22-April-2015: Collisions in Two Dimensions

The purpose of this lab was to determine whether not momentum and energy are conserved in a two-dimensional collision.

Set Up:

Intro:

To perform this lab, we needed two create an environment in which we could create, observe, and analyze a two dimensional collision. The collision would occur between two marble in two separate trials. The first would consist of two marbles with similar masses and the second would consist of two marble with different masses. To create a reasonable surface to conduct the lab, we used a leveled glass table. The idea was that we would leave one marble stationary at the center of the leveled glass table and another marble would be aimed directly at the stationary marble. Once ready to begin collecting data, we would flick/push our marble towards the stationary marble. Ideally, the balls would deflect off each other at some angle from one another. Theoretically, we should be able to conclude whether not momentum in the x- and y-directions was conserved and whether or not energy was conserved.

Actual Apparatus:

We setup the experiment just as describe above, with a leveled glass table and two marbles of similar mass and two of different. To observe and collect data from the collision, we had a camera setup directly above the leveled glass table. This camera would be hooked up to LoggerPro and would allow us to split up the velocity of the marbles into x- and y-components. This would in turn allow us to find the momentum in the x- and y-directions (Fig. 1)

Fig. 1
With one marble stationary, we would flick another marble at it and record the collision with our LoggerPro camera.

Part 1- Two Marbles of Similar Mass:

Fig. 2
When setting our origin, we tried to line up at least one axis with the path
of the flicked marble. This would leave us with only one value of velocity to
calculate in our initial stages before the collision. 

To begin the lab, we first  used the apparatus with two marbles of similar masses. The first marble, which we will call marble one, was measured to be 0.021 kg. The second marble, which we called marble 2, was measured to be 0.019 kg. We could conclude that these two marbles had relatively similar masses. As stated in the intro, we needed to flick one marble at the other while recording the collision. Once we were ready, we began collecting data and performed the collision. We analyzed the video capture in LoggerPro (see Fig. 2).

Fig. 3
This position graph shows us the position of the two marbles
 in both the x- and y-directions.

To analyze the video, we needed to set a scale and origin for the collision. Our scale was placed in the top right corner of the glass table, where we measured a small piece of it to be roughly 22.3 cm. We placed the origin at the stationary marble and attempted to line up the y-vertical axis to the path the flicked marble took. We then placed frame by frame dots at the positions of the marbles before and after they collided. We flicked marble one at marble two.

Once we analyzed the video, we were given enough data two obtain the graphs of velocity and position in both the x- and y-positions for the two marbles (see Fig. 3). For our trial, we would be mainly using the position versus time graphs. To find the momentum and energy of the marble, we needed to find the velocities before and after the collision in both the x- and y-directions. To do this we simply took the slope of the position graphs (see Fig. 4).

Fig. 4
If we found the slope of the position graphs, we could find our velocity in the x- and y-directions for both of the marbles.

Once we had the velocity of the two marbles, it was now time to manually calculate the change in momentum and kinetic energy of the two marbles. For momentum, we broke the problem into x- and y- components (see Fig. 5). For kinetic energy, we simply found the resultant velocity of the x- and y- velocities and used this value in the equation KE=(1/2)*m*v^2. (see Fig. 6).

Fig. 5
Calculations for the initial and final momentum. We lost 28.6 % of our initial momentum after the collision.

Fig. 6
 Calculations for the initial and final energies. We lost 64.7 % of our kinetic energy after the collision. 

Once we had our initial and final values, we compared our values for initial and final momentum and energy. In both instances, we wounded up with less in the final stage, which is reasonable given the fact that there are more forces acting on this system than we are accounting for. This will be further discussed in the error section.

Part 2- Two Marble of Different Masses:
The second part of the lab included the same apparatus, only this time the two marbles would have different masses. We would flick a small marble (0.005 kg) at our original marble one (0.021 kg). Once again, we captured the collision on video and analyzed it on Logger Pro.

Just like before, we set our origin, scale and points on the video. This in turn would allow us to analyze the velocity of the two marbles. Just like before, we were planning to use the slope of the position graphs to find our initial and final momentum and energy (see Fig. 7)

Fig. 7
Again, we found the slope of the position graphs to find our values for velocity.

Once we had our values for velocity in the x- and y-directions for both of the marbles, we then found the initial and final momentum and kinetic energy for the system (see Fig. 8)

Fig. 8
Once we had our values for velocity, we calculated our momentum and kinetic energy for the collision.

Again, we found that we had lost some energy and momentum in our collision, however, this may have been due to the error in our lab.

Conclusion and Error:

As we can see from the collected data, energy and momentum was not perfectly conserved in this collision. This may have been due to the following sources of error in our lab:

• The surface of the glass table and marble were not completely frictionless, which suggests the presence of an external force.
• As we can see Fig. 2, the video capture did not give us an accurate image of the collision. Instead, it was slightly curved. This may have altered our ability to find the true position of the marbles. 
• Again with the video capture, the marble moves quickly enough to make it difficult to find its pin point center in every frame, causing there to be some uncertainty in its position.
• All of these values become estimates as the sources of error pile on.

However, although we had all these sources of error, are results are relatively realistic. In every instance, the difference in energy and momentum is a loss, which in a real life situation is probable. Overall, in a perfect world this lab would have yielded us a more accurate depiction of the conservation of momentum and energy.

27-April -2015: Ballistic Pendulum Lab

The purpose of this lab was to find the initial velocity of some mass in a ballistic pendulum by combining both conservation of momentum and energy.

Setup:

Fig. 1
The setup

Unlike most of our other experiments, this lab was initially setup and ready to go the day of our lab. The apparatus consisted of a ballistic pendulum to which a spring gun was aimed at. From the spring gun a small mass projectile would be shot out. When done correctly, the mass would become lodged in the pendulum, causing it to swing upwards and to the right. The angle displacement of the pendulum could then be read by a protractor attached to the apparatus. Beginning at zero degrees from which the pendulum was stationary, the protractor would swing upwards as the pendulum pushed, and would then stay in place as the pendulum swung back. This would give us the angle of displacement. (see Fig. 1 and Fig. 2 for setup)








Measuring Data:
Once the projectile was loaded into the spring gun, we cocked the spring back, allowed the ballistic pendulum to be completely still, and FIRED! As expected, the pendulum swung to the right and was displaced by some angle. All we could measure from this was the angle of displacement, which measured to be 17.2 degrees (+/- 0.1 degree) and the length of the string(L) which was 21.7 cm (+/- 0.1 cm)

Fig. 2
The ball was lined up so that it would fit snug into the pendulum.

Calculating Initial Velocity:
We only had two equations at our disposal that we could have used to calculate the initial momentum of the projectile: Conservation of Momentum and Conservation of Energy equations. (NOTE: We were given the mass of the projectile(7.63g +/- 0.01g) and the mass of the ballistic pendulum(80.9g +/- 0.1g) For the conservation of momentum, we were missing the initial velocity of the projectile and the final velocity of the projectile and pendulum (see Fig. 3)

Fig. 3
We had to come up with an expression for the final velocity of the system.

Since our end goal is to find the initial velocity of the ball, we had to come up with an expression for the final velocity of the ball and pendulum. This is where our conservation of energy equation comes into play. Conceptually, we know that the initial energy will only consist of the kinetic energy of the ball and pendulum, and the final energy (gravitational potential energy) we would consider at the maximum height of the pendulum. (see Fig. 4 and Fig. 5)

Fig. 4
To calculate the height at which the pendulum went, we had to do some trigonometric work.

Fig. 5
Our final expression for the velocity of the system after initial impact.

Once we had our expression for the velocity of the ball and pendulum, we could plug that back into our original momentum equation. (see Fig. 6) At this point we could simply plug in our values for the mass of the ball, mass of the pendulum, length of the string, and the angle at which it was swung. This would eventually give us an initial velocity of 5.06 m/s. (see Fig. 7)

Fig. 6
Our final expression for the initial velocity of the ball.

Fig. 7
We found our initial velocity to be 5.06 m/s BEFORE finding our uncertainty.

Calculating the Propagated Uncertainty:

Although we had our value for the initial velocity of the projectile, we had to account for the fact that all of our measurements had some uncertainty. To do this, we did propagated uncertainty (see Fig. 8 for all calculations).

Fig. 8
Calculating the uncertainty in our entire expression.

Our final answer came out to be 5.060 m/s +/- 0.052 m/s for the initial velocity of the ball. The only major sources of uncertainty occurred when we had to read the measurement of something with our naked eye (ruler for length and protractor for angle). Other than that, however, our answer came out to be fairly accurate.

15-April-2015: Impulse-Momentum Activity

The purpose of this lab was to prove the Impulse-Momentum Theorem: The change in momentum is equal to impulse.

Part 1: Elastic Collision:
Setup:

Fig. 1

To set up this lab, we needed a few materials:

• Two carts, one with an extended spring.
• A track that would direct one of the carts.
• LoggerPro, force sensor, motion sensor.
• Rod and some clamps.

First, we secured the cart with the extended spring to the lab bench using the rod and some clamps. Once we had done this, we setup up the track into such a way that a cart moving on the track would collide with the stationary cart attached to the lab bench. Once we had this, we attached the motion sensor to the cart which would move on the track, facing the direction of impact. Finally, we hooked up a motion sensor at the opposite end of the track (see Fig. 1 for set up).

Conducting the Experiment:

Once we had our setup complete, it was time to begin the experiment. First, we zeroed our force sensor in both the horizontal and vertical direction. Once this was done, and LoggerPro was ready with both sensors, we began collecting data and gave the cart a gentle push towards the stationary cart. The cart collided and repelled back.

Analyzing Data:

To test the impulse-momentum theorem, we first had to calculate the change in momentum. To do this we first found used logger pro to find the initial velocity of the cart and the final velocity of the cart. By taking the mean value of the velocity of the cart before and after the collision, we would be able to find both initial and final momentum of the cart (see Fig. 2). The mass of the cart we measured to be .678 kg. Thus the change in momentum = mv(f)-mv(o). = .678(-0.3965-0.4796)= -0.594 kg*m/s.

Fig. 2
After we had completed the lab, we found that one of our runs had been misplaced. During lab, we obtained the real values for initial and final velocity, however the graph from which we obtained it from had been deleted or altered. Thus, Fig. 2 contains a sample graph of what it the velocity graph for the first trial should have looked like. THIS IS NOT THE ACTUAL GRAPH.

For Impulse, we took the area under the Force vs Time graph (since we knew that Impulse= F*t, or the integral of F*dt.) and found that to be -0.6137 N*s. (see Fig. 3) With a 3.2 % difference, we could confirm the impulse-momentum theorem (see Fig. 4 for all calculations).

Fig. 3
Since we did not reverse the direction of the force sensor, we obtained a negative force, which in turn gave us a negative impulse.

Fig. 4
We compared our calculated impulse to that of LoggerPro's and found that the two values were reasonably close.

Part 2: Elastic Collision with More Mass:

Fig. 5

It was now time to conduct the experiment with the same setup, only now we added more mass to the moving cart. The goal of this trial was to determine whether or not the mass would affect the overall outcome of the impulse-momentum theorem.

With the exact same set up, we simply added 0.2 kg to the cart (see Fig. 5). Once again, we gave the cart a slight push, and collected data using LoggerPro. 

We found the initial and final velocity of the cart, which in turn lead us to the initial and final momentum of the cart. We took mean of the velocity graph before and after the collision which gave us an estimate of both the initial and final velocities. Our new cart mass was .878 kg. However, in this trial we added a magnetic mini white board to the back of our cart (this was to allow the motion sensor to track the cart more easily). The mass of the mini white board was .072 kg, which lead to a new total mass of 0.950 kg. We calculated our change in momentum to be -0.883 kg*m/s.

We again used LoggerPro to integrate the area under the Force vs Time graph and found the impulse to be -0.8537 N*s. There was 3.2% difference between the values. (see Fig. 6 for graphs, and Fig. 7 for calculations). This again confirmed the Impulse-Momentum Theorem.

Fig. 6
Since the velocity was relatively constant before and after the collision we were able to find the mean of the two horizontal pieces to find a reasonable value for momentum.

Fig. 7
Again, we were able to confirm the impulse-momentum theorem due to such a minimal difference between our change in momentum and impulse.

 Part 3: Inelastic Collision:
For the third portion of the lab, we wanted to examine the impulse-momentum theorem under inelastic conditions. To do this, we change our setup slightly. We replaced the stationary spring cart with a secured wooden block. Attached to this wooden block was a piece of clay. We then replaced then added a rubber stopper to the end of our force sensor, to which we attached a nail.The nail would then attach to the clay upon impact, not allowing the cart to bounce back. This would make our inelastic collision. We left the motion sensor and track as was (see Fig. 8).

Fig. 8

Once all of our equipment was set up, we once again gave the cart a slight push and began collecting data. With the cart being stuck to the clay upon collision, we knew the final velocity would be zero. Thus, we were only left to find the initial momentum. Since our graph gave us a slight downward slope, we tried to use the velocity most near the moment of impact, since this would be considered the initial velocity (see Fig. 9).

Fig. 9
Since the velocity is not reasonably constant, we measured the velocity 
nearest the point of collision.

Once we had this, we calculated our momentum. For this trial, we left the added mass on the cart and added the mass of the nail and rubber stopper which totaled to be 0.967 kg. We found our initial momentum to be 0.3733 kg*m/s, which meant our change in momentum was -0.3733 kg*m/s. We once again used LoggerPro to calculate the impulse exerted on the cart (see Fig. 10).

Fig. 10
We again used LoggerPro to calculate impulse using our Force vs Time graph.

We found this to be -0.305 N*s. (Note that we did not include the integral beyond the initial impact. The other humps in the graph were due to the vibration of the cart once it initially hit the nail. We compared the impulse we calculated to the impulse LoggerPro calculated and found that the values had a 20.1% difference (see Fig. 11 for all calculations).

Fig. 11
For our inelastic collision, we found that the difference between our change in momentum and impulse was significant. This indicates that there was some possible source of error. This could be explained perhaps due to the fact that velocity was not completely constant and there was some significant outside force(friction) acting on the cart.

Error, Comparison, and Conclusion:

When comparing the Force vs Time graph of the inelastic and elastic collisions, we found the they were similar in height, but different in thickness. Since we kept the extra mass on the cart for the inelastic collision, we compared the Force vs Time graph in Fig. 10 to that in Fig. 4. The thickness of the elastic collision force curve was much larger than that of the inelastic. 

There were quite a few areas of uncertainty in our lab. For one, we could not get any of the trials to a perfect constant initial velocity. This was especially apparent in our inelastic trial, in which the slope of the velocity graph is extremely apparent. However, perhaps the most prominent source of error was our inability to create a perfectly inelastic or elastic collision. This prohibited us from obtaining perfect results.

Once completed with the lab, we had verified the impulse-momentum theorem for both elastic and inelastic collisions. The elastic collisions had a small percentage difference between the different impulses, while our inelastic collisions had a rather large one. However, we believe that the larger gap may have been due to the fact that we did not give the cart a sufficient enough push and the added fact that the cart vibrated back and forth after colliding with the clay.

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.