Magnetic Potential Energy Lab

The objective of this lab was to confirm that conservation of energy still applies in situations with magnets and to find an equation for magnetic potential energy.

In order to check conservation of energy still applies in situations with magnets, we created a system that would directly test magnetic potential energy without other forces such as friction complicating things.  We used an air track with a cart on it that was attached to a machine that would blow air through the holes in the track to lift the cart.  Most importantly, we placed a magnet on both the cart and the end of the track so the magnets would repel each other.  We then placed books and other objects beneath the track to raise it.

The reason we were lifting the track was to get the cart to be supported by the magnetic force parallel with the track.  We would then use the component of gravity parallel to the track to equate it to the magnetic force.  We need to find the angle that the track is making with the table which we will call our horizon.  For this, we used our phones which were accurate to a tenth of a degree.  After the cart settled in it's new elevated position, we used a ruler to measure the distance between each face of the magnets.  Also important was the mass of the cart.  Below is a table with all our findings for this part of the lab.

We then put all our data into Logger Pro.  In Logger Pro, we graphed the component of the force of gravity that was parallel to the track, calculated from our recordings, against the distance we measured between the magnets.  Below is a picture of the graph.

From here, we assume that the line is the graph of some power law and do a power fit on the line.  From here, we find our constants A and B and plug them into the equation that states that the negative of a force integrated over a distance is equal to the potential energy.  We now have an equation for magnetic potential energy for this system.  Below is our computation.

Next, we wanted to see if energy was conserved in the system.  We attached a motion detector behind the magnet attached to the now level track.  We then turned on the air and gave the cart a modest push so that the cart would be bounced back by the magnets and our number would turn out nice.  From the graphs below, you can see in the velocity vs time graph that velocity starts at a number then becomes it's negative after the magnets push each other away.  From the KE, magnetic potential energy and total energy vs time graph above it, we can see our magnetic potential energy depicted by the red line and our kinetic energy as the light blue line become mirror opposites of each other after the magnets push against each other.  Our total energy, dark blue, unfortunately spikes slightly and we see magnetic potential energy as being larger than kinetic energy.

In the end, my group and I were satisfied with the first part of the lab but wished our magnetic potential energy was more the mirror image of the kinetic energy.  This problem could have spawned from the first part of our lab with our constants relying on measurements taken with a ruler.  It could have been that I pushed the cart too hard and we got the magnets closer than we ever did in the first part of the lab and saw it respond with greater force than anticipated.

Collisions in Two Dimensions

The purpose of this lab was to simulate collision and determine if both momentum and energy are conserved through the ordeal.

The setup for this lab was both easy and difficult.  What I mean by this is that a lab technician had already set up the camera, rod stand, firewire cable and glass that we were to conduct the collision on which was the easy part.  The difficult came when the camera didn't want to communicate with the laptop we were using so our professor was kind enough to lend us his.  Below is a picture of the setup.

After following the instructions to setup our camera, we chose three marbles, two of equal mass and a smaller marble, and weighed them for their masses(seen below). The results were, m1=0.021kg, m2=0.019kg and small=0.005kg.  For our first collision, we used the two marbles of equal mass.  One we had stationary under the camera and the other we rolled into the stationary at an angle so they would go off in different directions.  Next, we collided the stationary large marble with the smaller marble with the same intent of them going in different directions.  Since there were groups waiting to use the station, we conducted both our collisions before we analyzed our video.

Seen below is the analysis of our video capture with the segment highlighted being before the collision.  The red dots represent the x component of motion for our moving marble and the blue represent the y component.  The y-axis is position and the x-axis is time.  The velocity components of our moving marble, displayed as the slope of the linearization,  are shown in the boxes respectively.  These are the initial velocities of the collision.  As a reminder the mass of the moving marble, m1, was 0.021kg.

Next below is a picture of the analysis with the highlighted portion being after the collision for m1.  As you can see by the slopes, the ball has slowed down and changed direction after the collision.

Displayed below is a picture of the same collision but this time, I have analyzed the motion of m2.  Before the collision, m2 was stationary and had no velocity.  Now though it has gained some velocity displayed in the boxes as the slopes thanks to the collision.  The mass of m2=0.019kg just to remind.

Next, we use our masses and recorded velocities to see whether or not both momentum and energy are conserved after the collision.

Above we can see that momentum is almost conserved in the elastic collision between the two similar marbles.  The velocities were obtained from Logger Pro after tracking their motion before and after the collision.

Seen above are the calculations for kinetic energy of the two similar mass marbles.  Most of the energy is in the Y direction as we chose to have M1 progress along this axis.

In the next experiment, we used marble M1 and S, a smaller marble.

In the picture above, we can see that M1 is the only marble moving with it's velocities displayed as slopes in the boxes above.

Now displayed above is the velocities of M1 after the collision with the smaller marble.

Seen above are the velocities of S after M1 runs into it.

Above are the calculations for momentum.  For this collision, we decided to use M1 again to collide with stationary S.  The velocities were obtained from Logger Pro after tracking M1 as it went through the action of colliding with S.

These are the calculations for kinetic energy.  For both initial components of kinetic energy, only M1 is moving with velocities obtained from Logger Pro.  After the collision, both M1 and S are moving but M1 is moving a little slower while S starts moving after the collision.

In conclusion, since these are considered elastic collisions both momentum and kinetic energy should be conserved.  However, kinetic energy can be seen to lose about half its value after the collision.  It could have been that M1 was projected too fast and didn't roll but skid along the glass surface creating a small amount of friction that took away from the already small values of kinetic energy.  It is also possible that the marbles were not tracked very well when the collisions were examined.

The major sources of error from this lab came from the tracking of the marbles.  Although great effort was put into obtaining data, the values that came from the data were less than desirable.  Another small source of error came from the weighing of the marbles.  The scale used to weigh the marbles was accurate to one gram.

Centripetal Force with a Motor

The objective of this lab was to find a relationship between the angle created by swinging a rubber stopper with an angular speed.

For this lab, our professor had setup a rotating apparatus and conducted the experiment while the groups recorded data.  To record the height of the swinging stopper, our professor attached a piece of paper to a ring stand and slowly raised the paper until the top of the paper was barely grazed by the bottom of the stopper.  It is also important to mention that there was a meter stick at the top of the rotating apparatus which the rubber stopper was attached to the end.  This created a distance from the apparatus that would form an angle with the vertical when swung.  We then proceeded to measure the distance of the string to the apparatus, height of the apparatus, and length of the string.

Our group collected six trials with our periods consisting of ten rotations.  Our group members individually timed the ten rotations with our cell phones and our periods generally agreed to a hundredth of a second.

From there, we drew a free body diagram of the rubber stopper and divided the forces into their x and y components.  From the x and y components we were able to derive a relationship between the angle the string makes with the vertical and its angular speed (shown below).

Using our measured heights, we were able to find the angles of each trial (shown below).

Next we calculated our angular speed using our trial periods (shown below).

Using our equation relating angular speed and the angle, we found values for calculated angular speed (seen below).

Here is a table with all our data for convenience.

Here is the error in our calculated rotational speed compared to our experimental rotational speed.

Where our error most likely comes from in this lab are our measurements taken with the meter stick and our period times taken with our phones.

Centripetal Acceleration vs. Angular Frequency

The focus of this lab was to determine a relationship between centripetal acceleration and angular speed from measurements.

Our professor set up and collected data for the class in a demonstration using a Heavy Rotating Disk, accelerometer, a photo gate, and an electric motor off of a scooter.  He attached a small piece of tape to the edge of the disk that would pass through the photo gate to count the number of times it passed through that point in a period of time.  He then used the scooter motor to rotate the disk at a certain number of volts and used Logger Pro to record data.

Our professor ran five trials and collected the acceleration and times for a number of rotations that  could be used to find the angular speed of the disk.  For example, in the first trial our professor set the voltage to 4.4v.  He used the photo gate to count the number of times the tape passed through it in a time period and Logger Pro to get the acceleration.

With this information, we are able to first get the period for the trial by dividing a set of time by the number of full rotations that happened in the time set.  After that, we can find the angular speed by dividing 2pi by the period for the trial.  Lastly, we are able to see the relationship between acceleration and the angular speed by the equation a=rw^2 which would be the radius of the point we are using to take data.  Our professor had previously measured the radius to be 13.8cm.  Below is a table with the trials and their data.

Here is an example of my calculations.















According to the equation, acceleration and the angular speed squared are related by the radius of the circle.  Taking my data and plotting it in a graph, we can see this to be true as the radius acts like the slope of a line equation.

Here we can see the slope of the equation to be 0.1388m or 13.88cm.

For this lab, the error would come out of the measuring of the radius using a ruler.  Everything else was measured by the accelerometer and photo gate.