4/13/15: Conservation of Energy Mass-Spring System Lab 12

Purpose: To verify that conservation of energy holds true for a vertically oscillating mass-spring system.

Procedure / Set up:

We will be performing two experiments: determining the spring constant and oscillating the spring to sum up the various energies involved in our system in order to find the total mechanical energy. 

Determining k

Mount a table clamp with a vertical rod to the table and to the vertical rod, mount a horizontal rod. Attach a force sensor to the horizontal rod and calibrate it using 1kg. Once calibrated, attach a spring to the force sensor and attach a hanging mass to the spring. Hit collect on Logger Pro and slowly pull down the mass so that the spring stretches. Logger pro will generate a graph of force vs. position and the slope of the graph will give us the value for k.

Our spring constant k was 8.057 N/m.

Stretch

We will need this value for the second part of this experiment because one of the various energies within our system is elastic potential energy in the spring, which includes the "stretch" as a variable. Set up a motion detector on the floor so it is right below the mass. Attach a notecard to the bottom of the mass so the motion detector can get an accurate reading. Adjust the spring so it is un-stretched and hit collect on logger pro to record the position of the mass. Our un-stretched value was 0.723 m

The stretch of the the spring = (un-stretched position - stretch position).

We will also need the mass of the spring which was .089 kg and the mass of the hanging weight which was 0.1 kg. These values will be used for the second part as well.

Total Mechanical Energy of the System

Part 2 will consist of the same set up as in part 1, only this time we will pull down the spring with the hanging mass attached and let it freely oscillate. We will verify that the total energy in the system is constant at any position. 

Various energies within the system, which will be input into logger pro as calculated columns:

EPE_spring­ = 0.5k(stretch)²

GPE_mass = (m_hanging)(“position”)g

KE_mass = 0.5(m_hanging)”velocity”²

GPE_{sping itself} = 0.5(M_spring)(“position”) We must include this as one of the energies in the system because the mass of the spring itself contributes to the potential energy of the system. Below is a derivation of how we can calculate this.

KE_{spring itself} = 0.5(M_spring/3)(V_end)² We must include the kinetic energy of the spring itself because the bottom end of the spring contributes to the kinetic energy of the system as it oscillates up and down. Using the same steps as in the image above, we can derive an equation for the KE of the spring itself. 

Now that we know our system will have 5 various energies, we can run the experiment and input our derived equations into calculated columns. We simply hung 250 grams of mass on the spring, pulled the spring down 10 cm hit collect and let it go. We made sure to record the position and velocity vs. time graphs because we will need those values for input into our calculated energy columns.

The main column we want to observe is the very last column, which is the total mechanical energy of the system.

TE = "EPE "+"GPE MASS"+"KE MASS"+"GPE Spring"+"KE Spring"

For the entire motion of the oscillating spring, the total energy should be constant.

We can verify this as well by summing the various energies at any point.

If we add up each energy at .58 seconds, the total energy is 0.985 J.

Below is a graph of the total energy.

In theory, the total energy graph should be a straight line, so it can be concluded that we have some error within our experiment. 

Possible sources of error:

The directions in the lab manual stated to hang 250 grams of mass when oscillating the spring, we didn't follow directions and hung 100 grams instead and only realized it until the end. 

As a result, the spring went past its equilibrium point when oscillating. Our equilibrium was 0.723 m, and the spring highest position above the ground was 0.732 m. Another source of error could have been made when measuring the equilibrium value of the spring. We measured the position with logger pro and then with a meter stick. Both values did not match. 

Conclusion:

The big picture of this lab was to verify that the conservation of energy theorem for a system holds true. To do this we stretched a spring over a distance and analyzed the graph logger pro generated in order to find the spring constant k. Our system included five various energies and by vertically oscillating a spring, we can see how these energies are transferred from one form to another throughout the motion of our system. We can also conclude that when the the mass reaches its lowest point, the elastic potential energy of the spring it at its max value. And when the mass reaches its highest point, the elastic potential energy of the spring is at its minimum value. We can also say that when the mass reaches its maximum and minimum height, the mass itself has no kinetic energy at that moment in time. Even though our results are not what was expected, we can conclude that all energy in the system was accounted for. 

4/08/15: Work-Kinetic Energy Theorem Lab 11

Purpose: By performing three different experiments, we are to determine how the work done on a cart by a spring compares to the change in kinetic energy of the cart. The work-kinetic energy theorem states that W = ∆KE.

Experiment 1
Apparatus:

The set up consists of a motion detector, track, force probe, spring, and cart.

We will be measuring the work done and spring constant when we stretch a spring through a measured distance. After setting up the apparatus, we made sure to calibrate the force sensor and to reverse the motion setting on the motion detector so the slope of our graph would be positive. We next opened the file called L11E-2 (Stretching Spring) to display a force vs. position axes. Once ready, we hit collect  and moved the cart slowly towards the motion detector until the spring stretch about 1.0 meters. Once logger pro generated our graph, we analyzed it and determined the spring constant is simply the slope of the force vs. position graph and the work done is the area under the force vs position graph.

We determined our spring constant k = 2.735 N/m and W = .4519 J

Experiment 2: We will be using the same set up, but the cart will be stretched a distance and the data will be collected once we let the cart go. Once logger pro generated our graph, we entered a formula in a new calculated column that would allow us to calculate the kinetic energy of the cart at any point. 

We are able to use the velocity vs time graph as well as the mass of our cart (.339 kg) to find the kinetic energy of the cart at any point. 

Once we had our calculated kinetic energy column, we then arranged the axes to read (kinetic energy and force) on the y-axis vs (position) on the x-axis. We used the integral fit on the force vs. position graph and compared it to the area under the kinetic energy vs. position graph. We analyzed the graph at three different section to make sure the work-kinetic energy theorem holds true and it certainly did.

Work done by spring = .3044 J and the ∆KE = .298 J

Work done by spring = .5326 J and ∆KE = .534 J

Work done by spring = .6354 J ∆KE =.638 J

This equation hold true. The work done by the spring is equal to the change in kinetic energy of the cart. 

Experiment 3: We were shown a video of a machine pulling back a large rubber band and the force exerted on the rubber band was recorded. It was then attached to a cart of known mass, stretched and released. The cart passes through two photogates, which are a known distance apart. The time it takes was also recorded. With the down distance and time, we can calculate the final speed of the cart and thus the final kinetic energy of the cart. 

Force vs. Position for the machine stretching the rubber band. 

By breaking the graph into various segments, we are able to calculate the work which was equal to 25.675 J. 

We then calculated the kinetic energy of the cart, which was equal to 23.8 J.

These values did not match exactly, but we still know the theorem holds true. 

Some possible errors for part 3 could be that since we estimated the heights and basses of our graphs, the end results will be slightly off and the equipment used in the experiment may not be as accurate as the equipment we are able to use today. 

Conclusion: By performing three experiments, we were able to determine a spring constant, analyze a force vs. position graph to find the work done from stretching the spring, verify that the work done by a spring is equal to the change in kinetic energy of a cart, and analyzed a video to further prove the work-kinetic energy theorem holds true.

4/1/15: Centripetal Force with a motor Lab 9

Purpose: Use our knowledge of centripetal force to derive an expression/model for ω and to compare our model to a measured value of ω in order to describe the relationship between ω and ø.

Apparatus: The apparatus was already set up before beginning the experiment. An electric motor mounted on a surveying tripod will spin a horizontal rod mounted on a vertical rod and as the motor spins with ω, a rubber stopper attached to the horizontal rod by a string will revolve around the central shaft. As we increase ω, the stopper revolves at a larger radius and ø increases. As the stopper spins, it will strike a horizontal piece of paper on a ring stand a distance h above the ground. 

Procedure: Before we derived our model for ω we needed to measure and record the distance from the ground to the horizontal rod (H), the distance from the vertical rod to the string (R), the distance from the ground to the horizontal piece of paper (h), and the length of the string to the stopper (L).

To begin deriving our model we will need to use Newtons 2nd Law on the rubber stopper. 

We were able to find ø by looking at the right triangle with hypotenuse L and height being H-h.

We next spun the stopper at six different ω's using six different voltages. Each time we increased the voltage we created and measured a new h. From this data we can find ø.

Once we derived our model for ω, we found the measured value for ω by timing how long it took the stopper to make 10 revolutions around the shaft (T) and dividing it by 10 to get the time for 1 revolution.

Below is a table of our gathered data that we can use to test our model for ω and see how it compares to the measured omega. The last two columns are the values we were most interested in. 

Below is a graph of measured ω vs. modeled ω. 

By using a best fit line, we see the graph increases linearly and has a slope close to 1.

This tells us that our experimental data compares well with our measured data and that our modeled equation for ω is reasonable for calculating angular speed. 

Conclusion: Although our model correlates well with the measured values of ω, there is still always some uncertainty in our measurements. The period T had an uncertainty of +/- .01 seconds and h had an uncertainty of +/- .5 cm. 

When we plotted measured ω vs modeled ω, each different value for T and h introduced some uncertainty within each of our points. This is why the slope of our graph is not exactly one. 







The final outcome of this experiment is that we were able to successfully derive an expression for ω from an apparatus that was set up in class and test it with a measured ω. It was found that as ω becomes larger, ø becomes larger.