By: William Berdugo
Lab Partners: Abdal Feileh and Cheng Lin Nie
Lab Performed from 6/3/2019 to 6/5/2019

Purpose:
The goal of this experiment was to derive expressions for the period of a variety of physical pendulums using angular momentum principles and physical measurements of dimensions to then verify these predicted periods using a photogate measuring the actual period of the oscillating object.

Introduction/Theory:
The method to mathematically determining the period of an object acting as a pendulum first involves ascertaining where the axis of rotation is located on the mass.
Using the moment of inertia about the axis of rotation, a step which involves taking the parallel axis theorem with d being the displacement from the axis of rotation to the object’s center of mass.

The fundamental equation for simple harmonic motion is Newton’s 2nd Law for rotational motion which relates torque to moment of inertia and angular acceleration.

When we rearrange the equation above to isolate angular acceleration, we get 𝛂 = – constant * displacement where the constant is equal to ⍵². The constant will contain the force of the object’s center of mass at the distance perpendicular to the axis and the center of mass divided by the moment of inertia of that particular rotation. This ⍵ is angular frequency, not to be confused with rotational velocity and has the units of rad/s². Finally, we use the equation for the period of an oscillating mass that uses this omega to calculate the theoretical value for period.

Apparatus/Procedure:
Each part of the experiment involves one particular object or objects swinging at various points of axes of rotation. Every part required a large vertical ring stand with two clamps; one holding the a horizontal rod that lets the rod end of the photogate to be clamped vertically and one pivoting the object at a particular axis. Every part has a very similar set up. Once the object is balanced on the pivot, we lift the object to an angle where it can swing across. To make sure the photogate records the period of the swing, a piece of masking tape is placed at the end of it with a thickness enough to go through the laser.

Prior to performing the actual experiment, all derivations for the moment of inertia and set up for the period equation were done before class and only required the measurements of the necessary dimensions to be plugged in. All objects were either already provided or only needed pivot clips to be attached to them.

Part 1: Meter Stick
The first mass to place on the ring stand is a wooden meter stick at a point on the 20 cm mark using a nail to act as the pivot point.

Part 2: Ring of Finite Thickness
The nail was used again to serve as the pivot for the ring of finite thickness.

Part 3: Isosceles Triangle & Semicircular Disk
For these objects, paper clips were bent and shaped as pivot clips for another paper clip (extended) taped to a horizontal rod to act as the pivot for the objects to be placed on. for the axis where a hole was not already made. The same procedure as before was performed for measuring the period. An isosceles triangle of base B and height H was pivoted at both the apex and at the mid-point of its base.

A semicircular disk of radius R was pivoted at the mid-point of the base and at the circular edge directly across (opposite) the midpoint of the base.

Data:
Below are all the values collected for the periods of each oscillating object using the photogate as the measuring tool along with the values for the measurements of the necessary dimensions.
Meter Stick
The length of the meter stick is 1 meter with the center of mass at 0.5 m.

Ring of Finite Thickness
The outer radius was measured to be 0.0404 m while the inner radius was 0.0337 m.

Isosceles Triangle
The base was measured to be 0.177 m and the height was 0.0147 m (measured from the center of the pivot hole at its apex to the base).

Semicircular Disk
The radius of the semicircular disk was measured to be 0.086 m.

Results/Analysis:
Below are all the calculations for moment of inertia and the calculations for period of oscillation for each object.
Ruler

Ring of Finite Thickness

Isosceles Triangle

Semicircular Disk

Conclusions:
Comparing the experimental results with what we derived, the percent errors for the periods for the meter stick was -0.46 %, the thick ring was 1.01%, the apex of the triangle was 2.41%, the midpoint of the base of the triangle was 1.73%, the midpoint of the base of the semicircular disk was 0.85%, and at the top of the semicircular disk was 2.45%.

We can assume that the extra mass of the pivot clips, which were located a small distance away from the edge, shouldn’t measurably affect moment of inertia measurements, but the periods measured for the isosceles triangle and semicircular disk where the holes we cut out in the actual shape of the object proved to be ones with the largest percent difference in comparison to the theoretical values. The masking tape was expected to have a bigger effect on the measured moment of inertia than the pivot clips because the tape has a larger surface area that covers more when passing through the photogate and the period was not measured from the actual object, but from an extension of it (an increased displacement from the axis of rotation to the center of mass).

The thick ring of an outer and inner radius had an indent to act as the suspension point and was considered equivalent to the distance halfway between the radii. Although, taking the mass of each object would have been a measurement that could have been performed, eventually the mass of the object cancels out within the set up of the angular acceleration equal to – constant multiplied by displacement. Therefore, it was not necessary to do this. The times where we had to redo our period measurements was when the swing was not perfectly perpendicular to the photogate

By: William Berdugo
Lab Partners: Abdal Feileh and Cheng Lin Nie
Lab Performed on 5/17/19

Purpose:
The goal of this experiment is to determine the moment of inertia of a right uniformly massed triangle around its center of mass for when its up right and on its side by rotating the entire system.

Introduction/Theory:
From the previous lab, we determined an equation for the moment of inertia of a disk with a hanging mass connected to it relying on 𝛂_average of the system. We use this equation because the complication in the experiment is that there is some frictional torque in the system from the pulley and the disk connected to the hanging mass with a tension in the string, thus using the 𝛂_average of the system.

Below is the approach taken for this experiment utilizing the parallel axis theorem to determine the moment of inertia of an object when the axis of rotation is shifted from the one of its sides to its center of mass and below that is the calculation of the moment of inertia using 𝜤 = 𝜤_cm+ Md² where d is the distance shifted to the new axis of rotation along the center of mass or the parallel axis displacement.

We will use this calculated moment of inertia for the center of mass as the theoretical value and use the data values obtained from the experiment to get our experimental value, which we will then compare the two.

Apparatus/Procedure:
The apparatus in this experiment is the same as the rotary motion machine, rotational dynamics apparatus, from the Angular Acceleration lab, where we will use the large torque pulley and top steel rotating disk. The difference here is that we will be mounting a uniform triangle either with the base at the bottom and the long edge facing vertically or facing horizontally acting as the base on a holder with a screw.

The first step is to take appropriate measurements of the equipment such as the sides of the triangle, its mass, the masses of the hanging object, the steel rotating disk, the large torque pulley, and the triangular plate holder. Thus, we can place everything except the triangle and let the entire apparatus rotate. Take an appropriate amount of time so that there’s enough data collection for the hanging mass to travel completely down and up a few times.

Then, place the triangle upwards onto the holder and let the entire apparatus rotate. Likewise, take data collection of the angular velocity.

Finally, place the triangle in the holder with the longer side serving as the new base value and obtain the data of its angular velocity.

Data:
Below, is one example of the three screenshots taken to calculate the magnitudes of 𝛂_down and 𝛂_up from the angular velocity vs time graph used to obtain a singular value for 𝛂_average.

The measurements taken for the triangular thin plate were 0.0985 m for the shorter side, B, 0.1493 m for the longer side, H, and the mass was 0.4569 kg). The radius of the large torque pulley was 0.0249 m and the total mass of system without the triangle being placed was 1.4111 kg from the combination of masses of the torque pulley (0.0363 kg), the steel disk (1.3527 kg), and the triangular plate holder (0.0221 kg). The hanging mass was 0.025 kg.

Results/Analysis:
The moment of inertia of the entire system with the triangle in the up position was 0.0030120 kg m² and with the triangle in the side position was 0.0033327 kg m². Using these moment of inertia, we can subtract from it the moment of inertia of just the system without the triangle to obtain our experimental values for the triangle in either orientation. Below is the calculations for calculating the experimental and theoretical moment of inertia for the triangle in both setups.

Conclusions:
After using the equation for moment of inertia for the center of mass of the triangle as the theoretical value and the moment of inertia of the disk as the experiment subtrahend, the percent error for when the triangle was orientated upwards with the base of 0.0985 m was 0.28% and when the triangle was orientated on the side with base value as the height of 0.1493 m it was 0.34%. Overall, the calculations were very successful and making sure taking the experimental values with the correct 𝛂_average value for the triangle orientation minus the moment of inertia of the apparatus without the triangle was paramount in making sure both approaches were correct. The conduct for this experiment went very smoothly and going through the angular acceleration experiment beforehand really made this experiment much easier to go through.

By: William Berdugo
Lab Partners: Abdal Feileh and Cheng Lin Nie
Lab Performed from 5/10/19 to 5/17/19

Purpose:
Using a rotational dynamics apparatus that utilizes sets of rotating metal disks and a pulley-hanging mass component that contributes a torque to cause an angular acceleration used to calculate the moment of inertia of the system.

Introduction/Theory:
To obtain an angular acceleration for when the hanging mass goes down and when it goes back up, we look at the angular velocity of the system, ⍵, versus time and take the linear positive slope as 𝛂_down and the linear negative slope as 𝛂_up.
When calculating the moment of inertia of a disk, we know that when it is spinning on an axis perpendicular to the center of the disk, on the z-axis of the disk, this is the equation for the moment of inertia.

This equation will be used as the theoretical value for the moment of inertia of the system, for each of the different experiments, based on the dimensions and masses of the disks.

From applying Newton’s second law to the system of the disk+pulley and hanging mass, we obtain an equation for the moment of inertia of the disk. This equation will be used as the experimental value for the moment of inertia of the system, for each of the experiments, based on the data and results obtained for the hanging mass, radius of the torque pulleys, and the average angular acceleration.

Apparatus/Procedure:
The apparatus consists of a set of two metal disks, either steel or aluminum, stacked on top of each other where the top disk is rotating (once where both top and bottom are rotating) and either a small or larger torque pulley cup is screwed in the center of the set of disks that is connected to a hanging mass on a string of negligible mass.

The first section of the experiment was to take measurements (the diameter and mass) of the disks and torque pulleys and the value of the hanging mass. There are set of six different experiments (which are detailed in the third image of the data section) with varying setups consisting of changing the hanging mass by adding slotted masses on top, changing the radius from the torque pulleys, and changing the rotating mass (switching out sets of the metal disks). Connect a tube to provide air to cause the frictionless disks to rotate and connect the rotational dynamics machine to a computer to create graphs of angular data. The second section deals with using all the data values obtained to calculate theoretical and experimental values for the moment of inertia of the system.

Data:
The initial data set collected for the diameter and mass of each equipment. The hanging mass value was 25 g (0.025 kg) and the slotted masses used to double and triple the hanging mass value.

Using the idea presented earlier for determining the 𝛂_down and 𝛂_up, we took the linear fit of both the positive and negative slopes of the angular velocity graphs.

Detailed below is the chart that describes each of the six different experiments with 1, 2, and 3 only changing the hanging mass with a steep top disk rotating and small torque pulley screwed, 1 and 4 changing the size of the torque pulley (change in the radius), and experiments 4, 5, and 6 changing the rotating disks (steel, aluminum, and two steel) with a constant torque pulley (same r). Using the method from above for determining the angular acceleration, we calculated the average angular acceleration, 𝛂_average, for each experiment.

Results/Analysis:
The r is the radius of the torque pulleys, which were calculated by dividing the small and large torque pulley diameters by 2. The hanging mass, m, the radius, r, and average angular acceleration, 𝛂_average, were used for the experimental portion and the mass, M, and radius, R, of the rotating disk(s) were used for the theoretical portion.

Using the same method from the photo above, the theoretical values for moment of inertia using the radius and mass of the rotating disk (or disk combination) and the experimental values for the moment of inertia using the hanging mass, the radius of the torque pulley, and the average angular acceleration for each experiment are in the chart below.

Conclusions:
The effect of changing the hanging mass by twice and three times its mass directly correlated with the angular acceleration of the system where they increased by a factor of 1 in each case. For example, the 𝛂_average went from 1.136 to 2.253 to 3.380 rad/s². The difference in changing the radius of the torque (switching from a small pulley of radius 0.0124 m to a large pulley of radius 0.0249 m), also increases the 𝛂_average by a factor of 1. Finally, when changing the mass of the rotating disks (from top steel, to top aluminum, and two steel), the 𝛂_average increases when the mass is smaller.

After noting the effects these changes cause for the acceleration of the system, we then determined the theoretical and experimental values for the moment of inertia of the disks to draw comparisons between the two. The percent error for experiment 1 is -0.52%, experiment 2 is 0.18%, experiment 3 is 0.02%, experiment 4 is 3.43%, experiment 5 is 4.80%, and for experiment 6 is 3.43%.

The sources for the uncertainty in the calculations for the moment of inertia either come from the data collected from the system rotating and the slopes of the angular velocity or the assumptions made for calculating the theoretical values using the measurement values of the mass and dimensions of the disks. We had to perform the second experiment again after we had completed all of them because the 𝛂_down and 𝛂_up were less than 2.000 rad/s².

By: William Berdugo
Lab Partner: Sam Meng
Lab Performed from 5/20/19 to 5/22/19

Purpose:
The goal of this experiment is to determine the frictional torque of a spinning apparatus connected to a mass traveling down a slanted platform using the moment of inertia of the apparatus and Newton’s 2nd Law with basic kinematics to predict how fast the mass would travel 1 meter down the platform.

Introduction/Theory:
When the total mass of the entire system is known and measuring the dimensions of each of component of the system to calculate each parts volume, we can determine the mass of each components by knowing the relationship of each part’s individual mass over its volume is equal to the total mass over the total volume of the system. The individual density is proportional to the total density. Therefore, to find the total moment of inertia of an irregularly shaped system, take each components mass and radius/length to determine each part’s moment of inertia and sum them together to fins the total moment of inertia. In this experiment, all the parts of the apparatus have the same equation for its moment of inertia. Also, to calculate the torque 𝝉, of the entire apparatus, take the total moment of inertia, I, and multiply it by the angular acceleration, 𝛂 (in the case of this experiment, its angular deceleration).

Apparatus/Procedure:
The apparatus consists of a large, metal, rotating disk on a central shaft where the disk splits the two sides of the shaft on each of its sides. The mass of the entire system is stamped onto the right side of the disk. The entire apparatus is elevated by two sets of supports bolted to the outer ends of the shafts placed on a horizontal metal slab.

The first set of procedures includes measuring each of the diameters and depth of each of the cylindrical shaft pieces and the rotating disk using a variety of metal and digital calipers. Then, a phone device that is capable of recording is to be placed in a selfie stick that is clamped on a ring stand so that the phone can take the entire area of one side of the rotating disk. A small piece of tape is placed on right side of the apparatus’s disk with a black mark and the back of a clipboard was used as a background to make video capture easier. Make the disk spin and begin recording. Once the video is imported into the computer, begin taking video capture of the black dot marked tape rotating until the disk stops rotating. After all the points are collected, determine the angular deceleration of the disk as it’s slowing down from either plotting the angular velocity vs time or the angle of rotation vs time depending on how the video capture was taken.

Use the angular deceleration from before to calculate the frictional torque acting on the apparatus. Next, bring the apparatus to the edge of the table and place a frictionless track at an angle to the floor with a set of weights keeping the track from shifting where a cart attached to a long string, over 1.5 m, wrapped around the first cylinder can be put on the track so that the string can be horizontal. After calculating the acceleration of the cart and the time it would take for it to travel down the track for 1 meter, we recorded three trials of the cart actually moving down the track.

Data:
Below are the measurements of each of the components of the apparatus with a diagram representing what each variable is. The total mass, 4724 g, was taken from where it is stamped onto the rotating disk of the apparatus.

Using the measurements above we used the formula for a volume of a cylinder, V=πr²h, where r = d/2 for each component of the apparatus. For part 1, the volume was 40.165 cm³, for part 2, the volume was 38.210 cm³, and for part 3, the volume was 477.522 cm³. Using the reasoning from the theory, our calculations for the each of the components’ masses are 341.322 g, 324.709 g, and 4057.970 g.

The total moment of inertia, after converting from g*cm², was 0.0204 kg*m².

Results/Analysis:
This Omega (Angle) vs Time plot is from taking the video capture at every (pi) and manually inputting increments of pi at each position. Using a derivative and taking the linear fit of that derivative, we obtained a function for angular deceleration shown to be the slope of the line. Angular deceleration of the spinning apparatus is -0.5804 rad/s².

Multiplying the total moment of inertia of the entire apparatus and the angular deceleration from the graph above we get the frictional torque to be -0.0118 Nm².

Using the free body diagrams of both the apparatus (the moment of inertia of the rotating disk, the radius of the first component, and the frictional torque) and the cart (the horizontal component of its weight), we can cancel out tension in the string to have an equation for the carts acceleration, which becomes 0.04 m/s². We take the positive direction as the one going down the track. All the values of the variables are shown below. Also, the angle, θ, the track makes to the floor is 51.5 ° or 0.898 rad.

Conclusions:
One assumption made was that the total mass we took, 4724 g, was the total mass of the entire system (the large rotating disk and the two connected cylinder holders), which may only show the mass of the disk. We had to redo our points for the video capture and had to calculate 𝛂 in the manner discussed before, thus causing a delay in our calculation for 𝝉_f. Once we placed the cart of the track, we assume that the track is completely frictionless which may have slightly change our calculations for acceleration and, ultimately, the time predicted.

The results from timing the the cart’s run down the track were 7.13 s, 6.89 s, and 6.98 s. Our prediction for the time was 7.07 s. Taking the relative difference of the predicted time as the accepted value and the trial runs as the experimental values, we get 0.85%, -2.55%, and -1.27% percent error for each trail. The way the weights were placed at the end of the track allowed for the top of the cart to reach 1 meter, thus providing an easy indicator of when to press stop on the stopwatch.

We can therefore conclude that all of our measurements, our video capture to determine 𝛂 and 𝝉_f, and our calculations for the cart’s acceleration and the predicted time are all accurate and completely acceptable, since none of the trials go ±4%.

By: William Berdugo
Lab Partners: Shawn Espinoza and Sam Meng
Lab Performed from 4/22/19 to 4/24/19

Purpose:
The goal of this experiment is to observe the motion before and after the collision of two spherical objects in a two-dimensional plane and determine if energy and momentum is conserved.

Introduction/Theory:
When determining if an elastic collision between two objects in one-dimensional system has a conservation of momentum and energy we use the various equations that depend on the type of collision that took place. The only difference in a 2D collision is that one must account for the momentum and kinetic energy of two dimensions. For elastic collisions, which is the case in this experiment, we can take calculations, assuming, for the most part, that kinetic energy and momentum are conserved. The law of conservation of momentum requires the masses of the two objects to be known and, for both momentum and kinetic energy, the initial and final should be equal at each point of velocity.
Below are the equation we are to use to determine the momentum in the x and y axis and the kinetic energy in both axes. m_A is for the glass ball, the object that will collide with the stationary ball, m_B, the steel ball. The velocities correspond with the speed at a certain point for each ball.

To produce positions in both axes we used the expression of center of mass from below. The point of taking determining the center of mass between two colliding objects is that it would result in a more accurate representation of the positions and the velocities of both balls from before and after the elastic collision.

Apparatus/Procedure:
The apparatus in this experiment involves a squared 0.65m, “frictionless” glass table raised on three movable stands that moves around underneath the edges of the table (to find a level) with a vertical ring stand and a selfie stick clamped horizontally to have a cell phone placed to record the collision of the two balls from a downward perspective. A glass ball is set on the opposite top side of the table and the steel ball placed as close to the center of the table.

To accomplish our goal of determining conservation, our procedure is to launch the glass marble ball directly at the stationary steel ball and record all observations using video capture which will then be imported from the moment it leaves the hand to some adequate time after the collision. We are then to click on each reference point of motion for each of the balls from before and after the collision. This will give us a position vs time plot with two sets of x and y position points. We can then use the equations from before to obtain graphs such as momentum in each direction, kinetic energy, the center of mass position, and center of mass velocity vs time.

Data:
The mass of the glass ball was 0.0146 kg and the mass of the steel ball was 0.0670 kg and the inner length of the table was 0.65 m. The y-axis was set corresponding to the direction of motion of the glass ball and the x-axis is perpendicular to the that axis at the origin of where the steel ball was placed.

Results/Analysis:
Using the width of the inner table as the reference frame for the distance the balls travel and taking the points of each ball at every reference frame we obtain the graph below from the video capture. There is also the linear fits of the plots after the collision. We can compare these to the center of mass position graphs.

Instead of calculations to determine conservation of both energy and momentum in the system, we created calculated columns for the kinetic energy and the change in momentum in the x and y directions. Using a linear fit from before and after the collision collision, we can see that in the x direction momentum is not evidently conserved.

Below are the additional produced graphs showing the center of mass between the two balls and we can observe that for the y-axis the position remains fairly constant, while we have a negative slope in the x-axis. This depends on the way we set up our axes from the video capture. The velocities in x and y are not consistently constant, but the overall trend is that they remain in the same range. The set of actual calculations are shown on the left side.

Conclusions:
One of the sources of uncertainty that we can see from the kinetic energy graph is that there was energy lost in sound. The glass table wasn’t completely leveled so this could also account for the change in the position slopes after the collision for the first ball since the table was slightly more slanted to the left from the image of the video capture.
We spent a lot of time trying to import the video of our collision and eventually ended up having one of our group members to complete the video capture portion of the experiment at home, the most essential part of the experiment.

By: William Berdugo
Lab Partners: Jessica Moody, Shawn Espinoza, and Sam Meng
Lab Performed on 4/19/19

Purpose:
The purpose of this experiment is to discover a model equation for the theoretical “magnetic potential energy” in a system involving magnets by examining the relationship between force and distance of an object on a track and testing the model to verify if energy is conserved in the system.

Introduction/Theory:
The approach we are to take with determining an equation for magnetic PE is to know that when two magnets will reach an equilibrium point where the magnetic repulsion force between them is equal to the gravitational force component in the x direction of the sliding object (parallel to the air track), which will be the same as F_g,x= mgsin(θ). The power law relationship between the force, calculated by the F_g,x, and distance, between the two magnets, is predicted to be F = Ar^B. After determining the values of the coefficient and the power, we are then able to create our relationship model equation for magnetic potential energy U(r).

Above is the predicted model equation for a non-constant potential energy that relates force (from using Newton’s Laws of motion) and separation (the magnets separation + 2*inner plastic offset in magnets). At r = ∞, we are to assume F = 0, because if the magnets are infinitely apart there’s ideally no magnetic force between them. At the closest separation between the magnets, r, the object’s kinetic energy is momentarily zero while all the energy is stored as magnetic potential energy, just for that moment. Afterwards, the energy is reverted back into kinetic. Ideally, the total energy of the system should remain constant and verify conservation of energy.

Apparatus/Procedure:
The apparatus used in this experiment was a red air track glider with an aluminum reflector on top and a magnet encased in a plastic holder attached to the side placed on an air track. At the end of the track, there is another magnet with a plastic casing that has the same polarity as the magnet on the glider.

For the first part of the experiment, we are to raise the entire air track to a height and determine the angle, θ, the track makes to the horizontal (we used a phone application). Turning on the air will make the glider travel down the air track and will reach the equilibrium point between the magnets. At that point we are to use plastic calipers to measure the distance between the magnets and also use metal calipers to measure the depth of the plastic holders. We are to repeat this procedure four more times at increasing heights and write down both the angle and caliper reading measurements. This gives us the five points to determine our values for the power law relationship.

For the next part of the experiment, instead of tilting the entire air track to an angle, we are to level the track to be as close to 0 degrees to the horizontal (also determined using a phone app) and place a motion detector at the magnet end of the track to measure the forward and backward motion of the glider with the air turned on over a period of time. This portion of the lab gives the necessary data for plotting KE, MPE, and Total Energy as a function of time.

Data:
To calculate force, we use (Mass/1000)*g (9.81 N/kg)*sin(angle) for each trial set. The mass is sum of the glider and the aluminum reflector attached above it. The offset is the inner plastic offset distance within the magnets multiplied by two and converted to mm from cm. The actual distance is the addition of the offset and distance converted to m.

Results/Analysis:
In the graph below is the Force (N) vs. Distance (m) graph, using the data above, that gives the power relationship between the two. Each point corresponds to the the actual distance between the magnets from the air track and glider as the x coordinate and the corresponding force calculation as the y coordinate.

The calculated column for Magnetic PE is equal to A/(B+1) *r (separation)^(B+1) from taking the integral of the derived power law equation from above F(r) = A*r (separation)^B. The calculated column for Kinetic Energy was 0.5*mass*velocity^2 using the mass and speed of the glider.

Conclusions:
The coefficient value of A and the power value B we obtained from the
force and separation graph was 2.850×10^-5 ± 8.664×10^-6 and -2.700 ± 0.08041, respectively. Each angle measurement had an uncertainty of ± 0.1° from using the phone app. The max value of the magnetic PE was a larger value than the minimum of the kinetic energy in the system thus resulting in a total energy plot that is not an ideal straight line.

There are a few sources of error that lead us to the results we got in our last graph. The reason as to why the kinetic energy was so large for the first 2 seconds was due the fact that when we placed the glider on the track and turned on the air, the separation between the magnets was not reasonably close enough. We had to perform the second part of the experiment a few times before getting the graph we obtained, because there was a delay to when we turned on the air to when the glider began to move which may have been a result of the friction between the glider and track. Even leveling the track to be as close to zero degrees was difficult because our phone app never gave a reading that was exact. I don’t think we recorded any other data to verify conservation of energy at each section of the “collision” between the magnets. Despite all that, our total energy for a specific interval of time was in the range from 0.012 J to 0.014 J and, overall, showed conservation of energy in the system.

By: William Berdugo
Lab Partners: Shawn Espinoza and Sam Meng
Lab Performed on 4/17/19

Purpose:
The goal of this experiment is to observe the different types of energy, gravitational potential, elastic potential, and kinetic, in a vertically-oscillating mass-spring system and demonstrate that energy is conserved throughout the process.

Introduction/Theory:
In this lab experiment, we use the concept of conservation of energy KE₀ + GPE₀ + EPE₀ + Work Done is equal to the final energy in the system. Another method that was used was the center of mass of a system to calculate new equations for the three energies.

The gravitational potential energy is the work done to bring an object to a final position starting from a defined zero and where the object will begin and end at rest. To represent a little bit of mass we designate a system with dm to be integrated to have an equation to represent GPE. Starting from dm = dy/L * M, where L is the length from subtracting the height where the mass-spring is at rest from the height of the horizontal rod, L = (h-y_bottom) , we are able to get a new equation for GPE to be Mg/2*(H+ y_bottom). Using the center of mass, we obtain the GPE equation below for this system. Likewise, we use calculus to determine the kinetic energy of the system starting with with dm = dx/L * M where the integrand becomes 1/2 * (dx/L * M) * (x*v_end)^2 from 0 to L to get the equation above for KE. Eventually, the equation becomes the way it is below.

In this scenario, the spring used doesn’t have a uniform shape so it was not ideal, therefore we did not use the general equation for elastic potential energy (1/2 * k*x^2), instead we used the relationship between the force it takes to move the system down and the stretch to get a relationship that resembles F = kx + F₀, where F₀ is the initial force, x is the stretch, and k is the spring constant.

Apparatus/Procedure:
The apparatus involved two rods attached perpendicularly where the larger vertical one is connected to the table and the smaller one is parallel to the floor. The mass-spring (m_hanging + m_spring) was attached to the hooked end of the force sensor attached to the smaller horizontal rod and the motion detector on the floor pointing upwards. The m_hanging used was the mass hanger plus a slotted mass to give an overall value of 0.250 kg.

The first part of the experiment is to determine the elastic potential energy of the system since the spring constant, k, is not a given value. We zeroed the motion detector at the resting place of the system and had the positive direction downwards, so when we slowly pulled on the hanging mass we determined the force of the stretch.

The initial setup and second setup were identical except, in the second part, the force sensor was disconnected and the motion detector was zero at the floor with the positive direction going up.

Data:
The force vs time and stretch vs time graphs to get the spring constant with a force vs stretch relationship. Also, the position and velocity plots vs time to observe their relationship in the oscillating system.

Results/Analysis:
The kinetic, gravitational potential, and elastic potential energies vs position and vs velocity plots. We omitted the first initial data points because they did not yield the same path as the entire system in the oscillating process. Purple is KE, red is GPE, blue is EPE, and the energy sum, E_sum, is green.

Conclusion:
We determined the mass of the spring by weighing the spring on a scale to get value of 0.0938 kg which was used as the m_spring. From the force vs stretch plot we obtained the value of 10.76 N/m for the spring constant, k. The value for F₀ from our data was 0.2654 N. The graphs of each of the energies vs time were very similar to our predictions that we made prior to creating the calculated columns.

From the graphs, the kinetic energy is inversely related to the position and velocity of the system. When the system reaches its mid position the kinetic energy is at its highest while its nearly zero when the velocity is nearly zero at the same position.
When the system is at its lowest position the GPE is at its lowest value and EPE is at its highest value. Inversely is also true with GPE being at the highest value and EPE being at its lowest when the system is at its largest position. All the energies are positive values during the period of the system’s oscillation.

We may have not determined the stretch to be as accurate it was when using the lab equipment for the motion detector is not lying directly on the surface of the floor. Since, the spring was not an ideal shape, the initial values of the force vs stretch graph doesn’t have a linear relationship therefore the spring constant is not as accurate to the system as it should be and the value of k used only depended on the linear fit of the data. The uncertainty in the stretch may have yielded an inaccurate spring constant. There is not a comparison that could be made to determine if our experiment was successful enough in demonstrating the conservation of energy in the system, yet, confined to the bounds of the experiment and our procedure, we obtained data for each of the energies that show conservation was done.

By: William Berdugo
Lab Partners: Kyle Perlin and Julio Rivera
Lab Performed on 4/3/19

Purpose:
The goal of this experiment is to create a relationship between the increasing angle a mass revolves around an apparatus spinning in a circular motion and the increasing angular speed at which the mass travels at different voltages.

Introduction/Theory:
The traveling motion of objects prior have dealt in solely translational motion either in one or two dimension. Another factor that comes into observing the motion of objects is when an object is in a circular, rotational motion. Angular velocity, represented by the lower case omega, ⍵, is equal to the amount of rotation/time (typically in radians per seconds). The object’s velocity (length of the circular path/time for one rotation) is equal to R⍵. When an object travels in a circular motion, a circle of some form, there is a centripetal force that accelerates towards the center. The acceleration of an object in uniform circular motion is the change in velocity over the change in time, however the instantaneous centripetal acceleration of an object is ⍺ = 𝑣²/R = R²⍵²/R = R⍵² where R is the distance to the center of the rotating disk or wheel (a circular rotating body).

When there is a free body diagram, as the one represented below, where a hanging mass is at a certain length away from the point it’s attached to the end of an extended horizontal radius, using the sum of the forces in the x and y directions allows for a relationship between the mass’s ⍵ and the θ.

Apparatus/Procedure:
The apparatus involves a surveying tripod setup where a one meter ruler is clamped onto a vertical rod and a hanging mass, a rubber stopper, is attached to the end of a string with the other end on the end of the one meter ruler. The vertical rod is directly connected to an electric motor that requires a certain voltage from a power source to provide the ruler an angular speed. Increasing the voltage, increases the angle the string makes to the vertical, and, ultimately, increases the angular speed. When this happens a piece of paper is taped onto a rod which is perpendicularly clamped onto a standing ring clamp from the floor and raised to a certain height when the stopper merely grazes the paper. The measurements taken are from the height where the stopper hits the paper and the time of ten revolutions for a set of five, incrementally increasing voltages causing an increase in angular speed.

All these measurements are necessary in approaching the creation of a model that relates angular speed, ⍵, and the angle, θ. This equation will result in angular speeds which will be labelled as the theoretical results. The measurements of its revolutions
will also determine the angular speed which will be described as the experimental results.

Data:
The calculations for the angles based from the height the paper was raised to just be grazed by the stopper is from the equation θ = cos^-1((H-h)/L). The angles were converted into radians from degrees.

Results/Analysis:
Below is the table comparing the two ⍵’s one from using the model as a function of θ, R, and L and the other one from taking the period of one revolution divided from 2*π.

Below is a plot where ⍵ from the theoretical is plotted against ⍵ from the experimental.
The slope of the plot is 1.049 with a correlation of 0.9915.

Conclusion:
Percent difference from theoretical and experimental results of the angular speed, ⍵, for the first set is 19.31%, for the second set is 1.87%, for the third set is 2.55%, for the fourth set is 2.87%, and for the fifth set is 4.23%. The theoretical results were regarded as the accepted values. The correlation from plot’s slope is 0.0085 from an exact 1 to 1 relationship.

There are many factors that cause this overall 4.9% of the slope (the ideal goal was for it to be 1.000) from the five theory and experimental points that were used. The professor handled the measuring of these h’s in both placing the paper and measuring the height with a meter stick. The first set of data when the height, h, was measured to be 19 cm. The first height proved to be much less than what was taken experimentally from the ten revolutions thus causing such a large percent difference that was not seen in the other runs. The second time the professor measured the height provided the most accurate results between the theoretical and experimental values. Other students in the classroom provided the heights h, from the third set and onwards, those measurements were most likely slightly less than the actual height at which stopper grazed the paper.

The one meter ruler that was perpendicularly clamped to the vertical rotating rod appeared to be slightly raised, perhaps if it was exactly horizontal (parallel to the floor) much larger voltage would have been needed to provide a more measurable angle, but would be too large that the grazing of the paper process would not provide the height, h. This would also affect the uncertainty in the exact measurements for R (80 cm) and the length of the string, L (188 cm). All measurements were in centimeters so the g used in the calculations were 980 cm/s².

The model equation was a good test for the majority of the runs, however basing this theory on such uncertain measurements as R, L, and θ which is determined by h, would only provide angular speeds that pertain to this certain apparatus.

By: William Berdugo
Lab Partners: Kyle Perlin and Julio Rivera
Lab Performed from 3/29/19 – 4/3/19

Purpose:
The purpose of the experiment is to derive a power law model for the relationship between the terminal velocity of a falling mass and the force of air resistance it experiences. To test this model and assess the how well it works we are to determine the terminal velocity from video capture and compare it to the experimental data from our model.

Introduction/Theory:
Building upon the Newton’s Laws of Motion, an object traveling through air experiences a force of air drag that opposes the direction of gravity. The main idea is that the force of air resistance directly depends on the velocity of an object moving through the air. The proportionality of this relationship is when the speed is larger, the drag force of air also becomes larger. The model equation we derived was F_{air resistance} = k*v^n where k is the share factor of air resistance and velocity and n is a value velocity is raised to.

Along with the relationship between the air resistance and speed, as an object travels down in air the net force acting on it lowers, the acceleration reaches zero, and the incremental increases of velocity (Δv) lowers since at a certain point the object has reached terminal speed.

The slope of the position versus time graph gives the terminal speed for each mass falling. At that terminal speed, the force of air resistance is equal to the weight of the mass.

Apparatus/Procedure:
The procedure for the first section of the experiment is to set up a ring stand with a “selfie stick” holding a phone in place to record the fall of coffee filters starting with one filter and increasing the mass each time by another filter until six are dropped. We were to then import the videos into the computer and capture the position at every frame of time of choosing (one that results in the most linear portion with necessary points). Basing off the power law model equation we created a F_air vs. terminal velocity graph to derive the k coefficient (the slope) and the n power (some fractional power of velocity).

The second part involved creating spreadsheets using the k and n values we obtained based on our mathematical model we developed and calculating the terminal velocities for each drop by manually inputting the values of a time interval (one that results in the most accurate model), the mass (kg) of coffee filters of each drop, and the value of gravity (m/s²). The goal was to find a Δt that was small enough to visualize a change in velocity that isn’t too large for each interval and to have the net force to essentially be constant for that interval since velocity, acceleration, and F_net change instantaneously.

Data:
Below are the position (m) versus time (s) plots taking each highlighted section and examining the slope which gave us the terminal velocity from the video capture of each of the filters.

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Results/Analysis:
Below is the power fit line of the F_air vs. terminal velocity graph we used to determine the k and n values and their uncertainties. The F_air is calculated by the mass of each filter set multiplied by the force of gravity since at each terminal velocity F_air =mg.
The values for five filters were neglected since this terminal velocity is less than the previous one resulting in inaccurate data.

Every spreadsheet of each mass of the filters using the k and n values from the F_air vs. terminal velocity graph at time intervals of 0.01 seconds. The highlighted row is the target value for terminal velocity that we used to compare to our video capture results.

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Conclusion:
From the slopes of our position vs time graphs for each filter set dropped we captured terminal velocity values of 1.630 m/s for one filter dropped, 2.253 m/s for two filters dropped, 2.884 m/s for three filters dropped, 3.079 m/s for four filters dropped, 3.052 m/s for five filters dropped (this value was considered to be invalid due to it being less than the terminal speed for four filters dropped), and 3.483 m/s for six filters dropped.
We derived the k value to equal 0.001639 ± 0.0006963 and the n value to equal 2.738
± 0.3655 using all the terminal speed and calculated force of air resistance values except for when five filters were dropped. Inputting these k and n values into the spreadsheets to experimentally find the terminal velocity from every value of mass dropped we got terminal velocity values of 1.842 m/s for one filter dropped, 2.372 m/s for two filters dropped, 2.750 m/s for three filters dropped, 3.055 m/s for four filters dropped, 3.313 m/s for five filters dropped, and 3.540 m/s for six filters dropped. The percent error between the the accepted values (the terminal velocity from our video capture) and the experimental values (terminal velocity from our spreadsheets) were as follows: 13.00% for one filter, 5.28% for two filters, -4.65% for three filters, -0.78% for four filters, 8.55% for five filters, and 1.64% for six filters. The most accurate our model was compared to the terminal velocity from our video analysis was for four filters where the model was just slightly below. For most of the runs, the video capture results were less than the ones calculated in the spreadsheets.

It was difficult evaluating at what point in each frame of reference a set of filters dropped because it was not a linear downwards path for each run. Our video analysis did not always evaluate the points from the section that was supposed to be evaluated from, the marking on the black sheet that was suspended indicating 1.69 m as seen in the first photo. We mistakenly only had the highlighted time interval of 1.05 seconds for every case in the spreadsheet for each run, where in fact the terminal velocity wasn’t at that precise interval every time. The professor may have dropped the wrong set of coffee filters during a run, which was evident when we found the terminal velocity of five filters being dropped to be less than when four were dropped. Not every filter was able to stick together enough which caused the mass at each run to differ from our calculated and assumed measurements. The second part of the experiment was dependent on the values of k and n determined from the video analysis portion which was influenced by a multitude of uncertainty sources to begin with, thus resulting in large differing terminal values. Our model may have been necessary enough to determine the terminal velocities, however, re-doing the video capture and analysis section to obtain different values for k and n may be the most integral part of making a successful relationship model.

By: William Berdugo
Lab Partners: Kyle Perlin and Julio Rivera
Lab Performed from 3/18/19 – 3/20/19

Purpose:
The goal of this lab is to successfully model frictional forces between their relationship with normal forces through a series of multiple experiments that involve a wooden block sliding on a flat surface.

Introduction/Theory:
In Newton’s Laws of Motion, we know there to be a force opposing the motion of an object when that object is on a certain surface, this force is called a friction force. Like the normal force, it is caused by direct contact between surfaces. However, while the normal force is always perpendicular to the surface, the frictional force is always parallel to the surface. 

If friction is present, a certain minimum horizontal force is required to move an object on a surface or another object. If a horizontal force less than this minimum force is applied to the main object, a force must act to counter the applied force and keep the object at rest. This force is called the static frictional force. The maximum static frictional force is expressed as:
f_{static,maximum =} 𝜇_S*N
In general, static friction is expressed as:
f_{static ≤} 𝜇_S*N
Once a force is applied to an object that exceeds f_{static,maximum} , the object begins to move, and static frictional forces no longer apply. The moving object experiences a frictional force of a different nature, called the kinetic frictional force. Therefore:
f_{kinetic =} 𝜇_K*N

Apparatus/Procedure:
The main tools needed to set up each of the five different experiments are the wooden block attached to a string with a linoleum side that we will continue to observe experiencing motion and the flat surface that will either be placed horizontally or at an angle.
The first setup is placing the surface on the table and attaching the block to a pulley at the end of the surface with a mass hanger on the other end of the string. Here we continued to add more mass on the wooden block while also adding more mass to end of the string stopping only when the block begins to slid. This involved static friction.
Part 2 just requires the block with the string and a force sensor placed at the end of the surface. Using the same four masses from Part 1 that was on the block, we were to pull of the block and collect the data from the force sensor. This involved kinetic friction.
In Part 3, the flat surface is to be raised to an angle at which the block with the string is about to start sliding. We used a phone application that showed the angle of elevation to the horizontal and clamped the phone to the surface. This involved static friction.
The last two parts involve kinetic friction. Part 4 continued the same set up as the previous section as it still required to raise the surface, but it needed to be supported and remain in place, so we used a rod. This time a motion detector was attached at the top with the block sliding away from it. We placed a sticky note on the block for detecting its position.

Finally, in Part 5, we use the pulley system again with the mass hanger at the end of the string connected to the block, but, this time, the motion sensor was placed at the opposite end of the surface.

Data:
The measurements of the wooden block and the corresponding added slotted masses to the mass hanger necessary for Part 1.

The graphs below are the measurements of the forces according to the different masses from Part 1 taken by using the mean. The data used for Part 2 is in the table after the graphs.

Part 3 data with the measurement of the angle at which the block begins to slip.

For Part 4, below is the data for the acceleration of the block sliding down a steeper incline from part 3.

Also there’s the velocity vs time graph which was used to determine the acceleration in this section.

Part 5’s values for the mass of the block, the hanging mass, and also the kinetic coefficient from part 4 and their uncertainties that are used to determine the acceleration. Also, the velocity and acceleration graphs are shown below with the actual result of acceleration by using the mean of the slope, being 2.268 m/s².

Results/Analysis:
The graph for coefficient of static friction between the felt and the track (wooden block and surface) with the increments of the mass on block in the x axis and the corresponding added masses in the y axis. The slope of the line is 0.532.

The graph for coefficient of kinetic friction between the felt and the track (wooden block and surface) with the increments of the mass on block in the x axis and the corresponding mean of forces in the y axis. The slope of the line is 0.356.

Conclusion:
The coefficient of static friction for Part 1 was 0.532 and the coefficient of kinetic friction for Part 2 was 0.356. The static friction from a sloped surface (Part 3) involved the angle at which the block begins to slip and the coefficient of static friction that was calculated was 0.404 ± 0.002. The kinetic friction from sliding a block down an incline (Part 4) involved a new angle and the measured acceleration from the motion detector. The coefficient of kinetic friction in this part was 0.525 ± 0.003. Part 5, “Predicting the Acceleration of a Two-Mass System” involved the previous calculated 𝜇_K, the mass of the block, and a hanging mass. The acceleration was 2.533 ± 0.272 m/s² compared to the measured acceleration of 2.268 m/s². The percent error in this part was +11.7%.

There were many sources of error that impacted the results from each experiment. I used MATLAB to graph the data from the first two parts to find an averaged value of the coefficients of static and kinetic friction and used a quadratic fit to create an equation for data 2 since the mean of forces didn’t produce a set of data where the slope was easy to determine. The acceleration that was used in Part 4 was taken from taking the points from the velocity graph instead of creating an acceleration graph and using that mean. Even though the app that was used to measure the angles of the incline, the uncertainty of the measurement could’ve been more than was used. A green ring would form indicating that the measurement was accurate, but the degree of accuracy is unknown. The block was placed at different areas on the surface in each experiment and may have not been wiped down with enough oil each time the block slid.