15-April-2015: Magnetic Potential Energy

The purpose of this lab was to find an unknown potential energy function, and to determine whether it obeyed the Law of Conservation of Energy.

Part one- Finding a Magnetic Potential Energy Function:
The first part of this experiment was to determine the potential energy function between two magnets. Since this was unknown, we had to derive it using the given formula F= -du/dr. To do this we set up our experiment as such (see Fig. 1):

• We placed an air track (attached with a magnet) on our lab bench with an air track glider (attached with a different magnet) on top of it.
• Connected to the air track was a vacuum which would slightly push the air track glider off of the air track, making it relatively friction-less.
• We had books on the side that would allow us to put the air track at an incline.

Fig. 1

Fig. 2
Using this free-body diagram, we could conclude that 
F=mgsinθ.

Since we had to find a function of force relative to position, we had to put the air track at an angle to get the glider moving. If we created a free-body diagram of the system, we would find that raising the air track, (increasing it's angle θ with the horizontal) would continuously change that magnitude of this force, increasing it each time (see Fig. 2). This meant that we would have to measure multiple angles of inclination, measure the distance between the two magnets at each incline, and calculate the force exerted at each distance. If we graphed the force vs distance in LoggerPro, we would have our force as a function of distance which would then allow us to derive an expression for the magnetic potential energy.

To collect data, we had to increase θ a number of times while recording the distance between the two magnets. To do this, we placed books at the end of the air track, causing it to incline. Using our phones, we measured the angle at which it was inclining. Once the magnet attached to the glider was at its maximum distance from the magnet attached to the air track, we measured how far apart they were with a ruler. We did this five times, which was sufficient enough to use in LoggerPro. (see Fig. 3 and Fig. 4)

Fig. 4
We had LoggerPro calculate the force.

Fig. 3
Using a ruler, we measured the
distance between the two magnets at each incline.

Once we had our data table set, we plotted force vs distance, which we curve fit and found an equation of  F = 0.0002036r^-1.871 (see Fig. 5). Once we had this function, we plugged it into F = -du/dr and derived it to find our magnetic potential energy function of U(r)= 0.0002338r^-0.871 (see Fig. 6).

Fig. 5
Using this graph, we were able to find a the magnetic force as a function of distance between the two magnets.

Fig. 7
Since we knew that F=-du/dr, we were able to find our function for U using integration.

Part 2- Testing Our Magnetic Potential Energy Function:

Once we had our magnetic potential energy function, we had to test if it actually worked. To do this, we would set up the air track horizontally with the glider on top as before. With an aluminum reflector attached to the glider, we set up a motion sensor that would measure position and velocity of the glider. If the glider slid across the air track at a constant speed, LoggerPro would be able to calculate our kinetic and magnetic potential energy. Since energy was to be conserved, the sum of kinetic energy and magnetic potential energy throughout the sliding of the glider should remain constant. That is, the total energy should remain constant. This would verify that our magnetic potential energy function was good. 

The setup is relatively simple (see Fig. 7):

• Place the air track vertical on the lab bench, again with the air glider on top.
• Place the motion sensor facing the glider at the end of the air track which had the magnet.
• Slightly push the glider across the friction-less surface and allow it to reach the other magnet and repel back.
• Using the data obtained from the motion sensor, calculate KE, U(r), and total energy.

Fig. 7

Once we had our setup complete, it was time to begin recording data. When putting our functions of kinetic and magnetic potential energy into LoggerPro, we had to find a definition for "r" in U(r). Since "r" was supposed to be the distance between the two magnets, and the motion sensor could only read the distance between it and the aluminum reflector, we had to come up with an equation that describe the distance between the two magnets. Placing the glider at some random distance from the motion sensor, we measured the distance from the motion sensor to the aluminum plate and the distance from the two magnets. We found the difference between these two distances and found that r="position"-0.3045 (see Fig. 8).

Fig. 8
Since the motion sensor could not calculate the separation between the two magnets directly, we had to find
an expression that could describe it.

When recording the data, we created a column that would calculate KE using 1/2m"velocity"^2, a column that would calculate the separation distance between the magnets (r) using "position"-0.3045, a column that would calculate U using 0.0002338r^-0.871, and a column that would calculate total energy by adding KE to U.
Once we had our columns ready, we gave the glider a slight push and began collecting data. We found that total energy was relatively horizontal toward the beginning and end of the run. However, once the magnet stopped the glider and pushed it in the opposite direction, there was a spike in the total energy. This meant that there was some uncertainty in our lab (see Fig. 9 and Fig. 10).

Fig. 9
The total energy rises as the velocity turns from negative to positive (the turn around point).

Fig. 10
Our energy graphs.

There were a few sources of uncertainty in this lab:

• In our function of force, we had uncertainty in both our measurement for theta (phone) and distance (ruler)
• Since the force function had uncertainty, the magnetic potential energy function U(r) had some uncertainty. 
• Also, the value of "r" had some uncertainty since we measured it with a ruler.
• Most importantly, however, we assumed that the air track was friction-less. Since energy was not completely conserved, we can also conclude that the air track did indeed have some friction acting on it. 

Conclusion:

Considering the imperfection of our environment, I would most certainly consider this lab a success. We were able to find a relatively decent function for magnetic potential energy, and although it didn't seem as if energy was conserved, this in fact makes sense. Considering that there must have been some friction acting on the system, it makes sense that the total energy spiked up when the glider was in its turn around period. The upward slope the total energy graph makes as it reaches this point suggests the presence of possible a frictional force acting on the glider. 

13-March-2015: Conservation of Energy: Mass-Spring System

The purpose of this lab was to see if Energy was conserved in a vertically-oscillating mass-spring system.

Setup:

Fig. 1

In order to measure the total energy in this spring mass system, we needed to measure the system's potential energy, spring potential energy, and the kinetic energy. To set up the system, we needed: a spring, a mass, a ring stand, clamps, a laptop, and our LoggerPro equipment. We attached the ring stand to our lab bench using the clamps. Once attached, we hung the spring via a bar connecting to the ring stand. We used a motion sensor and LoggerPro in order to find the velocity and position of the oscillating mass and spring (see Fig. 1).

Finding Spring Constant "k":
We needed to calculate the spring constant "k" since LoggerPro could not measure it for us. To do this, we planned to measure the "stretch" of the spring as we hung different masses from it. Since we knew that the spring constant was a ratio between force and length (N/m), if we graphed enough points of stretch and weight, we could use the slope of this graph to find the spring constant "k".
First, we began by measuring the length of the relaxed spring. Once we had this, we hung a 100g mass to the end of the spring and measured the new length of the spring, We subtracted the length of the relaxed spring to this in order to find the displacement of the spring. We repeated this twice, adding 100g each time. Once we had this, we created a data table in LoggerPro that found us the force in newtons of the hanging masses. We then graphed force vs displacement and found our slope to be 11.01. This meant that the spring constant "k" of our spring was 11.01 N/m. ( see Fig. 2-1 & 2-2)

Fig. 2-2
The slope of the graph was the value of our spring constant.

Fig. 2-1
Using a ruler, we measured the new length of
the spring every time we added more mass
to it. 

How to Express Kinetic, Gravitational Potential, and Spring Potential Energy:
Because the spring had mass, we needed to account for its kinetic and gravitational potential energy. The total energy of the system would consist of the kinetic energy of the spring and mass, the gravitational energy of the spring and mass, and the elastic potential energy of the spring. To find the kinetic energy of the spring itself, we needed to consider it as a large amount of littler pieces of mass(dm), each piece having its own velocity. Each piece of the spring had its own kinetic energy. since we had a ratio of the mass and length of the spring, we could assume that dm/dy, where dy was a little length of the spring, was equal to M/L (M being the mass of the spring and L being it's relaxed length.). We then integrated to find our spring's kinetic energy (see Fig. 3 for calculations).

Fig. 3
We found our final equation for the kinetic energy of our spring to be KE=1/2(M/3)v^2

In order to find the gravitational potential energy of the spring, we again needed to find a representation for dm (small piece of the spring.) Since we knew that the length of the spring would vary, we needed to come up with a function to represent it. We used "h" to be the length from the top of the spring to the motion sensor, "y(0)" to be the length from the end of the spring to the motion sensor. Thus, the length of the spring, L, would be "h"-"y(0)".  We integrated dm once we found it to be M/(h-y(0))*dy (see Fig. 4 for calculations).

Fig. 4
We found our final equation for the gravitational potential energy of our spring to be GPE=(M/2)gy

Once we had these two expression, we were able to find an overall expression for the total kinetic energy of our system. The total kinetic energy was the sum of the spring's and masses kinetic energy. The total gravitational potential energy was the sum of the spring's and masses gravitational energy. The total energy was the sum of all of these along with the spring potential energy (see Fig. 5).

Fig. 5

The Experiment:

Once we had all of our expressions, it was time to actually do the lab. We inputted our expression for total KE, total GPE, and EPE into LoggerPro. We used "position" as our y-variable for GPE, "velocity" as our v-variable for KE, and (0.887-"position") as our y-variable for EPE. (this was because we needed to square the spring's stretch, which was the difference in initial relaxed height and "position") Once we had all of this, we began to collect data as we set the spring into an oscillating motion. All of the graph took a sinusoidal type of shape. This was because at different points of oscillation, each energy was become larger and smaller. At high points, GPE was large, and at low points it was small. The velocity continuously increased and decreased, changing the KE. And the stretch in the spring continuously changed, changing the EPE. If energy was to be conserved, then the total energy of this experiment should have remained constant. To test this, we created a new total energy column, which was the sum of the KE, GPE, and EPE. What we found was that our total energy was relatively consistent. Ranging from 1.82 Joules to 1.88 Joules justified that energy was conserved throughout the oscillation. (see Fig. 6)

Fig. 6
The total energy graph remained relatively consistent for both a function of time and position.

 Conclusion:

Considering that our environment for the lab was imperfect, we gained a very pleasing result to our test of conservation of energy. The were a few possible sources of error within our experiment. For one, our measurement for our spring constant had some uncertainty due to our measurements using a ruler. This in turn affected our function of spring potential energy. Along with this, when we set the spring into an oscillating motion we had multiple trials to try to get it moving strictly vertically. Considering that the spring could sway in the horizontal direction, our values for kinetic and gravitational potential energy could have been affected. However, considering all of our uncertainty, we were able to conclude that energy was conserved throughout this motion.

01-April-2015: Centripetal Force with a Motor

The purpose of this lab was to come up with a relationship between the angle at which the mass is rotating and the angular speed.

Setup and Demonstration:
For this lab, there was only one setup that the entire class was to observe. The setup consisted of a long vertical rod. Attached to this vertical rod was a horizontal rod. At the end of this horizontal rod was a string to which a mass was connected. The horizontal rod was driven by a motor which set it into constant circular motion. The magnitude of this velocity was controlled by this motor. The entire apparatus can be seen in Fig. 1.

Fig. 1

We measured the the height of the vertical rod to be 2.00 meters, the radius at which the horizontal rod to be 0.97 meters, and the length of the string at which it rotated from to be 1.65 meters. We considered the angle the string made with the vertical to be θ, and the combined radius at which the mass rotated to be (0.97+1.65sinθ) meters. Since we could make a right triangle with the string, we could express a segment of the radius to be a trigonometric function of θ (Fig. 2).

Fig. 2
We needed to model our apparatus in order to find our expression for omega. 

Fig. 3-2
We used a FBD to find our expression for
angular speed.

Once we had a general idea of how the system would work, it was time to put some math to our ideas. We used Newton's Second Law to find an expression for the angular speed, omega. This omega would depend on the angle θ, created by the string and vertical. To find θ, we broke down the diagram into a right triangle from which we could solve for θ. (Fig. 3-1 and Fig. 3-2). Once we had this, it was time to begin recording data.

Fig. 3-1
Our expression for omega needed a way for theta to be measured.

Collecting Data:
The only thing we needed to (and could) measure from this experiment was the height at which the mass was rotating and the period at which it rotated. With the height, we would be able to find θ, which would in turn allow us to find the angular speed omega. To do this, we set the object into circular motion using our setup. Once it had reached some consistent circular motion, we placed an adjustable rod near the edge at which the object was rotating. attached to this rod was an extended piece of note card paper. As the object rotated, we slowly began to raise the note card. Once the note card hit the rotating mass, all we then had to do was measure the height of the note card relative to the ground. This height was the height at which the mass was rotating (Fig. 4). The period we mesured would allow us to determine whether our model for omega using Newton's Second Law was sufficient or not.

Fig. 4
As the rod spun, we adjusted the height of the note card to find the height of the mass. 

We observed the mass rotate for six separate trials, recording the height and period of the rotating mass for each trial. Our data table can be seen in Fig. 5. 

Fig. 5

Analyzing Data and Verifying Our Expression:

Once we had all of our data, it was time to begin using it to test the relationship we found between the angle at which the mass rotated and angular speed. As a reminder, we found the angular speed to equal the square root of (gtanθ)/(0.97+1.65sinθ). We plugged this formula into logger pro to produce our experimental value for omega. Along with this, we plugged our periods for each trial into LoggerPro to produce our "real" value for omega. We used the formula omega=2π/"period". (see Fig. 6)

Fig. 6
We used LoggerPro to calculate our experimental values for omega.

Once we had produced our two different angular speeds, we graphed them versus each other. The slope produced was 0.9947, which indicated that our model for angular speed was good. (see Fig. 7)

Fig. 7
Since the slope was practically 1, our values for omega were extremely close and similar, indicating that we
had found a good expression for omega for our apparatus.

There are two main areas of uncertainty that yielded us a slope that was not exactly one. First, our measurements of height of the apparatus, length of the string, etc. Using rulers, we could only be so close to the actual value of the dimensions of our apparatus. Also, our measured period was dependent on our reaction time for pressing our stopwatches. Since we did not know the exact spot where a rotation ended, we had to estimate. Overall, however, even with our uncertainty, we were able to yield pleasing results.

25-March-2015: Centripetal Acceleration vs. Angular Frequency

The purpose of this demonstration was to determine the relationship between centripetal acceleration and angular speed.

The Idea Behind The Experiment:
Given, we already have a relationship for centripetal acceleration and angular speed. a=rω^2, where a is centripetal acceleration, r is the radius at which the object is rotating, and omega is the objects angular speed. Given this relationship, our goal now was to prove it.

Materials and Setup:
For this demonstration, we used the following materials:

• A heavy rotating disk
• An accelerometer
• A photogate
• A motorized wheel
• LoggerPro

To setup the demonstration, the heavy rotating disk was placed flat onto a lab table. There, the accelerometer was secured near the edge of the heavy rotating disk, with its axis pointing towards the center of the disk. To secure it, a vast amount of tape was used. Once secured, we placed a photogate which was attached to a ring stand to the side of the heavy rotating disk. From here, a long piece of tape was put at the edge of the accelerometer. This piece of tape hung off the edge of the heavy rotating disk and would be used to activated the photogate. After this, we placed the motorized wheel parallel and to the side of the heavy rotating disk. This would ensure the the heavy rotating disk was spinning at a constant speed. Finally all of our LoggerPro equipment was attached to the accelerometer and photogate. For our final set up, see Fig. 1-1 and Fig. 1-2.

Fig. 1-1
Since we didn't have enough equipment for everyone to use, the entire class just witnessed one single demonstration.

Fig. 1-2
The hanging piece of tape would alert the photogate every time a rotation had occurred. 

Collecting Data:

Once we had our lab set up and our accelerometer calibrated, we proceeded to collect our data. Initially, we started the motorized wheel at some low voltage which gave us a low constant velocity. We used LoggerPro to measure the centripetal acceleration. However, since this was not a perfect environment, we got data that seemed a bit "noisy." In our acceleration data chart, we used the mean of the accelerations to find our value for centripetal acceleration (See Fig. 2).

Fig. 2
Since this was not a perfect environment, we had to choose the average value of our
recorded accelerations. 

Now that we had centripetal acceleration, we needed to find our value for angular speed. Since we did not have a device that could directly measure angular speed, finding our value for omega would be a little bit more tricky. Since we had the photogate, we were able to find the exact times in which the heavy disk rotated one time. We had a starting point of time which we considered zero rotations. We then allowed the disk to rotate ten times, giving us an ending time after ten rotations. When we took the difference of these two times, we had the exact time it took for the disk to rotate ten times. To find its period, we simply divided the time it took for ten rotations by ten. Once we had the period, we knew that angular speed was simply 2π/T, where T was the objects period. Thus we were able to find the angular speed for this trial. (see Fig. 3)

Fig. 3 
Although we were able to directly measure angular speed, we
were able to derive it with the period of the rotating object.

We repeated this process five times, each time using a different voltage to drive the mechanical wheel, which in turn gave us different accelerations and velocities. With all our data, it was now time to plot our centripetal a vs. ω^2 graph.

Analyzing Our Data:

Once we had all of our data (see Fig. 4),

Fig. 4

we opened a new LoggerPro file which we would use to graph our a vs. ω^2. Our graph appeared to be linear. Thus, we added a linear fit which would give us the slope of the graph. If our given model of  a=rω^2 was correct, we should see a slope that is similar to the radius at which we were spinning the accelerometer at. The actual radius we physically measured was 0.139 meters. The value our graph gave us was 0.1386 meters, which is extremely close to our actual value. (see Fig. 5)

Fig. 5

Possibility of Error and Uncertainty:

We calculated our percent error from the actual value of our radius to be 0.288%. This indicated that the relationship we had between centripetal acceleration and angular speed was correct. Due to the experiment being relatively straight forward, there were only a few areas where we may have hit some uncertainty. For example, when we measured the length from the center of the heavy rotating disk to the accelerometer, we had to essential guess from which point of the accelerometer the device was reading. So in a sense, we had to guess the actual value of our radius. Another possible source of uncertainty may have came from the mechanical wheel. We put all our faith into the notion that the wheel would keep the heavy disk rotating at a constant speed. If there was some flaw in the wheel, it would have affected our results. Another source of uncertainty may have also came from our mean value for acceleration. However, in all the experiment was conducted effectively and yielded us the relationship we were initially looking for. 

23-March-2015: Trajectories Lab

The purpose of this lab was to use our understanding of projectile motion to predict  the impact point of a ball on an inclined board.

Materials:

• Aluminum "v-channel"
• Small Steel Ball
• Wooden Board
• Ring Stand
• Clamp
• Carbon Paper
• Tape

Setup:

To set up the experiment, we had to make an apparatus that would allow the steel ball to roll off our table with some initial velocity. To do this, we used and connected to aluminum "v-channels." One of the v-channels would be horizontal and parallel with the surface of the lab bench, while the other would be inclined and connected roughly on the halfway point of the horizontal "v-channel." To get the incline, we secured the second "v-channel" to the ring stand using a small metal pole and tape. The edge of the horizontal "v-channel" would be in line with the edge of the lab bench. The final setup can be seen in Fig. 1-1 and Fig. 1-2.

Fig. 1-1
We ensured the velocity of the ball
would be the same by marking the
point at which we would release
the ball.

Fig. 1-2
The ring stand was used to support the inclined "v-channel."

Making Our Prediction:

Fig. 2
We had to drop the ball several times
to get a general idea of where it
would land. 

To make a semi-decent prediction of where the ball would land on an inclined board, we first needed to determine the initial velocity of the ball is it fell off the "v-channel." To do this, we would need to measure how far the ball would land horizontally from the edge of the "v-channel." To do this, we first let the ball go from a marked position on the inclined "v-channel," and allowed it to fall of the edge of the lab bench. Once we had a rough idea of where the ball would land, we placed a piece of carbon paper onto the floor in that general area (see Fig. 2). The carbon paper would have a mark indicating how far the ball fell. We let the ball fall several times. We then measured the distance from the parked point to the edge of the table. This was our horizontal ("x") displacement. (The ball landed in a certain range of locations, we used the average distance for our calculations). We then measured the height of the fall, which was the vertical distance from the table to the floor. At this point, we used kinematics to find the initial velocity of the ball. (see Fig. 3-1 and Fig. 3-2). The initial velocity of the ball was 1.079 m/s in the horizontal direction. 

Fig. 3-1
Data Table

Fig. 3-2
We could find time be using kinematic
equations relative to the y-axis. We
could then plug-in time into our
x equation to find the initial
velocity.

Once we had our initial velocity, it was time to make our prediction. We had to imagine that there was now an inclined board that connected to the edge of the "v-channel." This board would be at some angle "α". The distance at which the ball would hit the board at would be considered "d". Our goal now was to derive an expression that would allow us to calculate "d", given initial velocity and the angle "α". Once we had this expression, we measured our last unknown "α", which was 49.1 degrees. Our derivation and prediction can be seen in Fig. 4. Our prediction was that the ball would land 0.4190 m down the inclined board. 

Fig. 4
We used our kinematic equations to derive an expression for our predicted distance "d".

We finally set up the board and ran the actual experiment. The set up of the inclined board can be  seen in Fig. 5. Again, we used carbon paper to find a fairly accurate spot at which the ball landed on the board. After letting the ball fall several times, we found that on average, the ball landed 0.4285 m down  the ramp. We found our calculated value to be -2.22% off from the actual value. This indicated that we had made a good prediction.

Fig. 5
We finally set up our inclined board and once again placed carbon paper in the general area the ball was landing.

Fig. 6 
Once we had our value, we had to
calculate the combined uncertainty in
both our calculated value of "d", and
our actual measured value of "d".

Our uncertainty:

We used partial derivation to find the uncertainty in our predicted value of "d". (see Fig. 6). Our uncertainty was roughly 0.88 cm. Thus our final predicted value of "d" was 41.90 cm +/- 0.88 cm. The "actual" value of "d" was measured to be 42.85 cm. At first glance, it may seem as if we are not within that range, even with our certainty which yields us a maximum value of 42.78 cm. However, we must consider that our "actual" value of "d" had a range that left it with an uncertainty of 0.45 cm. Thus, our data may be considered acceptable, so long as we show our possible error.