Solving for Magnetic Potential Energy

Purpose: The purpose of the lab was to use conservation of energy laws in order to solve for a function of magnetic potential energy for the magnet-cart system.

Procedure: In order to get the magnetic potential energy we first had to set up the air track and the cart on top of it. The air emitted from the air track allowed the system to remain as frictionless as possible. On one end of the air track we had a motion sensor so that we could measure how much the air cart will move. In order to get the actual distance of the magnet from the other magnet on the glider we had to get the track level for it to be in equilibrium. We then let the motion sensor read how far away the magnet on the track was and then we subtracted how far the glider was. This gave us the real distance from the magnet. Then we changed the angle at which the track was placed 8 times so that we could get different readings for the distance the magnets were from each other. This info would then be used to plot a graph of Magnetic Force vs Distance which would give us a value of force. Then we completed the lab by plotting kinetic energy, potential energy and total energy to see whether the system enforces the conservation of energy.

Below is a picture of how the whole set up looked like.

Data: The data we had to record included the angles and distances apart the magnets were.

Below is the data table we had gotten for the different angles and the different distances.

We also plotted the values into Logger Pro in order to get a force and an angle chart. Using the angles and Weight of the glider we were able to come up with the forces.We then used this to make a force and distance graph.

We did decide to get rid of the last point to increase the accuracy of our graph as shown below.

Calculations: The calculations we did were for finding the force of the magnets. In order to do this we needed the mass of the cart and by using its weight and the angle it was elevated with we were able to get the force on the magnets.

The graph below shows what the force vs distance graph looked like after plotting the points.

After plotting the points we took the integral of the curve given by the graph in order to get a function for the magnetic potential energy.

After that we calculated the remaining energies using the given equations.

 Finally, we graphed all the energies.

 Summary: Overall the lab was a success because we managed to find a function for the magnetic potential energy between two magnets and we were able to show that energy was conserved throughout meaning that there is a valid function for the magnetic potential energy. By finding the forces between the magnets and the angles in relation to the weight of the glider we were able to come up with a function for the magnetic potential energy by finding the area under the force vs distance curve. The total energy on the last graph though is not straight and this might have to do with the fact that there was still some friction between the glider and the track. The whole experiment was not also exactly level because of the location of the classroom which was another contributing factor.

Momentum, Impulse and Collisions

Purpose: The purpose of the lab was to see whether the impulse - momentum theorem worked during different types of collisions shown by two carts and a piece of clay.

Procedure: We had three different types of collisions. Two elastic and one inelastic.

 The picture below depicts the elastic collisions. One cart was held in place while the other one was moving. Different weights were added so that we could see whether momentum was conserved even if the mass increased.

The picture below shows the inelastic collision where the moving cart will get stuck to a piece of clay and from there we would use momentum to see what would happen.

We would use the force sensor to get a force vs time graph and take the integral of the area under the curve and as a result that would give us impulse which would then lead us to get the conservation of momentum. We also had some hand calculations to see how far off we were from the actual answer. This is why we set up a motion sensor at the end of the track so we could calculate the velocity of the cart.

Data: The data we had gathered was all input into logger pro. From the motion sensor we acquired the velocity of the car which we used to get our calculations. The force vs time was used to calculate the actual conservation of momentum portion.

Below is the set up the professor wanted to use in order to get data for the lab and set up our graphs.

The graph below shows the result of the second collision. The result of the impulse was -0.2723 Ns when the mass was increased. The red portion represents whatever the motion detector acquired while the blue was the force sensor.

The graph below shows the result of the second collision. The impulse was -0.6706 Ns and this was when the cart was just by itself.

The last graph was that of an inelastic collision. The value of the impulse was -0.2351 Ns.

Calculations: The only calculations we had to come up with were the ones for the actual impulse. All of the values are listed below. We did mv(final) - mv(initial) in order to get the impulse. We then compared that to the impulse we had gotten from the one gotten from the lab. 

Summary: Overall, the lab was successful because the values gotten from the calculations were not far off from the actual values gotten from the lab. Some factors were not taken into consideration tough like friction, air resistance and the strength of the actual push the cart gave. Nevertheless, the lab gave accurate results and proved that the momentum-impulse theorem is accurate.

Work and Spring Relationship

Purpose: The purpose of the lab was to see whether we could prove that the conservation of energy theorem applies to a spring oscillating.

Procedure: In order to see whether energy is conserved we have to find the spring constant of the spring first. In order to do that we have to measure the unstretched position of the spring and then the stretched position of the spring. Then we will let the spring oscillate for about 10 seconds. There is a motion sensor at the bottom of the spring in order to capture the movement of the spring. This will allow us to get the position and velocity which we need for the kinetic and potential energies of the spring and mass. Then we will compare results and come to see whether energy is truly conserved.

Below is what the experiment set up looked like. There was a force sensor attached to the top of the spring in order for the spring to oscillate more efficiently. The notecard was attached to the bottom so that the motion sensor could get a better reading.

Data: The data we needed was captured by Logger Pro which included the position and velocity.

The data table below shows the different types of energies needed to prove that energy is conserved within the spring system. The energy columns are all calculated columns based off of the data gathered.

The graph below is that of the position vs time and velocity vs time. The graphs represent simple harmonic motion because it is a spring oscillating.

Below is the final graph of the energies. The graphs are not completely showing that the energy of the system was conserved. There must have been something wrong with our data because total energy (purple) should have been in a straight line.

Calculations: The calculations were all done in class with the professor. It shows how we came to know what data to acquire and how to solve the functions for each of the energies.

Summary: Overall the lab was not successful because we did not conserve energy within our graphs. This may have been because we did not get the spring constant k correctly. I had a little bit of difficulty knowing how to get it as well as my partner. Otherwise, the rest of the lab was successful because the energies were not exact but they were really close.

Kinetic Energy and Spring Relationships

Purpose: The purpose of the lab was to find the work done by the spring-cart system by using the kinetic energy and force with the position of the cart.

Procedure: The lab consisted of a spring attached to a force sensor and a cart on top of a track. On the other side of the track there is a motion sensor that measure position of the cart on the track. First, we calibrated the force sensor so that we could get accurate readings. Then we moved the cart a certain position back so that when we let go so that we could gather data and plot the data into graphs. Then we would find the area under the graphs in order to get the work.

Below is the set up of how the experiment looked like.

Below is a picture of how the force sensor looked like. 

Below is a picture of how the motion sensor looked like.

Data: All the data was input into Logger Pro in order to get the graphs.

Below is the chart for the different types of data we recorded. It has the force, position, velocity, acceleration an kinetic energy which was a calculated column, meaning that we used different values to come up with the kinetic energy column.

Below are the graphs of position vs time, velocity vs time and force and kinetic energy vs time. We mainly focused on the force and kinetic energy vs time for it was the one that we used to calculate work.

Calculations: In order to find the work done we had to take the integral from a certain point on the graph to the other.

Below is one of the integrals taken of the force and kinetic energy vs position in order to find work. The value was 0.896 Nm.

The next graph is another area of integration which was 0.790 Nm.

The last graph shows the last integral we did for the graph. The value was 0.625 Nm.

Below are just instructions on what the professor expected us to do in the lab.

Summary: Overall the lab was a success because we were able to get a result for the force by integrating the force vs position portion of the graph. Since the work-kinetic energy theorem states that the change in kinetic energy is equal work and the values we got for both were about the same, the lab proved to be successful. The only thing that was weird was the position of the graphs because they were supposed to look like they were increasing but we did not inverse the position and it made the graphs look like the inverse of what they were supposed to be.

Calculating Power of Physical Activities

Purpose: The purpose of the lab was to calculate the power used in two different types of activities.

Procedure: The lab consisted of two different types of activities. One of them was walking/running up a flight of stairs and the second one was pulling a weighted backpack using a pulley system.

One of the experiments that we did was pulling up a weighted backpack using a pulley system and checking the time it took us to pull up the backpack as shown below. We recorded the time and the mass of the backpack.

The second experiment was us running up the stairs and walking up the stairs and recording the time it took for us to go up those steps. We recorded the time it took us to climb those steps.

Data; The data we recorded included the mass of the backpack, the times it took for us to do the activities, and we also needed our weight which we got during our own time and we also had to acquire the height of what we were lifting.

Calculations: In order to find the power, we needed the time, height and the weight of the different objects we had to elevate whether it was the backpack or our bodies.

Below is the way I calculated the power of lifting myself up the stairs. First I got my weight and then I divided that over the time it took to get the power.

The second calculation below was for lifting the backpacks. It was the same process as the one before. I got the power to be 49.6 Watts.

Summary; Overall, the lab was successful because we were able to calculate power out of the measure variables we got. By knowing that P = W / t we were able to get the results we needed. The results were that walking up the stars requires more power than lifting something up the same amount of height.

Collisions (Meter Stick with Clay)

Purpose: The purpose of the lab was to analyze an inelastic collision which involved a meter stick striking a piece of clay and how to use conservation of momentum in order to find how high the meter stick rises.

Procedure: The whole lab involved the use of a meter stick, a piece of clay and a stand from which the meter stick was swinging. The meter stick was pinned at one end so that it could swing from a certain height. We also used Video Capture and Logger Pro in order to compare our experimental results from our derived results. First, we used calculations so that we could come to a result and see whether conservation of momentum would apply to the inelastic collision. Then we would raise the meter stick a certain height and use Video Capture to see how high the meter stick swung. Below is an example of how the experiment looked.

Data: In order to get the numbers for our experiment we measured the the length of the ruler (which was already given), the mass of both the ruler and the piece of clay. Below is the graph of the data that we used for our actual value. The y value showed how high the meter stick went but we had to subtract 20 cm because the initial height of the clay was elevated.

Calculations: The calculations used were to solve for how high the meter stick was going to go. In order to solve for it we used conservation of angular momentum which involved finding the moment of inertia of the meter stick at one end and using that to get the final height.

Summary: Overall the lab was successful because the experimental value was not too far from the actual value. The experimental value was 0.238 m and the one from the graph was 0.280 which was -0.420 meters difference. The error might have consisted of friction between the pin and the ruler and some small air resistance. Also the ruler was supposed to be released from the horizontal position so it might not have been exactly released from that position.

Conservation of Linear and Angular Momentum

Purpose: The purpose of the lab was to see if the angular and linear momentum in a series of different circumstances including both types of motion.

Procedure: Below is the set up of how we had the first and second parts of the experiment set up. The second one in the picture below involved getting a ball and rolling it down from an initial height. Then it would roll down the ramp until it flew it off it and struck a certain point on the ground. The first part involved will allow us to acquire a final speed which in result leads on to the second part. In the second part, instead of the ball falling to the ground, the ball will fall into the ball catcher which is attached to a rotating apparatus and a hanging mass attached to the system as well. From this we will be able to calculate the rotational velocity of the disk and compare and see whether the linear momentum and rotational momentum are conserved.

Data; Since this lab was done in class everyone had gotten the same results. The picture above shows the different measurements that the professor gave to the class. This allowed us to get a final velocity for the system and recorded it as the final rotational velocity. The same data for when the ball was rolling down the ramp was also used.

The graph below shows the results of the disk spinning and the way the acceleration differs every time.

Calculations: The calculations below showed how we came the answer. For when the ball was rolling down the ramp we used Kinetic and Rotational Kinetic Energy and then once the ball left the ramp we used kinematics. The final velocity when the ball was off the ramp was 1.4 m/s and the final speed of the ball before it hit the ground was 1.11 m/s. 

Using the final velocity when the ball was initially off the ramp we were able to get the final rotational velocity of the disk when the ball struck the ball catcher. Using the moment of inertia of the disk and conservation of angular momentum of the impact we were able to calculate the final rotational velocity of the whole system.

Analysis: The differences between the actual and experimental values were really close. If I remember correctly we were below 5% as a class because the lab was done collectively.Overall it was a success because both the speed of the ball at launch and during the collision were very similar. This further proves the fact that linear and angular momentum are conserved if the speed of the ball initially is equal to the same amount in the final velocity.