In this activity we carry out measurements of two important physical quantities: work and energy. We will test the prediction of the work-energy relation: in a given process, the change in energy equals to the work done on the system.
The work-energy theorem is one of the fundamental results of Physics. It relates several physical quantities: different types of energy, kinetic and potential, and work. Corollaries of this theorem are of great importance; for example, the conservation of energy for closed systems is a restatement of this basic result.
The simplest form of energy is associated with the motion of object. Kinetic energy K is defined in terms of the mass and velocity of an object.
K= ½ mv2.
Other forms of energy are called potential, and include gravitational potential energy, which for an object near the surface of the earth is
where h is the height of the object measured from a given reference point. The total energy of the system is the sum of the kinetic and potential forms of energy:
The total energy is a property of the object and can be calculated at any specific time.
Work is not a property of a system but is associated to a process. In the simplest case, when an object experiences a displacement d while under the action of a force F parallel to the displacement, we define the work done during the process as:
The work-energy relation states that in a process, the total work equals the change in energy of the system:
To prepare for this activity, please review in more detail these definitions and relations, and study a few simple examples of the application of the work-energy relation.
In this experiment, a set of measurements will allow you to determine the work done by one external force on an object (a cart), and the change in the cart’s kinetic energy. These measurements will allow us to test the statement of the work-energy theorem.
In this experiment it is important to note that we expect a systematic lack of agreement due to the presence of friction.
A. Interface Setup
For this experiment you will record the position of the cart with the pulley photogate and determine the force exerted on the cart with a force sensor. These measurements can then be used to determine work and kinetic energy.
The motion sensor plugs must be connected to digital channels 1 and 2 of the machine. The force sensor should be connected to the analog input 1.
B. Software setup
Open the Pasco Capstone program in the computer and click on the file folder icon, or use the File menu to open a new activity.
Find, in the main hard drive of the computer, the file “Wrok-Energy.cap”. If not available, download it from blackboard.
This file will display the graphs of velocity against time and the force against position.
C. System setup.
The dynamics track must be firmly lying on a horizontal surface. Adjust if necessary, checking that the cart does not roll one way or the other.
Measure and record the total mass of the cart, including the force sensor.
Tie the hanger set to the cart with a string. Place the cart on the track and take precautions to make sure that the moving cart will not hit the pulley when the hanging mass is at ground level.
Adjust the pulley so that the string segment from the pulley to the cart is horizontal.
You will use several hanger mass values. Use masses from 0.1 to 0.3 Kg.
D. Data recording
Put the cart close to the force sensor, providing ample space for it to run before getting close to the pulley. Hold the string so that there is no force exerted on the cart (the string must be lax where it is attached to the cart). Tare the force sensor, by pressing the tare button. Carefully let the string tense again while still holding the cart. Start recording the run, wait for moment, and then let the cart go. Let the cart run until after the hanger set hits the ground but make sure to stop the cart before hitting the pulley. Stop the recording.
Capstone will display the force and velocity graphs in separate pages. After each run, check the graphs to see that your run made sense (see sample figure below).
The force graph shows one region of approximately constant force. Note that the force goes to zero when the hanging mass hits the ground.
The work done in the process is the area of this graph. To obtain it, use the select region feature and the area calculation.
Note that the run starts at position zero. The pulley only records motion when the cart is moving.
The velocity graph shows a clear maximum. To determine it, highlight the region of the maximum and use the calculations feature. Under its menu select the Maximum item.
After identifying the region of interest, analyze the individual graphs as shown in the figures. From the velocity graph, determine the maximum velocity reached during the run. This will allow you to estimate the final kinetic energy of the cart.
From the force graph, obtain the area of the graph from the moment of release to the moment the hanging mass hits the floor. This area is the work done on the cart.
This pair of values, kinetic energy and work, is the result of each of your runs. Record these in your notebook and in an excel table.
D. Data analysis.
Create a table containing the following data (or a similar set of variables as needed):
Mass of cart and sensor, mass of hanger and hanging pieces, measured maximum velocity, kinetic energy, work, and any other intermediate quantity needed.
Collect the relevant data for different runs with different hanging masses. Use a minimum of 5 different hanging mass values.
In your excel table compare the values of work and kinetic energy. Characterize their relation.
1. Theoretically determine the expected magnitude of the tension (assume no friction). Compare this value to the measured force. For this, draw the free body diagrams of both masses: the cart and the hanger.
2. Predict the final result in a different way: calculate the expected final kinetic energy when the hanging mass falls a known distance, using the concept of potential energy.
3. Compare the values of the kinetic energy change and the work measured, for every run. Which one is larger? Is this expected? Report the percentage of discrepancy between these two quantities.
4. Write down the work-energy theorem in such a way that includes the friction force. Use this relation to determine the amount of work done by friction on the cart during each experiment. Enter these values in your table.