15-Apr-2015: Magnetic Potential Energy

Purpose: Verify that conservation of energy applies to a magnetic system.

Procedure: Unlike with Gravitational and Elastic potential energy, we were never given an equation for Magnetic potential energy. So we set up a lab where energy from a frictionless moving cart transfers its kinetic energy completely into a magnetic field as magnetic potential energy and rebounds back, where theoretically all the energy should be conserved. Using this experiment, we want to first find an equation for magnetic potential energy in order to verify that conservation of energy in the system. The lab apparatus consisted of a frictionless cart with a strong magnet on one end placed onto a track with a fixed magnet of the same polarity placed on one end. 

Lab apparatus, air holes on the track to make cart glide over track.

Any system with a non-constant PE, the PE (U) is caused by an interaction force F. The relationship between the two is U(r) = the negative integral of F(r)dr from -infinite to r, where r is the separation distance. So we have to solve for  F(r). (assume F=0 when r=infinite).

Raising one end of the track to a height h makes the cart reach an equilibrium position with the magnetic field, where the magnetic repulsion force between the two magnets equals the gravitational component on the cart parallel to the track. Then we tilted the track at various angles so that we could plot a relationship between the magnetic force F and the separation distance r. Next we plot the graph of F vs. r and assumed that the relationship takes the form of a power law: F = Ar^n. We are able to to solve for F by drawing a free body diagram of the cart at its equilibrium and inputing the corresponding theta and r with a meter stick.

data of theta and corresponding r and F

graph of F vs. r

Using a curve fit for F(r) = Ar^n, we got  A = .0001092 and n = -2.003. By integrating the function, we get a function for magnetic potential energy.

calculation for derivation of magnetic potential energy

Data set for all energies in system

graph of KE, PE, and TE vs. time

Results: Total energy should have been a constant line, our results showed that energy was fairly conserved within the system but not exact.

13-Apr-2015: Conservation of Energy_Mass-Spring System

Purpose: To prove that energy is being conserved in a vertically-oscillating mass-spring system.

Procedure: We begin with the set up for the lab, which consisted of a table clamp holding up a vertical rod with a horizontal rod clamped at the top. A spring hangs from the raised horizontal rod and a motion sensor is set up to measure position from under the spring. Before beginning the lab, we had to measure the height of the unstretched spring from the motion sensor and the weight of the spring.

Lab Apparatus

To begin the experiment, we had to solve for the spring constant k of our spring. In order to do so we put a force sensor on the horizontal rod and calibrated it using zero mass and a 1 kg weight, then removed the weight. Then we attached the spring to the sensor and zeroed the sensor. We made sure that our motion sensor was set u to reverse direction in order to get a positive value for position. Next we placed a 50 gram mass hanger on the string but held it at that position where the spring was still unstretched in order to get an initial position. Then we began collecting our data for force vs. time and stretch vs. time and plotted a graph of force vs stretch. The slope of the graph gives us the spring constant k for our spring, which came out to 9.123.

graph of force vs. stretch

Now that we have our spring constant, we have to find the stretch of the spring to use for our elastic potential energy equation. We took off the force sensor and kept the motion sensor on the floor to measure the unstretched spring with hanging mass distance from the sensor and also the stretched distance with an additional 200 grams hanging on the spring. We had to make sure that the the measurements verified our spring constant that we just solved for. Then we made a calculated column with the formula (xunstretched -xstretched) to find the "stretch".

Having calculated the mass of spring, mass hanging, spring constant, stretch from no hanging mass to hanging mass, we can now see if the total energy of the system is constant at any position of oscillation. In order to do so, we must utilize the equations of the various energies involved.

KEmass = 0.5(mhanging)”velocity”^2

GPEmass = (mhanging)(“position”)g

EPEspring­ = 0.5k(stretch)^2

GPEspring itself = 0.5(Mspring)(“position”) - The mass of the spring itself adds to the potential energy of the system.

KEspring itself = 0.5(Mspring/3)(Vend)2 - The kinetic energy of the spring itself adds to the kinetic energy of the system as it oscillates up and down. Below are the calculations for the GPE and KE of the spring itself.

Calculations for deriving GPE spring and KE Spring

With the equations inputed in the calculated columns, we can now begin collecting data again. We hang a 250 gram mass to the mass hanger and pull down about 10 cm and let go. We record the position and velocity vs. time graphs in order to plug in the data to our calculated columns. A 6th calculated column was then created to calculate the total energy of the system by adding all the energies at any given position together. If conservation of energy holds true for this Mass-spring system, then the Total Energy for any given position should be constant.

Graph of each energy in color and total energy in black

Results: Looking at our graph, you can see the separate energies in different colors oscillating as the spring oscillated. The black line indicates the total energy of the system and theoretically should be a straight constant line. Although the line is not perfectly straight, we can say that our experiment verifies the conservation of energy theorem with some degree of error.

Conclusion/Uncertainty: Through experimenting with a vertical mass-spring system, we were able to prove that energy was being conserved throughout the oscillation of the spring and mass. We solved for the spring constant and measured the stretch of the spring when adding a mass to the system. Then we pulled on the string to oscillate the system in order to measure the energy of the system over a period of time. Energy was being transferred from kinetic to potential to elastic, but the total amount of energy in the system remained fairly constant. We can conclude that the amount of kinetic, potential, and elastic energies all depended on the height of the oscillating spring and that total energy was conserved. A few uncertainties were apparent to us when experimenting such as when we pulled the spring and mass down, we could not get a perfectly vertical oscillation which would effect the position over time of the weight. Also some of the measurements including the length of the spring having an uncertainty of +/- 1 cm and mass of spring having +/- 1 gram. 

08-Apr-2015: Work-Kinetic Energy Theorem

Purpose: We were to measure the work done in 3 separate experiments and find the relationship between the work and change in kinetic energy in the 3 systems.

Procedure: 
Experiment 1: For the first experiment we were to observe the work done by a non-constant spring force. We begin with a cart rigged to a spring connected to a force probe, which is on a track with a motion sensor set up on the other end of the track. Before collecting data we calibrated the force probe to 4.9 N applied and reversed the position of the motion sensor to get a positive slope on our graph. It is important that the motion sensor can read the cart's location so we rigged a flat panel to the cart to give it more area for the sensor to read. First we were to record the force applied by stretched spring vs. distance spring stretched and plot a graph of force vs. distance. Then we can find the work done by finding the area of the graph.

Lab apparatus with cart, spring, force probe, track, and motion sensor

Cart attached to spring and force probe

Now that our lab apparatus was completely set up, we opened the experiment file L11E2-2 (stretching spring) to display the force vs. position axes. Then we hit collect and slowly pulled the cart about 1 m towards the motion sensor, making sure to keep our hand out of the site of the sensor. Logger pro analyzed the data and plotted our graph below.

data set and graph of force vs. position.

The slope of the graph gives us the spring constant, which came out to 3.045. Then using the integration routine in the program to find the area under the the graph gave us the work done, which came out to 0.6561 J.

Experiment 2: For the next experiment we used the same set up as the last experiment and measured the mass of the cart, which was .573 kg. We then created a new calculated column and entered an equation that would give us the KE of the cart at any point. The equation was KE=1/2*m*v^2. Again we make sure that the cart can be seen by the motion sensor. This time we begin collecting data when the cart is already stretched 1 m and then let go of the cart, allowing the spring to pull it back to an unstretched position. After a couple trials we got a good graph and saved it. The blue graph plotted was the force vs. position and we found the work done by using the same integration routine as the last experiment. The purple graph gives us the KE at any given position. We then analyzed the change in KE from initial position to 3 different final positions and compared those results to the work done by the spring up to those positions.

Work done = .2205, Change in KE= .236

Work done = .3141, Change in KE = .318

Work done = .3533, Change in KE = .36

After analyzing our data, we found that the work done on the cart was equal to the the change in KE for all three final positions. The work-energy theorem states that W=1/2*m*v^2final - 1/2*m*v^2initial. The equation verified our results.

Experiment 3: Part three of the lab required us to open up a movie file on the laptop entitled Work KE theorem cart and machine for Phys. 1.mp4. The video shows a professor using a machine to pull back on a large rubber band. The force being exerted on the rubber band is recorded by an analog force transducer onto a graph. Then the stretched rubber band is attached to a cart with known mass, so that when the cart is released it passes through two photogates at a given distance apart. The distance and the time interval between the front of the cart passing through the first photogate and then the second photogate gives us the necessary variables to solve for the final speed and the final KE of the cart.  The force vs. position graph was plotted below in order to find the work done by the machine in stretching the rubber band.

By finding the area of smaller segments then adding them together, we were able to get a value for total work done at 25.675 J. Then solving for velocity using change in position over change in time and using the change in KE equation we were able to solve for KE, which came out to 23.88. Although the values are not exactly the same, we can say that the Work-Energy theorem holds true. There were a couple things that can create uncertainty in our final results such as the fact that we estimated the height and base of our graph to solve for the area. Also the video is pretty old so the equipment may not be extremely accurate.

Conclusion: After performing 3 separate experiments, we were able to verify that the work done was equal to the change in KE of the system. By finding the spring constant, measuring the area under the graph, and utilizing our KE equation, we were able to prove the work- KE theorem to be true.

01-Apr-2015: Centripetal Force With a Motor

Purpose: Using our knowledge of centripetal force, we had to come up with a relationship between theta and omega.

Lab Apparatus

Procedure: The lab apparatus consisted of an electric motor mounted to a surveying tripod, connected to a vertical rod which spun a horizontal rod around the tripod. Tied at the end of the horizontal rod was a string with a rubber stopper attached to the opposite end so that as the angular speed (omega) increased, the radius from the rubber stopper to the center of the tripod and angle theta also increased. A ring stand with a horizontal piece of paper sticking out was placed just outside the radius of the stopper and the tripod in order to measure the height (h) of the stopper from the ground.

Measurements of height (H), radius (R) and length (L) from lab apparatus. 

In order to find a relationship between theta and omega, we first had to make a few measurements, including total height (H), radius of horizontal rod (R), and length of string (L). Using Newtons second law, we were able to derive an equation for omega that we used later. We then found the height of the stopper from the ground using 6 different angular speeds and found the angle theta by looking at the right triangle with height (H-h), and hypotenuse L.

Newton's second law to derive equation for omega, right triangle to solve for theta

Measurements of heigh (h) from 6 runs, each at different sec/10 rev.

Next, we went onto logger pro in order to create a data chart and use a calculated column in order to easily find omega for each respected run. We were able to find omega using the derived equation of omega = sqrt( g*tan(theta)/ r) and test the model by measuring omega using 2pi*10/t(10 rev).

data set of modeled omega

Using the data of modeled omega and measured omega, we were able to create a graph in order to see just how similar the two results were. If our model is correct, then the measured and modeled omega graph should have a slope of about 1.

Measured over modeled omega graph with slope.

Results: Our results show that the linear fit slope of the measured vs. modeled graph was 0.9892, pretty close to 1 which is what we were looking for.

Conclusion/Uncertainty: The results showed us that the model for omega was very consistent with what we got through our measurements from the lab. It proves that the model is an excellent way of solving for angular velocity. We were able to derive a equation for omega that fits the model and figured out that as the angular velocity of the system increases, so does the radius from the mass to the center. The few uncertainties in the lab include the period T having an uncertainty of +/- .01 seconds and h having an uncertainty of +/- .5 cm. Having six runs is better than just plotting a couple runs, but maybe having a few more runs would have put the slope of the measured and modeled omega closer to 1. Human error in our stopwatch measurements and the paper technique to measure height (h) also could have affected our final results.

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

Purpose: To determine the relationship between centripetal acceleration and angular speed using an accelerometer and a photo-gate.

Disk with accelerometer being rotated through photo-gate by scooter tire.

Procedure: A scooter wheel rigged to a motor is used to rotate a disk with an accelerometer mounted flat on top. The accelerometer passes through a photo-gate in order to measure the seconds it takes to make one rotation.  Measurements we made include how long it takes the disk to make some number of rotations at a range of rotational speed, the accelerometer reading corresponding to the rotational speed, and the distance of the accelerometer from the center of the disk. Using six different speeds from six different voltages, we were able to measure the acceleration (a) of the disk between the time (t) of 10 rotations and the time t0(beginning of first rotation) to t10(end of tenth rotation) it took for the disk to make 10 rotations.

Data table for voltage, acceleration, time t0 and t10.

Using a=v^2/r, where v=rω (linear speed relation to angular speed), we get a=rω^2. Solving for r, we should be between 13.8 - 14.0 cm. We then solved for our omega using ω = 2π rad*10/(time for 10 rotations). We used a calculated column in logger pro to quickly solve for omega and omega^2. After recording all our data in the data table, we plotted an acceleration vs. angular speed^2 graph.

Data table and graph of a over ω^2.

Results: The slope of the graph gives us the radius, which came out to 0.1371 m or 13.71 cm. Not exactly within our measured range but still pretty accurate.

Conclusion/Uncertainty: As the theory predicted, we were able to show that centripetal acceleration and angular speed are directly affected by the radius of the disk or object rotating. During the course of this experiment, there were a few uncertainties that we became aware of. First there could be many points of friction that can alter the speed of the disk, and secondly the accelerometer's distance from the center is not perfect, which is why we had a range from 13.8 to 14 cm. The readings from the devices should be fairly accurate.