Writing Lab Report: Conservation of Energy

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Physics 161

Conservation of Energy

Introduction

In this lab there are two parts, both of which will measure how energy is conserved in a system. Part I will involve measuring the velocity and position of a cart on an inclined track in its path up and back down a dynamics track. These measurements are taken so that potential and kinetic energies and therefore total energy, may be calculated. For Part II, you will track the motion of a parachute falling and observe the effect of air resistance and its relationship to the terminal velocity of the parachute. You will use the Video Point software for your data.

Reference

Young & Freedman, University Physics, 13th Edition: Chapter 7, section 7.1-7.5;

Theory

Conservation principles play a very important role in physics. If the value of a physical quantity is conserved, then the value of that quantity stays constant. The total energy of a system is the sum of its kinetic energy and potential energy. In today's lab, the potential energy is gravitational potential energy given by PE = mgy. Thus:

Total Energy = Kinetic Energy + Gravitational Potential Energy (1)

constant (2)

(3)

If the total energy is conserved, a graph of E vs. time should be a horizontal line. Gravity is a conservative force, so if it is the only force involved we expect the total energy to be conserved. (For this lab, we will assume that the force of friction is negligible.)

Theory for Part I: You will use a coiled-spring launcher to launch a dynamics cart at the bottom of an inclined track and the cart will go up the track, reverse its motion and come back down. If the friction is essentially zero, then energy should be conserved, and you can analyze the data from the standpoint of energy conservation.

Energy_fig1

Figure 1: Experimental Setup showing launcher

The motion sensor is mounted on the track so the position it records is the distance from the top of the track. The position data are therefore taken along the track which makes the calculation of the velocity of the cart straightforward. However, it complicates the calculation of the vertical displacement, h, which is necessary to calculate the potential energy.

Figure 2

In Figure 2, P is the distance to the cart as measured by the position sensor and Z is the distance the front of the cart has moved up the track. You can calculate Z by the equation:

(4)

The vertical height of the cart, h, above the level of P0 is given by:

(5)

Where is the angle of the track's incline. The reference point,, is the point where the cart leaves the launcher, that is, when the spring is fully extended.

The graph of position vs. time that you plot will be parabolic. The minimum value of the curve corresponds to the maximum height of the cart, where potential energy is at a maximum, and will represent your maximum for kinetic energy.

Theory for Part II: An object with a large surface area and low density behaves differently than objects in free fall because it is subject air resistance. For a falling object like a parachute, air resistance acts counter to the acceleration of gravity. A velocity vs. time graph will show increasing velocity over time until the point at which the velocity becomes constant, because the downward force of gravity is balanced by the upward air resistance. This velocity is referred to the terminal velocity. The force of air resistance is proportional to the square of the velocity of the parachute:

~V2 (6)

Initially, the air resistance is small but as velocity increases it will increase to a point where it equals the acceleration due to gravity. At that point, the net force on the object is zero and velocity is constant. This constant, maximum velocity is known as the terminal velocity and is related to the force of air resistance by a constant,, that depends on the shape of the object:

(7)

The cumulative work done by air resistance can then be written as:

(8)

where Δy is the change of position along the y direction. We will use these formulas to test the work-energy theorem, which states that the net work done by all forces acting on an object is equal to the change of kinetic energy. When non-conservative forces like air resistance are present, the total mechanical energy is not constant but changes according to

(9)

Noting that Wair is negative (from equation 8), we can rewrite the above equation as

(10)

Procedure

Part I: Conservation of Energy in the Laboratory

1. To take data using the setup shown in Figure 1, set up the Capstone software with a motion sensor set to take data at 25 Hz and the switch at the top of the motion sensor set to record motion at short distances. Hold the track firmly so it will not recoil when the cart is launched. First measure the mass of the cart with the weight in it.

2. Press the Record button and launch the dynamics cart up the track so that it will reverse its direction of motion before getting too close to the motion sensor. You may need to do this a couple of times to practice. If it goes too high, the launcher can be adjusted.

3. Press the Stop button after the cart has bounced off the launcher.

4. Make a graph in Capstone of the position vs. time and inspect the data. You want to have a position graph that looks somewhat like Figure 3, which shows the cart bouncing off of the launcher. The horizontal line shows the position of the part when it is sitting on the launcher after it has been sprung.

Energy_plot

Figure 3

On the far left part of the graph, the data traces a horizontal line, because the cart is at rest on the coiled spring launcher. When the cart is launched it moves toward the motion sensor, it then reverses its course land moves away from the sensor. (Remember, the motion sensor records itself as position zero so when you move toward it, you get closer to zero, and as you move away, you get larger numbers. The minimum value of the curve corresponds to the maximum height of the cart, where potential energy is at a maximum, and is where kinetic energy is at a maximum.) The points reached a minimum as indicated by the arrow in Figure 3. At this minimum, the vertical height of the cart was at a maximum and it began to reverse its direction back down the ramp.

5. In order to calculate Z you must find the first point after the cart just leaves the launcher. This is indicated by in Figure 3. This is your initial position and it corresponds to your reference height,, shown in Figure 2. To do this you can take some data with the cart sitting on the launcher with the spring extended.

6. After you have found, highlight the data points for all points below it, giving you a cut-off parabola. Make sure you don't choose any points that are higher than on either side of the parabola.

7. Copy the data to Excel and create a table with the following format

Time (s)	P (m)	Z (m)	h (m)	v(m/s)	PE (J)	KE (J)	E (J)

8. Calculate and record Z for each point by the following equation (equivalent to using Equation 4 with an appropriate sign correction):

Z = - P

9. Multiply the values of Z by the sine of the angle of the incline to find and record the height, h, at each point in time. (See Equation 5 and Figure 2)

10. Use your original position data (P) to calculate the velocities of the cart along the track in Excel. (Remember, because the motion sensor records along the track, there is no need to find x or y components.)

11. With the heights and velocities that you calculated in Steps 8 and 9, use Equations 1-3 to determine the potential energy, the kinetic energy and the total energy in Excel.

12. Create a single graph of E, KE, and PE versus time in Excel.

13. Using the trendline equation of E versus time, determine Einital and Efinal where Einital=E(tinitial) and Efinal=E(tfinal) . Calculate the %Eloss= 100%*(Einital-Efinal)/ Einital

14. In the discussion part of your lab report, based on the value of %Eloss comment whether the total energy in your system was conserved.

Part II: Parachute Drop

This measurement is made from the analysis of a video file of Knott's Berry Farm's "Sky Jump".

1. Start up the Video Point software. Under File, select Open. Click on Computer in the left panel, then go to Desktop Videos_shortcut. Choose the Quick Time (.mov) option in the lower right-hand corner, then open the 33 file.

2. To calibrate your data, you can use the fact that each colored square on the tower is 1.70 m in size. Set the yellow calibration bracket to cover ten squares, and set the calibration distance to 17 m.

3. Take data from the video by clicking on the image. It is important to use a consistent reference point: for example, always click on the center of the basket. (It may be helpful to use the zoom option.) Take one data point every five frames. You will have to delete some time values in Excel, and keep only those times that correspond to every 5th frame. This is because Video Point software exports all time stamps.

4. Go to File Export and save the data as an Excel spreadsheet on the desktop. From your data, calculate.

5. Plot vs. time. Does this velocity approach a terminal value as expected? What is the terminal velocity? Find the uncertainty of the terminal velocity by finding the standard deviation of the constant section of the data.

6. Use the terminal velocity found above to determine a value for κ using equation 7 and a mass of 280 kg. Calculate the uncertainty in κ from the uncertainty in the terminal velocity and mass of the parachute system using equation 7. Assume a 5% uncertainty for the mass and 2% for gravitational acceleration. Do a sample calculation of this.

7. Calculate the KE, PE, and the E for each time step.

8. Find the work done by the air, for each time step, using the κ value from step 6.

9. In another column, calculate the cumulative work done by the air, Abs(Wair) (see equation 8), by summing all the values up to and including that time step.

10. In another column, add the total energy E to the cumulative work.

11. Create a new graph with time on the horizontal axis and KE, PE, Etotal, Abs(Wair), and Etotal + Abs(Wair) on the vertical.

12. By trial and error, make minor adjustments to the value of κ until the Etotal + Abs(Wair) is approximately constant. (It will help if you put the value of κ in a separate cell in your Excel table, and reference that cell in your calculation of Wair.) The slope of the trendline should be between -0.5 and 0.5. If you are successful, you have verified that air resistance is κ Vy2. Is κ from this part close to the value calculated from the terminal velocity? Find the % difference between the two.

For your Lab Report

Be sure to record your values of mcart (total mass of the cart and the load) and θ and report them in the data section of your lab write-up. Include a sample calculation of finding E, and % Eloss for Part I. Make sure you include your calculation of h from the position data. For part II, include a sample calculation for KE, PE, E and Wair. Show your calculation of κ and the error propagation to find the uncertainty for κ.

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Abs

mgy