3: Conservation of Energy
- Page ID
- 114094
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)To explore whether the principle of conservation of total mechanical energy of a system holds good by determining the gravitational potential energy, kinetic energy, and the total mechanical energy of a free-falling ball.
I. Introduction:
The purpose of this lab is to demonstrate the conservation of total mechanical energy. The principle of conservation of mechanical energy is that the total energy of an isolated system remains constant (i.e., if no nonconservative forces act on it). If a conservative force does act on a system the total energy remains the same but the kinetic and potential energy in the system may change individually. The equations that show the relationship between the total mechanical energy, kinetic energy and gravitational potential energy are listed below
\[E=K E+U \quad K E=\frac{1}{2} m v^2 \quad U=m g h\]
| Symbol |
|---|
| E= Total Mechanical Energy |
| U= Gravitational Potential Energy |
| g= acceleration of gravity (9.8) |
| m = mass |
| h= height of the object from a level of r |
This lab will make use of an ultrasound sensor that will record the position and velocity of a falling object. The recorded data can then be used to calculate the mechanical energy, kinetic energy and gravitational potential energy of the falling object. With the use of a spreadsheet those calculations can then be plotted into a graph showing the constant total energy can change in kinetic and potential energy over time.
II. A summary of the method
1. Use a motion sensor to measure the position and velocity of a falling basketball over a period of time.
2. Copy the measurement on a spreadsheet.
3. Calculate the kinetic energy, gravitational potential energy, and total mechanical energy and plot them as a function of time.
4. Analyze your graphs to determine whether the mechanical energy of this system was conserved.
III. Apparatus
|
What |
Why |
|---|---|
|
A computer with Vernier LoggerPro software. (or a Vernier LabQuest Mini LabQuest2 or LabQuest3 data logging solution). and a spreadsheet |
To control the motion sensor and record data |
|
Spreadsheet (Excel or Numbers or OpenOffice etc.)/ Google SHEETS |
Data analysis |
|
Vernier Go! Motion Sensor |
To measure position and velocity as function of time |
|
Basketball |
To be used as the object in a free-fall motion |
|
Metal wire cage |
To protect the motion sensor from the falling ball |
|
Wood blocks |
To raise the motion sensor above ground within the metal cage and minimize the recording of signal reflected from the cage. |
|
Step stool |
To drop the ball from a greater height or to climb on the table |
- Units of all physical quantities are important. ALWAYS.
- Read the instructions.
- It is important to focus on achieving good quality in your experiment and data set. Repeat the experiment as many times as needed until you are satisfied with your work and can explain what you observe. You can work with your lab partners outside the classroom to finish the lab report at home (you have one week’s time).
- Ask for help (if/as needed) in step 3 of the “PROCEDURE”
- The laptop that is connected to your experimental set up should only be used for the experiment. Do not use it for anything else during the experimental run.
- The Go! Motion sensor is sensitive and will record any movement near its line of sight. So it is important to stay away from it and keep other movements to a minimum when the sensor is recording measurements.
IV. Conceptual Understanding
Sketch Your Predictions for the Motion Graphs. Before doing the experiment with the falling basketball, imagine that it is held high above the ground and released from rest so that it falls vertically toward the ground. Discuss with your partners what the position vs. time, velocity vs. time, and acceleration vs. time plots should look like. Ignore any rebound. Sketch each of your predicted plots on the graphs below using a dotted line. Please do not change your predictions after seeing the actual motion plots.
Now, also draw your predictions for the Kinetic (KE), Gravitational Potential (U) and total mechanical energy (E).
V. Procedure
We will do the analysis on a spreadsheet. You have two options: create your own spreadsheet (highly recommended) or use the provided spreadsheet and copy your data on it.
- Measure the mass of the basketball and record it here (you can record directly on your spreadsheet if you want).
Mass of the basket ball: ________________ g = _______________ kg
- Prepare the Vernier Go! Motion sensor. If you flip open the sensor plate, you will see two options. Switch the button to to the “walking/ball” option. Add photo
- On the ground place both blocks (either on top of each other or side by side vertical) and then place the motion sensor on top the blocks. Secure the blocks and the motion sensor on the floor with tape to minimize movement. Ensure that there is still access to the output port on the Go! Motion sensor.
- We will first test that all components are working before placing the wire cage on top of the motion sensor and securing it to the ground with a tape.
- To test: thread the motion sensor cable through the wire cage and connect it to the motion sensor (keep the cage on the side and not on top of the sensor yet). Connect the other end of the cable to the computer.
- Open the Logger Pro software on the computer. If all components are connected properly, the Logger pro Program should detect the device and setup appropriate graphs and tables. A small green light will also turn on the Go! Motion. Below is that the Logger Pro window should display:
- Place the basket over the motion sensor. Ensure that the wire basket does not obstruct the motion sensor. Test this by going on Logger Pro and triggering the sensor by clicking on the “green arrow” at the top of the Logger Pro display. The Go motion should emit clicking sounds and data should populate the graphs and tables on Logger Pro. The position of a hand wave or other moving item should be reflected on the graph when the device is on and running. Now secure the cage to the ground using tape. Do note that when the sensor is running, the “green arrow” turns into a “red square” and that can be used to stop the sensor any time.
- Now you are ready to run your experiment.
- With the help of a step stool have a participant raise and hold the basketball at some height (~ 2 m) directly above the motion sensor. Once your team is ready, trigger the Go! motion and have the participant drop the ball. The sensor will automatically stop when finished or you can stop it any time by clicking on the red “stop” button.
- Logger Pro should display the data on its table as well as graph. Observe the graph and identify the region of the position vs. time and velocity vs. time curves that correspond to the free-fall motion of the basket ball. Do note that we must focus only on the free-fall and therefore, we must ensure that we do not include the data points before the fall or after the ball bounces back from the ground. The graph below (experiment run by Jenand Joseph on 07/23/2024) shows an experimental run.
- The region of the curves highlighted in grey corresponds to the free-fall motion. Notice that when we highlight the curve, the corresponding data values in the table are also highlighted. That is the dataset we need to copy from the table for our analysis on spreadsheet.
- Repeat the experiment by dropping the ball and collecting data. Try to create smooth slopes on the graph for better accuracy. When you are satisfied with your experiment, copy the data and paste it in your spreadsheet. Do it for at least three trials.
VI. Analysis
If you are using the sample spreadsheet provided here, follow the instructions therein. It will automatically create and display the graphs that you can then analyze.
If you are creating your own spreadsheet, make sure to have a cell for the mass of the basketball. Also create three columns where you can record the time, position, and velocity data you copied from Logger Pro. You will need to make three calculated columns for the gravitational potential energy, kinetic energy, and total mechanical energy.
For each experimental trial:
- Enter the mass of the basketball in a cell (use the data in any specific unit system. Here we are using the SI unit)
- With the Logger Pro data collected in a spreadsheet, such as EXCEL or Google SHEETS (or any other spreadsheet), organize the data. Have a cell dedicated for the mass of the basketball and have labels for each trials data for Time, Position and Velocity.
- Create 3 new columns beside the data labeled as Kinetic Energy, Potential Energy and Total Energy.
- Potential energy: on the first cell of this column, calculate the gravitational potential energy using the equation from the Introduction section of this document.
- Kinetic energy: on the first cell of this column, calculate the kinetic energy using the equation from the Introduction section of this document.
- Total energy: calculate this as the sum of the potential and kinetic energies.
- Once you have calculated the top cell of each of these energies, copy the calculations for the rest of the cells for each column. An example of a dataset is shown below (Jenand Joseph, 07/23/2024)
- Then graph the Kinetic, potential, and total energies as function of time. The simplest way to see the details is a scatter graph with a smooth line joining the data points.
|
Mass [kg] |
Time [s] |
Position [m] |
Velocity [m/s] |
Kinetic Energy [J] |
Potential Energy [J] |
Total Energy [J] |
|
|---|---|---|---|---|---|---|---|
|
0.496 |
3.000 |
1.580 |
-0.038 |
0.000 |
7.683 |
7.684 |
|
|
3.050 |
1.578 |
-0.116 |
0.003 |
7.678 |
7.681 |
||
|
3.100 |
1.574 |
-0.333 |
0.027 |
7.657 |
7.685 |
||
|
3.150 |
1.552 |
-0.708 |
0.124 |
7.551 |
7.675 |
||
|
3.200 |
1.506 |
-1.158 |
0.333 |
7.326 |
7.658 |
||
|
3.250 |
1.437 |
-1.627 |
0.656 |
6.989 |
7.645 |
||
|
3.300 |
1.344 |
-2.100 |
1.094 |
6.535 |
7.629 |
||
|
3.350 |
1.227 |
-2.571 |
1.640 |
5.966 |
7.606 |
||
|
3.400 |
1.086 |
-3.040 |
2.292 |
5.283 |
7.575 |
||
|
3.450 |
0.923 |
-3.525 |
3.082 |
4.492 |
7.574 |
||
|
3.500 |
0.735 |
-3.956 |
3.880 |
3.575 |
7.455 |
||
|
3.550 |
0.520 |
-4.039 |
4.046 |
2.527 |
6.573 |
||
|
3.600 |
0.280 |
-2.969 |
2.186 |
1.362 |
3.548 |
||
|
3.650 |
0.160 |
-0.636 |
0.100 |
0.779 |
0.879 |
A graph is shown below
- Do the calculations for each of your experiment trials.
The concepts of gravitational potential energy, kinetic energy and conservation of energy.
- What did you observe during this lab – explain with relevant equations used to calculate the different forms of energies (unless you already did that in part (a).
- What should happen to the potential and kinetic energies as the ball is dropped from a certain height? Does your experiment support your understanding?
- What happens to the total energy of the basketball? Elaborate your answer.
- Did this simple experiment demonstrate the principle of conservation of energy? If yes, explain why you came to this conclusion. If no, explain what factors may have caused any discrepancy.
- What results would you expect if you had used a coffee filter instead of a basketball?
- Your lab report must contain the names of your team members. You must also specify in one or two sentences what each person contributed specifically (yes, you worked together but what was contributed by each person).
Note: these questions will be best answered after you have finished the experiment and data analysis as explained next.