11/28/16 - Physical Pendulum Lab (Lab 22)

PURPOSE:
Derive expressions for the period of various physical pendulums. Verify your predicted periods by experiment.

THEORY:
In this lab, we created three physical pendulums using a thin metal ring and a thin isosceles triangle. The period of each pendulum is given by the equation: T = 2π/ω. Using Newton's second law, we created an equation for the sum of torques of each pendulum, This allowed us to find the angular speed of the system at the bottom of the swing, which we plugged into the original equation to find the period.

APPARATUS:
The apparatus we used consists of a ring stand, C-clamps, a metal rod, and a photogate connected to a computer with a Logger/Lab Pro set-up. The photogate is  positioned such that it can track the movement of the small piece of tape attached to the bottom of our swinging objects.

Physical Pendulum - fully assembled

EXPERIMENTAL PROCEDURE:
We used a photogate to measure the period of three physical pendulums: a thin metal ring, a thin isosceles triangle hung by its apex, and a thin isosceles triangle hung by the midpoint of the base.

Trial 1: Thin Metal Ring

Trial 2: Isosceles Triangle - Pivot at Apex

Trial 3: Isosceles Triangle - Pivot at Midpoint of Base

Notice how we attached the triangle

DATA/GRAPHS:
Trial 1:
Diameter of inside ring: 27.3 +/- 0.1 cm
Diameter of outside ring: 29.2 +/- 0.1 cm
Average diameter of ring: 28.25 cm

Trial 2 & 3:
Base of triangle: 14.4 +/- 0.1 cm
Height of triangle: 16.1 +/- 0.1 cm

Period prediction calculation for metal ring

Period prediction calculation for Isosceles Triangles 

Trial 1: Period Data

Trial 2: Period Data

Trial 3: Period Data

Period Data Results (Final)

ANALYSIS:

In order to verify that our theoretical calculations for the period of each pendulum were consistent with reality, our group had to use Logger Pro to collect experimental values for the period. Our group used a photogate to measure the  period of each pendulum, and we also used Logger Pro to calculate the mean value for each data set. We used the mean values of the period for the experimental values because we thought that it would be a more accurate representation of what the period would be. Each period data graph showcases individual data points for a given trial as well as the mean value of those data points.
CONCLUSION:
Our lab results turned out to be much better than expected. We successfully verified that our theoretical values for the period of each physical pendulum matched very closely to their real-life counterparts. In fact, our period data illustrates that our theoretical values and experimental values never exceeded a margin of error above 1%. For example, in Trial 1, our values for the period were about 0.37% different for one another. While this margin of error is clearly more than satisfactory for our circumstances, it is nevertheless a prudent endeavor to evaluate possible margins of error. One possible source or uncertainty in our calculations could have come from the tape we attached at the end of the object. As we realized in practice trials of this experiment, any added weight a distance away from the pivot can alter the angular speed of our system, thereby affecting the accuracy of our period. Another source of error in our methodology was the wobbliness of the moving object. This wobbliness indicates that some of the energy in the system is being lost due to friction at the pivot.

11/7/16 - Angular Acceleration (Lab 16)

PURPOSE:

To determine and compare the moments of inertia for a set of rotating disks.

THEORY:
Rotational inertia (aka, moment of inertia) refers to an object's resistance to undergo an angular acceleration. The harder it is for an object to spin and keep spinning, the more rotational inertia it has. In this experiment, we applied torque to an object that can rotate, measured the angular acceleration of the object, and used this information to find its moment of inertia using both Newton's laws and a calculus derived equation. 

APPARATUS:
Note: The apparatus was assembled beforehand for this lab.

Pasco rotational sensor attached to Lab Pro/Logger Pro setup

EXPERIMENTAL PROCEDURE:
Part 1:

1. Measure diameter and mass of relevant materials. (Listed under Data/Graphs)
2. Plug the power supply into the rotational sensor.
3. Connect the sensor to Lab Pro/Logger Pro.
4. Set up the computer and calibrate the sensor on Logger Pro to read 200 ticks per rotation.
5. Make sure the hose clamp on the bottom is open so that the bottom disk will rotate independently of the top disk when the drop pin is in place.
6. Turn on the compressed air such that the disks rotate separately.
7. With the string wrapped around the torque pulley and the hanging mass at its highest point, start the measurements and release the mass. 
8. Use a linear fit of the angular velocity vs. time graph to measure the angular acceleration as the mass moves up and down.
9. Draw some conclusions about your data and how the values are related.

Part 2:

1. Derive an expression for the moment of inertia of the disks using Newton's laws.
2. Calculate the experimental values for moment of inertia of the disks for each experiment using the equation: Idisk = (mgr/α) - mr^2.
3. Calculate the theoretical values for the moment of inertia of the disks using (1/2)MR².

DATA/GRAPHS:
Part 1:
***Diameter and mass of:

• Top steel disk: 126.3 mm, 1358.4 g
• Bottom steel disk: 126.4 mm, 1345.7 g
• Top aluminum disk: 126.3 mm, 465.1 g
• Smaller torque pulley: 49.7 mm, 18.1 g
• larger torque pulley:24.8 mm, 65.1 g

mhanging = 44.1 g

***Uncertainty in mass: +/- 0.1 g
    Uncertainty in diameter: +/- 0.1 mm

Recorded Data

Trial 1 - Angular speed vs time

ANALYSIS:
Part 1:

Data Analysis & Conclusions - Part 1

For part 1 of the lab, we conducted six experiments that demonstrated how changing the hanging mass, radius of the torque pulley, and the mass of the rotating disks affected the angular acceleration of the system. For example, in experiment 1,2, and 3, we found that when we doubled the hanging mass, the angular acceleration doubled as well. When we tripled the hanging mass, the alpha of the system also tripled. In experiments 1 and 4, we discovered that doubling the radius of the torque pulley doubled the alpha of the system. In experiments 4,5, and 6, we concluded that increasing the mass of the rotating disk decreased the alpha of the system. More specifically, when we tripled the mass of the rotating disk--with the alpha of the system with aluminum disk as our baseline--it reduced the alpha of the system to one third of its initial value. Moreover, when we increased the mass of the rotating disk sixfold, the alpha of the system receded to one sixth of its initial value.

Part 2:

Data Analysis & Conclusions - Part 2

Moment of Inertia Results - Part 2

After gathering data, our group calculated the theoretical and experimental values for the moment of inertia of the disks. We used Newton's laws to derive an equation for the experiment values and we used calculus to derive a simpler equation for the theoretical values. While the expressions we used to determine the moments of inertia look radically different from one another, their values were nearly identical. In fact, according to our data, the largest margin of error we encountered was about a three percent difference.

CONCLUSION:
Our experimental results for this lab were satisfactory because they lied within a reasonable margin of error to our theoretical values. As previously stated, the highest margin of error we encountered in our calculations was a three percent difference. While the small level of error in our moments of inertia indicate that our group's uncertainty in our calculations were fairly minimal, it is still in our group's best interest to pursue the most precise measurements and accurate calculations whenever possible. The first step in doing so involves recognizing the sources of uncertainty within our methodology. For example, one of the largest sources of uncertainty in our lab stemmed from human error, such as imprecisely measuring the radii of the disks. Another source of error in our lab came from assumptions we made about our procedure. For instance, we assumed that there is no frictional torque in the system. This assumption implies that the angular acceleration of the hanging mass is the same regardless whether it is ascending or descending. In reality, however, the system does indeed have some frictional torque because there is some mass in the frictionless pulley. Therefore, the angular acceleration of the system is not exactly the same when the mass is descending as when it is ascending.

GROUP MEMBERS: Xavier C., Billy J., Matthew I.,
with special guests: Christian R. & Haokun Z.

9/28/2016 - Centripetal Acceleration vs Angular Frequency (Lab 8)

PURPOSE:
To determine the relationship between centripetal acceleration and angular speed.

THEORY:
We know that the net centripetal force of a rotating object is given by F = mrω². If we compare this equation to Newton's second law, F = ma, then it becomes apparent that if the net force equal mass times acceleration, then the net centripetal force must be equal to mass time centripetal acceleration. Therefore, a_cent = rω² = v²/r

APPARATUS:
The apparatus was assembled beforehand for this lab.

Wooden rotating disk with power supply and voltage regulator.

Setup:

1. Place the wireless accelerometer on the wooden disk.
2. Adjust the voltage on the power supply, turn the scooter motor on, and let the disk come to a constant speed.
3. Collect period and acceleration data for a variety of rotational speeds by varying the voltage.

EXPERIMENTAL PROCEDURE:

1. Use a photogate to measure the rotational period of the given mass.
2. Determine the mass of the object on the rotating disk.
3. Measure the radius of the object from the center of the disk.
4. Record the mean force exerted on the rotating mass.
5. Calculate the angular speed of rotating mass.
6. Repeat these steps 7-8 more times w/ varying ω and record them on a spreadsheet.

DATA/GRAPHS:

Recorded data

F vs. mω² 

F vs. rω²

F vs. ω²

Sample Calculation:
F = mrω²
F/(mr) = ω²
ω = [F/(mr)]^1/2
F = 2.35 N
m = 0.2 kg
r = 0.58 m
ω = 4.5 rad/s

ANALYSIS:
For this experiment, we made graphs of F vs. mω², F vs. rω², and F vs. ω². Ideally, each graph was supposed to have a slope of r, m, and mr respectively. If the slopes of the graphs match the variable that it is missing from the equation for centripetal force, then we have verified that the relationship between angular speed and centripetal acceleration is credible. The reason this method works is because it allows us to see how a change in a single variable can effect the others. For example, from F vs. rω², we can see that the slope of the linear fit of the graph is 0.2026, which is very close to the mass of the object on the wooden disk: 0.2 kg. The same hypothesis held true for the F vs. ω² graph, which yielded a slope value of  0.1249; a very close approximation of mr = 0.116. On the other hand, when we tested it with the graph of F vs. mω², we found that our prediction did not hold to be true. In fact, our data shows that the slope of the graph was 0.8249 when it  should have been closer to 0.58: the measured radius of the spinning mass.

CONCLUSION:
The data that we collected allowed us to analyze the relationship between centripetal acceleration and angular speed. More specifically, the graphs we created allowed us to illustrate how each of the variables we collected -- m,r, ω -- can effect this relationship. By comparing the slope of each graph to its corresponding "missing variable", we were able to successfully prove that centripetal acceleration did in fact depend on angular speed, among other factors. This mathematical relationship, however, was not entirely consistent with our data. Therefore, there must have been some uncertainty that was influencing our results. The most likely source of this discrepancy was human error. After comparing our group's data to another, we discovered that one of our data points for force varied by .05 N. Additionally, the other group used more data points in their graph for F vs. mω², which yielded a slope that was relatively more consistent with the radius of the mass around the disk. Another likely source of uncertainty in our calculations was our assumptions about the experiment. For example, when we began the experiment, we assumed that friction was negligible, when in reality there were many sources of friction because the wooden disk was in contact with several wheels while spinning.

GROUP MEMBERS: Xavier C., Billy J., Matthew I.

8/5/2016 - Work-Kinetic Energy Theorem Activity (Lab 11)

PURPOSE:
Prove that the work done when you stretch a spring through a measured distance is equal to the change in the kinetic energy of the spring.

THEORY:
If there are no external forces acting on the system, then ideally, the total work done to the system should be equal to the change in the kinetic energy of the system. For a non-constant force, such as a cart+spring on a ramp, the total work done can be calculated by finding the area under the force vs. distance graph. In a perfect world, the calculated area and the change in KE should be equal. In the world we live in, however, the accuracy of our calculations will depend on a variety of factors, such as the accuracy of our assumptions, the preciseness of our equipment, and the diligence of the people who perform the lab. In this experiment, we will explore how well our theoretical model of energy fits with reality.

APPARATUS:
We set up a ramp, cart, motion detector, force probe, and spring as shown in the diagram. One end of the spring is attached to a metal rod which is clamped to the table.

cart+spring system

EXPERIMENTAL PROCEDURE:
Part 1: Force vs. Distance

1. Calibrate the force sensor.
2. Set up the apparatus as shown.
3. Open the apt experiment file to display a force vs. distance diagram in Logger Pro.
4. Zero the force probe, verify the motion sensor is measuring toward the detector as the positive direction.
5. Sketch/Capture an image of the graph. Find the spring constant. Find the work done in stretching the spring using the integral function on Logger Pro.

Part 2: Kinetic Energy vs. Distance

1. Measure the mass of the cart.
2. Create a New Calculated Column for the KE of the system with respect to position.
3. Zero the force sensor and the motion detector at the desired starting position. 
4. Pull the cart back about 0.6 m, let it go, and begin graphing your KE data.
5. Compare the ΔK from points a,b to the area under the F vs. x graph through points a,b.

DATA/GRAPHS:
mass of cart = 0.756 kg
spring constant, k = 5.794 N/m

Force vs. Position Graph

KE vs. x and ∫ F(x)dx through points (a,b)

KE vs. x and ∫ F(x)dx through points (a,c)

KE vs. x and ∫ F(x)dx through points (a,d)

Δx - Work - Kinetic Energy - Data Chart

ANALYSIS:
We calculated the work done by the force of the spring by integrating the force function with respect to position. This method works because it is the equivalent of calculating the area under the force vs distance curve, which yields W = force*distance. After comparing it to the change in kinetic energy, our group found that the KE at the final position for a given range of values was approximately equal to the work done over the same set of values. For example, between the position values x = 0.51 m to x = 0.17 m, we found the work done between those points to be about 0.63 J, while the change in KE was about 0.83 J. In other words, we found that the change in KE was relatively close to the total work done on the system between those two points. On average, however, there was about a 24% difference between the two values, indicating a few sources of uncertainty in our experiment.

CONCLUSION:
Our data clearly illustrates that the change in the KE of our system did not entire coincide with the total work done over two points. However, while the reality of our calculations did not totally match up with their theoretical counterparts, this does not mean that the theory we tested is discredited. In reality, our data showcases how small changes in our methodology can influence irregularities in our calculations. In other words, there are several reasons why there was such a large disparity between the calculated work and the change in KE of the system. For instance, the lab equipment we used is not the best in the world. If we decided to buy and use more expensive and precise lab equipment, then we could have measured the mass, velocity, and force of the system with minimal error, which undoubtedly could have greatly increased the accuracy of our calculations.

GROUP MEMBERS: Xavier C., Billy J., Matthew I.

8/5/2016 - Conservation of Energy -- Mass-Spring System (Lab 12)

PURPOSE:
To determine whether or not energy is conserved in a vertically-oscillating mass-spring system, where the spring has a non-negligible mass.

THEORY:
In this experiment, we have a mass hanging from a uniform spring a certain distance above the ground. When the spring is a certain height above the ground, it has GPE. When it is released from this position, some of its GPE is converted into KE. If there are no other forces acting on the system, then the GPE will be converted into KE until the spring is in its equilibrium position. As the spring-mass system reaches its maximum displacement, all of its KE is converted into EPE. After the spring has stretched its maximum distance, the hanging mass will rise as its EPE is converted back into KE. As the spring-mass system oscillates, this energy cycle continues until external forces such as friction retard the spring to a standstill. If energy is conserved, then the total energy of the system will be constant. However, in order to take the mass of the spring into account for our energy calculations, we must derive new formulas for GPE and KE using calculus.

APPARATUS:
We hung a 200 g mass on a hanger for a total mass of 250 g. The bottom end of the spring is holding the mass, while the top side is hanging from a horizontal metal rod, which is fixed to another metal rod that is clamped to the lab table.

We also attached a force sensor to the horizontal metal rod, such that the spring could hang from it with the hanging mass attached. Additionally, we placed a motion sensor beneath the spring-mass system. Lastly, we taped a flashcard to the bottom of the hanging mass to make it easier for the motion sensor to detect.

Apparatus: Part 2

EXPERIMENTAL PROCEDURE:
Part 1: Determine the Spring Constant

1. Calibrate the force sensor.
2. Set up Apparatus: Part 2, as shown. Note: Remember to zero the force sensor when spring is attached.
3. Calibrate motion sensor.
4. Collect Force vs. Time data and Stretch vs. Time data. 
5. Plot a Force vs. Stretch graph to obtain an equation for the force of the spring. (F = kx + F0)
6. Holding the mass+spring system from its equilibrium position, use the motion detector to determine its position relative to the "ground." (i.e. from the front of the motion sensor).
7. Make a new Calculated Column called "Stretch." This equation will be based on the motion sensor reading.

Part 2:

1. Hang 200 g on the mass hanger for a total mass of 250 grams.
2. Pull the spring down about  10 cm and let it go. 
3. Record position and velocity graphs and sketch them in your lab module.
4. Make prediction sketches of KE vs. time, GPE vs. time, and EPE vs. time.
   
   

   

   Energy Prediction Sketches

5. Now, set up Logger Pro to calculate the various energies in New Calculated Columns.
6. Use Logger Pro to produce plots of KE vs. y, KE vs. v, GPE vs. y, GPE vs. v, EPE vs. y, EPE vs. v.

Here are the equations we used for various energies, *Note: We derived them in class using calculus:

KE = (1/2)[mhanging + (1/3)mspring]v^2

GPE = [mhanging + (1/2)mspring]gy

Elastic PE = (1/2)k(stretch)^2

DATA/GRAPHS:
Part 1:
mspring = 86.0 g
mhanging = 350 g
k = 5.794

Part 2:

GPE vs. t - KE vs. t - EPE vs. t 

KE vs. y

KE vs. v

GPE vs. y - EPE vs. y

GPE vs. v - EPE vs. v

GPE vs y -EPE vs. y - Energy Sum (for each)

GPE vs. v - EPE vs. v - Energy Sum (for each)

ANALYSIS:
After analyzing my energy data and comparing it to my predictions, it is apparent that I made a few incorrect assumptions. For example, when I created my sketches, I predicted that GPE would increase at the same rate EPE would decrease and vice versa. I made this prediction because GPE should be at its lowest when EPE is at its highest. This, however, was not the case at all according to our data. In fact, my data illustrates that my prediction is the antithesis of what happens because both the GPE and EPE oscillate together at very similar rates. Another error I made in my prediction was that I took the EPE of the system to be positive when in reality it is negative because it acts in the opposite direction of the spring force (which is in the positive direction for this experiment). In order to verify that our data was consistent with our physical model, we also analyzed our energy vs y and energy vs. v graphs. Our KE vs. v,y graphs made sense because they they were both consistent with the increase in the speed of the system. Our GPE, EPE vs. y graphs were logically sound because they both increased and decreased respectively with respect to the height relative to the motion sensor. Moreover, our GPE, EPE vs. v graphs had warranted values because they are both reach their max when the KE decreases to zero.

CONCLUSION:
Overall, our group was able to successfully demonstrate that the energy of the mass+spring system was conserved because our data showed that the total energy of the system remained nearly constant throughout. However, the lines that Logger Pro displayed were not as straight as we hoped they would be. Therefore, there could have been a few sources of uncertainty in our experiment that we did not account for. One of the many possible sources of error we encountered were external forces acting on the system. This includes, but is not limited to: air resistance caused by the flashcard, damping caused by dissipated heat due to friction, and permanent spring deformation. Another possible source of uncertainty was through human error. For instance, if we inaccurately recorded length measurements, then the preciseness of our initial data would be negatively affected, thereby decreasing the accuracy of our energy calculations.

GROUP MEMBERS: Xavier C., Billy J., Matthew I.