uncertainties might be that when we marked the clay to determine the maximum height in Logger Pro, it might not have been the same exact location on the clay, which could be a difference from the video. Overall, the lab was successful that our theoretical and experimental maximum height of the clay and the meter stick are close to each other. Therefore, we can also see that this experiment confirms the conservation of angular momentum and the conservation of energy.

22-May-2017- Lab 17: Finding the Moment of Inertia of a Uniform Triangle about its Center of Mass

Lab 17: Finding the Moment of Inertia of a Uniform Triangle about its Center of Mass
May Soe Moe
Lab partners: Ben Chen, Stephanie Flores, Steven Castro
Date: 22-May-2017

Objective: To determine the moment of inertia of a uniform triangle about its center of mass through experiment and compare the experimental and theoretical values

Introduction:
To determine the moment of inertia of a uniform triangle, we will orient the triangle into two perpendicular orientations in this lab. We will determine they moment of inertia based on its orientation. The triangle will be mounted on the apparatus, which will rotate by using air source. A string is going to be wrapped around a torque pulley to a hanging mass, which will exert torque. We will measure the angular acceleration of the rotating disk using Logger Pro, by graphing an angular velocity vs. time graph. We will get the slope of the graph by using linear fit, in which the slope will be our angular acceleration. For experimental approach, we will use our derived equation for the moment of inertia. For theoretical approach, we will use the parallel axis theorem to get our moment of inertia equation. Once we have our equations ready, we will plug in the measurements such as mass of the hanging mass, radius of the torque pulley, average angular acceleration, mass of the rotating disk, and so on, to calculate the moment of inertia of the triangle around its center of mass. Then, we will compare the results.

Experimental Procedure:

(1) Our apparatus is as below:

(2) We set up our triangle into two different orientations as in the pictures above.

(3) We connected the rotational sensor and Lab Pro to our laptop and used Logger Pro to record the angular acceleration of the rotating disk.

(4) We measured the height of the triangle, the base of the triangle, the masses of the rotating disks and torque pulley, their diameters, the mass of the hanging mass, the mass of the holder that holds the disk and triangle together.

(5) We ran the experiment two times orienting the triangle differently.

(6) Using logger pro, while the disk along with the triangle was rotating, we graphed the angular velocity vs time graph.

(7) We used linear fit to figure out the slope of our graph, in which the slope is our angular acceleration. We found our ascending angular acceleration and descending acceleration, which were not the same due to the presence of the frictional torque in the system.

Experimental Data and Calculations:

Measured Data

Experimental Calculations for Moment of Inertia of an Uniform Triangle in Two Different Orientations

Experimental Calculations For Moment of Inertia of an Uniform Triangle in Two Different Orientations

Theoretical Calculations:

Calculating x-Center of Mass of triangle to find Moment of Inertia

Finding the Moment of Inertia of the triangle rotating around its edge

Moment of Inertia of the triangle around its center of mass

Theoretical Calculations for the Moment of Inertia of a Triangle in Two Different Orientations

Comparing Results- Theoretical Vs. Experimental:

Comparing experimental results vs. theoretical results

Conclusion:
When comparing our theoretical and experimental calculated results of the moment of inertia of the triangle in two different orientations, there is a difference of 11.8% and 11.2%. Percent errors are within 11%, which should still be in acceptable range as long as it is under 15% percent error. Therefore, the lab was successful. 

17-May-2017, 22-May-2017: Lab 18: Moment of Inertia and Frictional Torque

Lab 18: Moment of Inertia and Frictional Torque
May Soe Moe
Lab Partners: Ben Chen, Steven Castro, Stephanie Flores
Dates: 17-May-2017, 22-May-2017

Objective: To determine the moment of the inertia of the system, to determine the angular deceleration, to determine the frictional torque, and to determine the time it takes to descend for a cart to get to one meter mark of the ramp

Introduction:
Our apparatus consists of a disk and two cylinders, which are all attached together. We will consider the system to be consisted of three cylinders. The moment of inertia of the system is the sum of the moment of inertia of three cylinders. Since we do not know the mass of the two cylinders from the sides, we will calculate the volume of the cylinders after measuring their radius and height, and the total mass of the whole system. After calculating the volume of the system and each of the cylinders, we will calculate the mass of each cylinder. Once we ind out the mass of the cylinders, we will find out the moment of inertia of each cylinder by using I=1/2MR^2. For part 2 of the lab, to figure out the angular deceleration, we will take a slow-motion video of the system rotating. We will use logger pro and to dot around the rim of the disk rotating, which will give us the angular velocity vs. time graph. We will find the angular deceleration by finding the slope of the angular velocity vs. time graph. Then we can find torque by using T=Iα. For part 3 of the lab, we will wrap a string around the middle disk and connect it with a 500-gram dynamics cart. Using the Newton's Second law to get a representative expression for acceleration of the cart. After that we will use the kinematics equation to figure out the time it takes to reach 1 meter mark of the ramp.

Experimental Procedure:

Part 1:
(1) Our apparatus looks as below:

Front view of the apparatus

The apparatus consists of one big disk, and two cylinders on the sides as in the picture.

(2) We took measurements of the diameters of the cylinders, the height of the cylinders. The total mass of the cylinders are stamped onto the big rotating disk.

Part 1 Data and Calculations:

Measured data of the system

Volume Calculation of the three cylinders to figure out the mass of each cylinders

Calculation of Mass of each Cylinder to find the moment of inertia of the system

Calculation of the Moment of Inertia of the System Consisting two cylinders and a disk

 Experimental Procedure:

Part 2:

(1) We put the ring stand in front of the apparatus and set up an iPhone to the ring stand.

(2) We align the iPhone's camera to the center of the system.

(3) We took slow-motion video of the disk rotating around its center. We made sure it was 240 frames per second.

(4) We opened our slow-motion video in Logger Pro and dot around the rim of the disk. The dots around the rim would produce the angular velocity vs. time graph.

(5) We used linear fit to find the angular deceleration, which is the slope of the angular velocity vs. time graph.

Part 2 Data:

The dots around the rim of the disk and how the dots produce graph 

The angular acceleration of the rotating disk around its center, which is -0.4395 rad/s^2-- slope of the graph.

 When we did the dots around the rim of the disk, each dot was 30 frames per second, which is 8 times faster than the actual time, since we shot the video at 240 frames per second. Therefore, in the picture and graph above, our x-axis is actual time, and we divided the time we got for 30 frames per second by 8. Our angular deceleration for the disk came out to be -0.4395 rad/s^2, according to the graph.

Now we can calculate the frictional torque.

Part 2 Calculation:

Frictional Torque Calculation using the Angular Deceleration we got from the graph

The result from calculating frictional torque using the calculated moment of inertia from part 1 and the angular acceleration from part 2 came out to be -9.348x10^-3 kgm^2/s^2.

Experimental Procedure:

Part 3:

(1) We did the set up as below:

Our apparatus set up for part 3of this lab

The ramp is 50 Degree above horizontal

(2) After the set up, we measured the horizontal angle using the level from iPhone, which came out to be at 50 degree.

(3) After that we wrapped a string around the large disk, connecting to the cart.

(4) We started the rotation, and timed the time it took for a cart to reach 1 meter mark.

(5) We did this process three times. Our average time for the cart to reach 1 meter mark was 6.76 seconds.

Part 3 Calculation:

Calculation for time it took for the cart to reach 1 meter mark

The calculated time based on our previous calculated results such as the frictional torque and moment of inertia of the system, came out to be 6.16 seconds, while the average timed time of the cart to reach 1 meter mark was 6.76 seconds. There is a 0.6 seconds difference. Therefore, I calculated the percentage error for our result.

Percentage error for the difference between experimental time and theoretical time

The percent error for the time difference was 9.74%.

Conclusion:
For part 3 of the lab, where we compare the experimental time--the average time we got by using the timer for the cart to reach one meter mark-- and the theoretical time--the time we calculated by using the previous calculations from part 1 and 2 of the lab, which are the moment of inertia of a system and the frictional torque, we found that there was 0.6 seconds difference between the two. I calculated the percent error and it was 9.74%, which is less than 10% and we could say this is within acceptable range. Errors could be because the ramp moved every time the cart reached to the bottom, near 1 meter mark. Although we used several masses at the end of the ramp to make sure it stays still, it didn't work and we might not have readjust the angle to 50 degree as in the beginning after each trial. There could be some friction on the ramp that we did not consider in this lab and in our calculations. The measurement of the angle was done with the phone app, which might have some uncertainties in its measurements. Overall, the lab was successful that the error was within 9.74%.

8-May-2017, 15-May-2017- Lab 16: Angular Acceleration

Lab 16: Angular Acceleration
May Soe Moe
Lab Partners: Ben Chen, Steven Castro, Stephanie Flores
Dates: 8-May 2015, 15-May 2015

Objective: To measure the angular acceleration of the rotating object by using torque and to determine the moment of inertia of the rotating disks used in this lab

Introduction:
In this lab, we will collect data in part 1 and figure out the angular acceleration of the rotating disk. In part 2, using the part 1 data, we will determine the moment of inertia.We are using two disks that will rotate and two different torque pulley on top that will hang a hanging mass with a string. For the first part to figure out the angular acceleration, we will apply various changes to the experiment. By applying these changes in six experiments, we can figure out how these changes affect the angular acceleration. In the first three experiments, we intend to observe the effect of changing the hanging mass. In the first and fourth experiment, we use two different hanging pulley with same mass. In the last three experiments, we will observe the effect of changing the rotating mass (steel disk to aluminum disk). We will use logger pro to record the angular acceleration during the disk rotation. For part two of the experiment, we will use the angular accelerations we got from the part 1 of the lab to plug into the moment of inertia equations for the disks that we derived in class.

Experimental Procedure:

Part 1:

(1) Our apparatus is as in the picture below:

Apparatus to measure angular acceleration

(2) We measured the masses and diameters of the rotating disks and torque pulleys.

(3) We used compressed air to let the disks rotate. As the disks rotate, the hanging mass that is wrapped around the torque pulley moves up and down.

(4) We connected the Pasco rotational sensor and Lab Pro to Logger Pro on a laptop to help us record the angular acceleration of the rotating disk as it is rotating and as the hanging mass goes up and down.

(5) After setting up, we did six experiments by changing the hanging mass, by changing the torque pulley with different radii, by changing the rotating disk between steel disk and aluminum disk.

(6) During this six experiments, we recorded the angular velocities of the rotating mass.

(7) We used linear fit to find the slope of the angular velocities and found the slope to be its angular acceleration. When the hanging mass goes up, the angular acceleration of the disk on the graph goes down. When the hanging mass goes down, the angular acceleration of the disk on the graph goes up.

Part 1 Data: 

Measurements of the rotating disks and torque pulleys

Hanging mass used and angular accelerations from our graphs during six experiments

Angular Acceleration of the Rotating Disk (slope m) from the Angular Velocity Vs. Time Graph [Experiment 1]

Experiment 2- Angular Acceleration of the Rotating Disk

Experiment 3

Experiment 4

Experiment 5

Experiment 6

Conclusion For Part 1: 
From our data, we found that the average angular acceleration of the experiment 2 and the experiment 4 are similar. We doubled the hanging mass for the experiment 2 and used the large torque pulley which has twice of larger radius than the small pulley. Increasing the hanging mass increases the angular acceleration of the rotating disk. In experiment 6, when both the top and bottom steel disks rotated, the angular acceleration slowed down by half than just the top steel disk rotating. When the rotating disk has smaller disk, in this case Aluminum disk, and the torque pulley with twice radius, the average angular acceleration of the disk got around three times faster.

Part 2:

Procedure:
The professor showed us how to derive the moment of inertia equation, involving the hanging mass and torque pulley in class. The expression we derived for the moment of inertia of the rotating disk is as below:

Derived Moment of Inertia of the rotating disk Equation

There is some frictional torque in the system since the rotating disk is not totally frictionless and we cannot neglect the mass of the pulley. We can confirm this by checking our angular acceleration values. We see that the ascending angular acceleration were not the same as the descending angular acceleration. Therefore, we used the first equation of the moment of inertia from the picture. The second equation is for when there is no friction involved in the system.

Part 2 Calculation: 

Calculation of Moment of Inertia of different disks in six experiments 

Conclusion:
When we looked at our experimental results of the moment of inertia of rotating disks, the first four experiments should have approximately the same results due to using the same torque pulley and the rotating steel disk. But the calculated moment of inertia of the disk for the first experiment was off by half of the moment of inertia of the disks of the second, third, and fourth experiments. The calculated moment of inertia of the second, third, and fourth experiment came out to be pretty close and consistent. The fifth and sixth experiment's moment of inertia of the disks also seemed reasonable.