8/10/2016 - Magnetic Potential Energy (Lab 13)

PURPOSE:
To verify that conservation of energy applies to a system with magnetic potential energy (MPE).

THEORY:
Consider a cart with a strong magnet sliding along a level air track with another magnet of the same polarity fixed at the other end. As the cart moves along the track, it will have some KE. However, as the cart approached the end of the track, its KE will reduce to zero and all of the energy will be stored in the magnetic field as MPE, then it rebounds back. Since the MPE is not constant, we will use the relationship U(r) = -∫ [r,∞] F(r)dr to relate our MPE to the separation distance, r.

APPARATUS:
We used a glider on a frictionless track. The glider has a strong magnet attached to one end and at the end of the track there is another magnet with the same polarity. On top of the glider, there is a thin aluminum plate that is used to facilitate the collection of position data with a motion sensor, which is located at the same end of the track as the magnet.

EXPERIMENTAL PROCEDURE:
Part 1: Force Equation

1. Prepare apparatus as shown.
2. Weigh cart+reflector.
3. Connect vacuum hose to air track.
4. Calibrate and connect motion detector with Lab/LoggerPro.
5. Tilt the air track at various angles in order to create a relationship between the magnetic force, F and separation distance, r.
6. For a given angle θ, use calipers to record the separation distance, r between the two magnets.
7. Plot a graph of F vs. r. We assume that their relationship is a power type equation of the form: F = Arⁿ. 
8. Get the A and n values from the curve fit of your F vs. r graph. 
9. Integrate the force function to get your equation for the MPE. 

Part 2: Verification

1. With the air turned off, place the cart+reflector reasonably close to the fixed magnet at the end of the track. Run the motion detector. Determine the relationship between the the distance the motion detector reads and the separation distance between the two magnets. Assuming that the distance between the the reflector and magnet of the cart is negligible, the distance r = P - k.
2. Use the motion detector and LoggerPro to measure the speed and the separation between the two magnets.
3. Start the cart at the far end of the track, turn on the air track, and give the cart a gentle push.
4. Record data necessary to to verify that energy is conserved.
5. Create a graph of KE, MPE, and total energy as a function of time.

DATA/GRAPHS:

Data from Part 1

Sample Calculation of Fmag (Part 1)

Curve Fit of Force vs. Time

Derivation of Magnetic Potential Energy Function

KE-MPE-TE vs. Time Graph

ANALYSIS:
In order to verify that energy was conserved for this experiment, all of the KE of the cart must be transferred into MPE when the cart is just about to rebound. As the cart travels down the air track at a constant speed, the kinetic energy remains constant. Moreover, as the cart approaches its minimum distance between the two magnets, the KE exponentially decreases as the MPE increases at an identical rate. Since it appears that the KE reaches zero at the same time the MPE reaches its maximum, it is safe to conclude that energy was conserved for this experiment. Another way we know that energy was conserved is if the total energy (TE) of the system remains constant. If the TE of the system decreased over time, then this would imply that a net external force acted on the system, causing some of the initial KE to be lost in the process. However, since our line for the total system energy stayed relatively constant throughout, it is safe for us to conclude that energy is conserved.

CONCLUSION:
Our group was able to successfully create an equation for the MPE of the system that showcased the energy of the system was relatively conserved. However, there is some uncertainty in our calculations because our line for the total energy of the system is not as straight as we had hoped. This tells us that there was most likely a net external force acting on the system from sources we did not take into account. For example, if the track was unleveled or was not uniformly frictionless, then the speed of the cart could have been hindered by friction or gravity. Furthermore, the inaccuracies in our energy calculations could have also resulted from the imprecise measurement of the separation distance, r, the distance between the motion detector and the magnet, the angle of incline, or even the mass of the  cart+reflector. If we used more precise equipment, a perfectly level and frictionless surface, and minimal human error, then we could expect to see a more straight TE line, indicating that the energy of our system was indeed fully conserved.

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

8/12/2016 - Ballistic Pendulum Lab

PURPOSE:

Determine the firing speed of a ball form a spring-loaded gun.

THEORY:
The ball, mass m, undergoes an inelastic collision with a nylon block, mass M. We assume that the collision happens so quickly that the strings stay vertical throughout the entire collision. This allows us to use conservation of momentum to write an equation for the speed of the system after the collision.

After the collision, the block+ball system rises, losing KE and gaining GPE until it reaches its maximum height. We can use the conservation of energy to write an expression that relates the maximum height to the initial speed of the block.

APPARATUS:
In this lab, the apparatus comes pre-assembled. A spring-loaded gun fires a ball into a nylon block, which is supported by four strings. The ball is absorbed into the block, and the ball and block rise together through some angle, which is measured by the angle indicator.

EXPERIMENTAL PROCEDURE:
Part 1: Initial Velocity

1. Measure/record the mass of the ball and block.
2. Level the block and apparatus.
3. Pull back and lock the spring into one of the notches. (Keep this notch consistent)
4. Fire the ball and record the max angle it travels through.
5. Repeat this 4-5 times to get an average.
6. Calculate the firing speed of the ball.

Part 2: Verification

1. Move nylon block away. 
2. Prepare the apparatus to launch the ball off the table.
3. Place a piece of carbon paper over another piece of paper close to where you expect the ball to land.
4. Determine the actual launch speed of the ball.

DATA:
Part 1:

Part 2:

height of barrel from floor: 0.97 meter

distance traveled by ball: 2.52 meters

ANALYSIS:
Initial Velocity:

Energy and Momentum Approach

Propagated Uncertainty Calculations

Verification:

Kinematics Approach

Propagated Uncertainty Calculations 2

ANALYSIS:

The ballistic pendulum lab is a classic example of an inelastic collision, i.e. a collision between two objects that stick to one another afterwards. Since this collision is inelastic, the momentum of the system is conserved before and after the collision. Energy, on the other hand, is not conserved before and after because there is an inherent loss of KE in the system when the two colliding masses merge with one another. However, since there is presumably no net external force acting on the system after the collision, energy is conserved after the collision, as the bullet+ball system rises with KE and reaches its maximum height with nothing but GPE. After calculating the initial speed of the bullet with our energy and momentum equations, we yielded 6.02 +/- 0.15 m/s. After calculating the initial speed using kinematics, however, we yielded 5.64 +/- 0.00364 m/s. These results have an unacceptable margin between each other because they are not within even the respective upper and lower ends of our propagated uncertainty calculations.

CONCLUSION:

Our results for this experiment were unsatisfactory. After calculating the initial speed of the bullet with our energy and momentum equations, we yielded answers for the initial velocity of the bullet that had an unacceptable margin between each other. There are several factors that could have contributed to this uncertainty. If, for instance, the nylon block remained unbalanced, then our calculations for initial velocity would be inherently flawed because momentum would not have been conserved. Another source of uncertainty is in human error. In our group's case, we had up to twelve different people trying to work together while a handful of people made their own measurements. This chaos increased our chances of miscommunication, potentially leading to imprecise measurements for the whole group. 

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

10/19/2016 - Collisions in Two Dimensions (Lab 15)

PURPOSE:
To look at a 2-D collision and determine if momentum and kinetic energy are conserved.

THEORY:
In collisions where there are no external forces acting on the system, both momentum and kinetic energy are conserved. In this experiment, we want to see if these principles apply to a collision between two balls on a frictionless surface. To test this, we calculate the kinetic energy (KE = .5mv^2) and momentum (p = mv) before and after the collision. If the before and after values are nearly identical for the respective quantity of motion, then the collision is elastic.

APPARATUS:
We used a leveled glass table as our frictionless surface. For our collision objects, we used a glass marble and a steel ball. To capture the collision in slow-motion, we used a ring stand to suspend a smartphone with the capability to record at 120-240 fps.

EXPERIMENTAL PROCEDURE:

1. Gather appropriate materials.
2. Set up apparatus as shown.
3. Make sure glass table is leveled.
4. Measure length of glass table.
5. Place marble in the center of the table.
6. Set up smartphone to record in slow-motion.
7. Mount phone onto ring stand.
8. Record the two following collisions separately:
9. Aim and roll a steel ball towards the marble.
10. Aim and roll another marble towards the marble.
11. Transfer these video files to LoggerPro.
12. Input your table length measurement to calibrate your video tracer.
13. Adjust the click rate to 4-8 frames.
14. Trace the position of the each ball separately for each collision.
15. Get the velocities of each ball before and after each collision.
16. For each collision data set, create two calculated columns that show the momentum in both the x and y axes.
17. Finally, create graphs for the position and velocity of the center of mass in both the x and y axes.

DATA/GRAPHS:
mass of marble (middle): 19.6 g
mass of second marble: 19.8 g
mass of steel ball: 66.9 g
Length of glass table: 0.625 m

Marble-Marble Collision

Marble-Marble Position Graph

KE-Px-Py for Marble-Marble Collision

CM Position for Marble-Marble Collision

CM Velocity for Marble-Marble Collision

Steel Ball-Marble Collision

Steel Ball-Marble Position Graph

KE-Px-Py for Steel Ball-Marble Collision

CM Position for Steel Ball-Marble Collision

CM Velocity for Steel Ball-Marble Collision

ANALYSIS:
By simulating a collision between two balls with different masses, we were able to analyze their behavior to determine whether or not momentum and energy were conserved. However, instead of doing the calculations by hand, our group decided to derive equations that could be inputted into a calculated column for both momentum (x and y) and kinetic energy for each collision respectively.

KE-Px-Py Equations for LoggerPro

If momentum is conserved, then the the momentum in the x-axis should be equal but opposite to the momentum in the y-axis. In other words, if our graphs for Px vs. t and Py vs. t are nearly identical but opposite in value, then momentum is conserved. If kinetic energy is conserved, then we should expect to see that the KE vs. time graph should be nearly constant.

After analyzing our graphs, it becomes apparent that the momentum for both collisions appear to be conserved. However, it is less apparent to see that the kinetic energy is conserved. The collision between the two marbles yielded a relatively straight line, but the collision between the steel ball and the marble had an erratic line, indicating that the kinetic energy throughout the collision varied greatly.

CONCLUSION:
Our results for this experiment were somewhat bittersweet. On the one hand, we successfully demonstrated that the momentum of both collisions were conserved. On the other hand, we had difficulty verifying the same was true for the kinetic energy of both systems. We assumed that there were no external forces acting on the system, although there is bound to be some inherent uncertainty to this statement. For example, if the glass surface we used was unclean, it could have gathered residue, which could have inadvertently hindered the speed of the ball. Another possible source of uncertainty is human error. The motion tracking software on LoggerPro utilizes human input, which can lead to the imprecise collection of data for the position, velocity, and to that end, all of our momentum and KE computations as well.

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

10/3/16: Centripetal Force with Motor (Lab 9)

PURPOSE:
To create an expression that models the relationship between angular velocity, ω, and angle, θ, for a mass revolving around a central shaft.

INTRODUCTION:
As the motor spins at a higher angular speed ω, the mass attached to the central rod revolves at a larger radius and the angle θ increases. From this experiment, we can quantitatively define a relationship between angular speed and angle measure. The goal of this lab, therefore, is to derive an expression that defines ω in terms of θ. After this is done, we will test the accuracy of our model by comparing it to another method for calculating angular speed ω in terms of time, t.

APPARATUS:
Note: the apparatus was pre-assembled for this particular lab.

The real apparatus (top) and a diagram (bottom):

• An electric motor mounted on a surveying tripod 

• A long shaft going vertically up from the motor.

• A horizontal rod mounted on the vertical rod. 

• A long string tied to the end of the horizontal rod. 

• A rubber stopper at the end of the string. 

• A ring stand with a horizontal piece of paper or tape sticking out.

We utilized the apparatus to derive some key variables before proceeding with the experiment. For example, our group:

• Derived θ by looking at the right triangle -- from the diagram above -- with hypotenuse L and height H-h. Equation: θ = arccos((H-h)/L)
• Derived ω from timing the duration for the mass to make a number of revolutions around the shaft. Equation: ω = 20π/Δt (20π because we did ten rotations)
• Derived h by putting a horizontal piece of paper on a ring stand and slowly raising the piece of paper until the stopper just grazed the top of it as it passed by. Equation: None, just a ruler.

PROCEDURE:

1) Use a free body diagram (FBD) to create centripetal force equations of the system. Use these equations to create a mathematical model for ω in terms of θ. Here is our derivation of ω:

2) Use the apparatus to gather a sufficient amount of data to test your model. More specifically, you must collect values of h at a variety of values for ω. The professor adjusted ω by increasing the voltage to the motor driving the system.

3) Create an Excel Spreadsheet with the following variables: t, h, H-h, θ, ω (t), ω (h).

4) Create a graph of angular speed with respect to time t vs. angular speed with respect to angle θ.

DATA/GRAPHS:
Uncertainty in  r, L, and H: +/- 0.1 cm
Uncertainty in t: +/- 0.25 s (due to reaction time)
Uncertainty in h, H-h: +/- 0.1 cm

ω(t) = angular speed with respect to time.
ω(h) = angular speed with respect to θ.

***In chart "r" = "R" on the apparatus diagram.

Data Table with Relevant Variables and Constants

ANALYSIS: 

There are two forces acting on the stopper as it is spinning: tension and gravity. The horizontal component of tension is providing a net centripetal force on the rubber stopper, helping it accelerate towards the center and rotate at a constant speed. Since the stopper is not moving in the y-direction, Newton's first law dictates that the net force must be equal to zero. Therefore, we now know that the vertical component of the tension force is equal in magnitude but opposite in direction to the weight of the rubber stopper. Using these principles, we formulated our force equations, manipulated them, and solved for angular speed ω. The slope of the previous graph, therefore, represents the accuracy of our derived value of ω with respect to θ. The slope is in the form: 1 + uncertainty in ω(θ), (where ω(θ) is angular speed with respect to theta). Additionally, R² represents the correlation value of our graph. In other words, it quantifies how close our linear fit of the data came to matching all of our data points. According to our graph, we accumulated an uncertainty of about 4.82%.

CONCLUSION:

Overall, our calculation of angular speed with respect to an angle was very accurate. This is because our expression for ω(θ) calculated a value for angular speed that was within 5% of the angular speed we calculated with respect to time. While some 4A students would be satisfied with this minute margin of error, I for one find it much more satisfying to explore the ways in which our group could have mitigated a multitude of these myriad mistakes. The first step in minimizing this uncertainty is identifying the root causes. One of the main causes of uncertainty in our lab was human error. For example, it is possible that our group mate inaccurately measured the period of rotation due to our inherent lag known as "reaction time." Moreover, it is also possible that we could have inaccurately measured distances on the apparatus by a small margin as well. This is because the tick marks on the meter sticks can be difficult to distinguish at times and different perspectives can lead to inconsistencies over the most precise measurement. Another source of error in our lab were variables we omitted from our calculations. For instance, our class decided that air resistance due to drag was negligible. Thusly, we did not take drag force into consideration when creating our FBD and force equations. Another source of uncertainty in our calculations arose from the reverberation of the ruler atop the central shaft. As the motor spun the central shaft, the ruler would slightly oscillate, causing the radius of rotation to change as well. For the simplicity of the lab, however, our class decided that this effect was negligible as well and we treated the radius r as a constant for a given angular speed ω. In an ideal world, our group would redo this lab with top notch equipment and extreme attention to detail, but in recognition of the technical limitations and time constraints of a 4A class, our group accepts that our current experimental results will suffice. 

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

7-Sep-2016: Free Fall Lab

PURPOSE:

"To examine the validity of this statement:
In the absence of all other external forces except gravity, a falling body will accelerate at 9.81 m/s/s."

In other words, we will try to determine the value of g while simultaneously learning about Excel and some statistics for analyzing data.

INRODUCTION:

The goal of this lab was to derive a value for the acceleration due to gravity on planet Earth. Using the Spark Generator Free Fall Apparatus 9000, our group was able to directly measure and record the displacement of an object in free fall. With this information -- and a few other calculated variables -- our group made a spreadsheet on Excel and plotted a position vs. time graph, as well as a mid-interval speed vs. time graph.

APPARATUS:

This is the Spark Generator Free Fall Apparatus 9000, with corresponding electromagnet and variable power supply. Its main purpose is to provide students with a permanent record of a free falling body. The free falling body (blue thing) is held in place at the top by an electromagnet. When released, a spark generator precisely records the fall of this object onto a thin strip of spark-sensitive paper, which can be used to create a position vs. time graph and a velocity vs. time graph to calculate acceleration.

While it is always a wise convention to know how to use your equipment before doing lab, today I will make an exception in the interest of time. In my particular class, our professor decided to just hand us the strips of "pre-sparked" paper. For more information on how to use this device, refer to your lab module.

Adult supervision required

PROCEDURE:

Equipment needed: Spark tape from teacher; computer with Microsoft Excel; meter-stick

1) Lay your spark tape on a long, flat surface (such as a table). Notice the dots and their relative distances. Make sure that the distance between the dots becomes larger from left to right. These dots correspond to the position of the falling mass every 1/60th of a second:

spark tape with series of dots

2) Place a meter stick next to the tape. Line up the 0-cm mark with the first dot and record the position of each dot from the 0-cm mark. In other words, record the displacement of each dot.

3) Create an Excel Spreadsheet with the following columns: time, distance, delta x, mid-interval time,  and mid-interval speed. Like so:

The time column is in increments of 1/60th of a second. The distance column shows the relative displacement of each dot from the origin -- the 0-cm mark. Although unmarked, it should be noted that each distance has an uncertainty of +/- 0.1 cm. The delta x column shows the distance between two consecutive dots on the spark tape. For example, the value in cell C2 is obtained by =(B3-B2). The mid-interval time column gives the time for the middle of the each 1/60th s interval. The mid-interval speed is essentially (delta x)/time, where time = 1/60th of a second.

4) Create a mid-interval speed vs. time graph using your data from columns D and E. Make sure to do a linear fit afterwards to obtain an equation with a correlation value of R². 

5) Create a position vs. time graph using columns A and B. Do a polynomial fit of order 2 in order to obtain a polynomial equation with a correlation value of R² as well.

Here is a picture of our graphs:

Distance v. Time (left) and Mid-interval speed v. Time (right)

QUESTIONS:

1) Consider an object in free fall with an acceleration constant, a = 10 m/s/s. The object was initially at rest and fell a certain distance for three seconds. If I arbitrarily choose my interval to be three seconds, then my mid-interval would be defined as:

V_mid-interval, @t_1.5 = V₀ + at 

= 0 m/s + (10 m/s/s)(1.5 s) = 15 m/s

If I wanted to know the average velocity for my interval, then I would simply use the equation: 

 V_avg = (V₀ +V_f) / 2

Where

V₀, @t₀ = 0 m/s,

V_f, @t₃ = 30 m/s,

Plugging everything in, we find:

V_avg = (0 m/s + 30 m/s) / 2 = 15 m/s

Therefore,

V_avg = 15 m/s = V_mid-interval

2) The slope of the velocity vs. time graph for the free falling body yields the acceleration due to gravity. This means that our graph gave us a value for gravity, g ≈ 9.54 m/s/s. This corresponds to roughly a 3% error compared to the accepted value of 9.81 m/s/s.

3) Since the equation of the position vs. time graph is a polynomial of the form y = ax² + bx + c, we can assume that it corresponds to the kinematic equation: Δy = V_0yt + (1/2)gt². Therefore, we can conclude that:

(1/2)g = 4.7591 m/s/s

g = 9.5182 m/s/s

This means that our graph gave us a value for gravity, g ≈ 9.52 m/s/s. This corresponds to roughly a 3% error compared to the accepted value of 9.81 m/s/s.

CONCLUSION: 

After carefully analyzing our results, I think it is safe to say that our results for the theoretical value of g were fairly accurate. This is because both of our derived values of g were within 3% of the accepted value of 9.8 m/s/s. However, there is no doubt that there are bound to be sources of uncertainty inherent to even our most precise calculations. One possible source of error in our calculations was the variation in significant figures between the distances we measured. For example, in cells B2 through B7, we only used one to two significant figures, while in cells B8 through B11, we used three significant figures. Furthermore, it is entirely possible that we could have misread numbers in our measurements of distance, leading to inaccuracies in our derivation of g. 

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

21-9-2016: Friction Lab

PURPOSE:

Use experimental data to derive models of frictional force and use these models to predict the acceleration of a two-mass system.

INTRODUCTORY STATEMENT:

Our group performed five (5) different experiments involving friction. This lab report is designed to give a detailed analysis of each experiment.

(1) STATIC FRICTION:

The first experiment involved measuring static friction on a flat surface.

Apparatus:

wooden board on a table top

1) Use a C-clamp to secure flat surface to a table. We used a wooden board.

2) Secure pulley to the wooden board.

3) Attach hanging mass to wooden block using string.

4) Hang string off of pulley with the wooden block on the table and the hanging mass off the edge. Note: Make sure that the metal side of the block is on the table.

Procedure:

1) Add weight a little bit at a time until the wooden block begins to slip. Record the mass of the block and the corresponding hanging mass required to get the block to start to move.

2) Incrementally add mass to the wooden block. Record this new mass.

3) Reattach the hanging mass to the wooden block with string and hang it over the pulley.

4) Once again, add weight a little bit at a time until the block begins to slide.

5) Record the appropriate data and repeat these steps for a total of four data points.

6) Use your data to plot an N vs. F_{s, max} graph.

The goal of this experiment was to derive the coefficient of static friction, μ_s between a block and a table. Given the model of static friction, we know that the coefficient of static friction between the block and the table equals the maximum value of static friction, Fs, max divided by the normal force N that squeezes them together:

μ_s = (F_{s, max} )/ N, or F_{s, max}  = μ_sN

As previously stated, we incrementally added small weights to a hanging mass until the block just started to slip. According to Newton's first law, since the the block is not changing its motion, all of the forces acting on this object are balanced. Therefore, one can infer that there must be a static frictional force equal, but opposite to the weight of the hanging mass that is pulling on the other side of the block. In essence, we can conclude that:

F_{s, max} = weight of hanging mass = M_hangingg

(where g = 9.8 m/s/s, the acceleration due to gravity)

Furthermore, since the net force on the vertical axis must also be 0, we can infer that the normal force between the block and the table must be equal in magnitude, but opposite in direction to the weight of the block itself. In other words, we can also conclude that:

N = weight of block = M_blockg

Using this principle, we found the corresponding F_{s, max} values and N values for blocks of different masses and plotted an N vs. F_{s, max}  graph on Lab Pro. This was our graph:

The red column lists the different block masses we tried. We started off with a block mass of 191.5 +/- 0.1 g, adding 100, 200, 300, and 500 g respectively. The black column lists the corresponding hanging masses which caused the block to slip. The original hanging mass --without additional weights-- had a mass of 50.0 +/- 0.1 g. The blue column lists the max. static friction force, F_{s, max} and the green column lists the normal force, N. The slope of this graph equals the coefficient of static friction, μ_s between the block and the table. Therefore, our value of μ_s ≈ 0.269

(2) KINETIC FRICTION:

The second experiment dealt with the measurement of kinetic friction on a flat surface.

Apparatus / Procedure:

It is essentially the same as the first experiment but with one major change: the hanging mass is replaced with a force sensor. Note: make sure the string is not too long because it needs to be pulled by someone at a constant speed.

1) Attach force sensor to wooden block with string. Note: make sure it is calibrated first. Also, we decided to keep the pulley as a guiding mechanism for the path of the block.

2) Use Lab Pro to collect force data as someone slowly pulls horizontally, moving the block at a constant speed. Store this run and record the mean value of the pulling force used to move the block at a constant speed.

3) Incrementally add mass to the wooden block and repeat the above steps. Store this run as well.

4) Repeat these steps for a total of four different masses for the wooden block.

5) Use your data to plot an N vs. F_k graph.

The goal of this experiment was to derive the coefficient of kinetic friction, μ_k between the block and the table. Given the model of kinetic friction, we know that the coefficient of kinetic friction between the block and the table equals the kinetic friction force divided by the normal force N that squeezes them together:

μ_k = (F_k)/ N, or F_k = μ_kN

In this experiment, we used a Force sensor to directly measure the average magnitude of force necessary to keep the block sliding at a constant speed. The block may be moving this time around, but this does not change our mentality towards the problem. In fact, Newton's first law dictates that since the block is not accelerating, all of the forces acting on this object are still balanced. Therefore, we can conclude that there must be a kinetic frictional force equal, but opposite to the force that is pulling on the other side of the block. In essence, we can conclude that:

F_k = F_sensor
(where F_sensor equals the mean force sensor reading for a given trial)

Furthermore, since the net force on the vertical axis must also be 0, we can infer that the normal force between the block and the table must be equal in magnitude, but opposite in direction to the weight of the block itself. In other words, we can also conclude that:

N = weight of block = M_blockg

Using this principle, we found the corresponding F_k values and N values for blocks of different masses and plotted an N vs. F_k graph on Lab Pro. This is the graph of our force sensor readings:

Force vs. Time graph (four runs)

Below is our N vs. F_k graph:

Normal force vs. Kinetic friction force

The black column lists the different block masses we tried. Just like the first experiment, we started off with a block mass of 191.5 +/- 0.1 g, adding 100, 200, 300, and 500 g respectively. The red column lists the kinetic friction force, F_k and the blue column lists the normal force, N for each corresponding mass. The slope of this graph equals the coefficient of kinetic friction, μ_k between the block and the table. Therefore, our value of μ_k ≈ 0.305

(3) STATIC FRICTION ON AN INCLINE:

The third experiment required us to measure static friction on a sloped surface.

Apparatus / Procedure:

1) Place a wooden block on a horizontal surface and slowly raised one end of the surface until the block just started to slip. Record the angle at which this happens as well.

2) Use a ring stand and C-clamp to secure the flat wooden surface in place. Ignore the motion detector attached at the top for now.

3) Use the angle at which the slipping begins to determine the coefficient of static friction between the block and the sloped surface.

Much like the first experiment, the goal was to derive the coefficient of static friction, μ_s between a block and a table. Given the model of static friction, we know that the coefficient of static friction between the block and the table equals the maximum value of static friction, Fs, max divided by the normal force N that squeezes them together:

μ_s = (F_{s, max})/ N, or F_{s, max} = μ_sN

In order to determine the coefficient of static friction between the block and the surface, we must use the aforementioned equation. However, we still have two variables missing: F_{s, max} and N.

Therefore, we must draw a free body diagram (FBD) of all the forces acting on the block in order to determine μ_s. In this FBD, our coordinate system is centered at the block with the axes parallel and perpendicular to the slope:

According to Newton's first law, since the the block is not changing its motion, all of the forces acting on the system are balanced. For this to be true, the net force for the x axis must be equal to 0. Thusly, the static friction force must be equal but opposite to the horizontal component of the weight of the block. Essentially, we concluded that:

Fs, max = mgsinθ (where m is the mass of the block)

Furthermore, since the net force on the vertical axis must also be 0, we can infer that the normal force between the block and the table must be equal in magnitude, but opposite in direction to the vertical component of the weight of the block itself. In other words, we can also conclude that:

N = mgcosθ

Using these equations, we were able to substitute them into our original equation for μ_s to find that:

     Fs, max = mgsinθ

μ_smgcosθ = mgsinθ

        μ_s = tanθ

Our value for μ_s = tan19° ≈ 0.344

(4) KINETIC FRICTION ON AN INCLINE:

The fourth experiment required us to measure kinetic friction on a sloped surface.

Apparatus / Procedure:

In this experiment, we essentially used the same apparatus as the previous experiment. There are two main differences: (a), we put our incline at a higher angle than before and (b), there is now a motion detector at the top of the incline.

1) Increase the angle of incline such that it will facilitate the acceleration of the block down the slope.  Record this new angle.

2) Use a C-clamp and a metal rod to secure the motion detector onto the ring stand above the sloped surface. Make sure that the motion detector's line of sight is parallel to the incline's surface.

3) Tape a notecard to the back of the wooden block. This will make it easier for the motion detector to "see" the block.

4) Place the block near the motion detector with the notecard facing it. Hold the block a reasonable distance away from the motion detector to avoid inaccurate readings. We held it about 8 inches away.

5) Use the motion detector to create a velocity vs. time graph on Lab Pro. Use this graph to find the acceleration of the system.

With these changes in mind, we measured the angle of incline and used the slope of the velocity vs. time graph on Lab Pro to find the acceleration of the block. Here is our graph:

Linear fit of Velocity vs. Time graph

We found our system acceleration to be a = 2.829 m/s/s

Much like the second experiment, the goal was to derive the coefficient of kinetic friction, μ_k between a block and a table. Given the model of kinetic friction, we know that the coefficient of kinetic friction between the block and the table equals the kinetic friction force divided by the normal force N that squeezes them together:

μ_k = (F_k)/ N, or F_k = μ_kN

In order to determine the coefficient of kinetic friction, μ_k, we drew a FBD of the system, created force equations for the x and y components, and solved for μ_k. Here is our derivation of μ_k:

Since the block's velocity changed, our strategy for finding μ_k changed as well. Newton's second law dictates that since the block is accelerating down the slope, our net force can be written as the product of the system's mass and acceleration. More specifically, the net force in the x axis can be rewritten as the horizontal component of the block's weight minus the kinetic friction force equals mass times acceleration. Similarly, since the net force in the y axis is equal to 0, the vertical component of the block's weight must be equal in magnitude, but opposite in direction to the normal force. Using these equations, we were able to create a new expression for μ_k:

μ_k = (gsinθ - a)/(gcosθ)

(where theta equals 30 degrees)

After plugging everything in, we found that μ_k ≈ 0.244

(5) PREDICTING THE ACCELERATION OF A TWO-MASS SYSTEM

Using the coefficient of kinetic friction from experiment 4, we had to derive an expression for what the acceleration of the wooden block would be if we attached a hanging mass that was sufficiently heavy to accelerate the system. This is the system we had to model:

To find the acceleration of the system, we drew FBDs of the system, converted the information into force equations for each axis, and manipulated these equations to create an expression for the acceleration. Here is our derivation of this expression:

After plugging everything in, we found that the system acceleration, a = 2.964 m/s/s. To test the accuracy of our prediction, we ran the experiment ourselves. After one trial, we obtained a velocity vs. time graph and derived the acceleration from the slope of the linear fit of the graph. Here is the graph we obtained: 

As you can see, we found the experimental acceleration, a_actual ≈ 2.629 m/s/s.

CONCLUSION:

After finishing the lab, we asked our professor to look over our results to verify if it was reasonable. He not only agreed it was reasonable but also concluded that our theoretical value for the system acceleration was much closer than he had anticipated his students to have. However, no matter how close our models may come to real life situations, there will most likely be uncertainties which can contribute to error in our calculations. One possible source of error in our calculations was human error. In experiment 4, for instance, placing the wooden block too far or too close to the motion detector could have resulted in inaccurate readings for the velocity of the system. Another source of error could have been the preciseness of our measurements and other quantities. For example, we decided to round the coefficient of kinetic friction from experiment 4 to three significant figures in our calculations of the system acceleration in experiment 5. Similarly, we decided to use 9.8 m/s/s for the acceleration due to gravity instead of 9.81 m/s/s. While such practices may be considered trivial by some individuals, an astute 4A student would be able to recognize the propagated uncertainty inherent to these small changes and would be encouraged to mitigate this uncertainty to the highest degree. Despite these erroneous conventions, I still maintain that our lab was an overall success.

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