Experiment 8
Conservation of Energy
Objective:
Experimental verification of the law of conservation of mechanical energy
Equipment:
A computer with Internet connection, a calculator (The built-in calculator of the computer may be used.), paper, and pencil
Theory:
Consider an object of mass M moving at constant velocity V on a horizontal and frictionless surface. Its K.E. is 0.5MV^{2}. If this object climbs up an incline, its velocity keeps decreasing as its elevation increases. This means that it loses K.E. as it gains P.E. . The amount of K.E. loss is equal to the amount of P.E. gain. This is according to the Law of Conservation of Energy. Its total energy remains constant in the absence of energy losses due to friction.
The applet used in this experiment has a car with a certain amount of initial K.E.. The energy balance for the car (at the bottom and top of the incline) may be written as:
K.E. at the bottom of the incline = P.E. at the top where it comes to stop
This can mathematically be written as:
0.5MV^{2} = Mgh
In the absence of friction, we may cancel M from both sides and solve for h, the highest elevation it can reach.
h = v^{2}/2g (Verify).
Note: If the car is not at zero elevation to begin with, then its initial P.E. is not zero and its initial P.E. must be added to its initial K.E. The formula for final height calculation, in this case, becomes:
Mgh_{i }+_{ }0.5MV^{2} = Mgh_{f}
where h_{i} and h_{f} are its initial and final elevations.
Procedure:
Click on http://www.mhhe.com/physsci/physical/giambattista/roller/roller_coaster.html . The applet has a screen with a grid. The distance between every two neighbor points in the vertical direction is the equivalent of 20m. The zero in the vertical direction is not on the lowest row of dots. Each dot on the lowest row has a y-value of 20m. If you want to start from zero potential energy (or a zero height), you need to start from below the first row of dots. As a quick practice, below where it says "Construction Tools", click on "Straight." Then place the mouse on the first row at the leftmost dot and draw a straight line (uphill road) all the way to the right and end it at the 4th dot from the bottom. Set the initial speed at 32.5 m/s and let the mass be at 500kg. Then click on "Start." You will see that a car appears at the lower left and goes up the incline. On the information bar, you will notice an initial height of 20m. This is because the lowest row has an elevation of 20m in this applet. This elevation gives an initial potential energy to the car. If you want the initial potential energy to be zero, you need to start the car from zero elevation. Zero Elevation, is 20m lower. It is the borderline of the grid. Now click on "Reset," and then click on the incline you just drew. It will turn red. It means it is "selected." Then click on the "Delete Selection" button. The incline will disappear.
Now, try to start another incline from the leftmost point on the borderline (0,0) and extend it to the rightmost point that has an elevation of 100m. An elevation of 100m means the fifth point from the lowest point (5 spaces from the bottom borderline).
If you click "Reset," you will see an initial height of (0 m) on the information bar. Select a mass of 500kg and an initial speed of 40m/s, and calculate the initial K.E.. Run the applet and double-check your result with the value shown on the information bar and assure its correctness. If you start the applet, you will see that the car climbs the incline until it runs out of K.E. energy. If you look at the value of P.E. on the information bar, you will see a P.E. greater than expected. This applet has some glitches in it. The P.E. can not be more than the initial K.E. You can also change the simulation speed for easier and slower tracking. The "pause" key allows you to stop the simulation momentarily for taking or recording your readings.
You may try some examples of your own before starting the actual experiment. This will help you explore what the applet can do. We will not use the curved path options in this experiment because the applet is not bug free and results in wrong values. We will use the straight line option only.
The Experiment:
The matrix of dots in the applet is a (20-row) by (29-column)-matrix. Note that the lowest row of dots is the bottom borderline itself. The leftmost column of dots is the left borderline itself as well. In each case, select the path (as determined by the given coordinates), mass, initial velocity, and the initial height according to the values in Table 1. Also, have the simulation speed at its slowest setting and read the highest elevation (Height) the car reaches as the applet runs in each case. The highest elevation may be read from the information bar of the applet. The reading will be your measured value of the Height, each case. The accepted value is what you calculate and expect it to happen in each case.
Data:
Given and Measured:
Table 1
Trial | Straight or Zigzag Path
Coordinates (20m, 20m) |
Mass (kg) |
Initial Speed (m/s) |
Initial Height (m) |
Measured Final Height (m) |
Calculated Final Height (m) |
% Error |
1 | (0,0 - 24,5) | 500. | 20.0 | 0.0 | |||
2 | (0,0 - 18,5) | 500. | 20.0 | 0.0 | |||
3 | (0,0 - 12,5) | 500. | 20.0 | 0.0 | |||
4 | (0,0 - 06,5) | 500. | 20.0 | 0.0 | |||
5 | (0,0 - 24,5) | 500. | 40.0 | 0.0 | |||
6 | (0,0 - 18,5) | 500. | 40.0 | 0.0 | |||
7 | (0,0 - 12,5) | 500. | 40.0 | 0.0 | |||
8 | (0,0 - 06,5) | 500. | 40.0 | 0.0 | |||
9 | (0,0 - 24,5) | 500. | 44.3 | 0.0 | |||
10 | (0,0 - 18,5) | 500. | 44.3 | 0.0 | |||
11 | (0,0 - 12,5) | 500. | 44.3 | 0.0 | |||
12 | (0,0 - 06,5) | 500. | 44.3 | 0.0 | |||
13 | (0,5 - 24,11) | 500. | 25.0 | 100. | |||
14 | (0,5 - 18,11) | 500. | 25.0 | 100. | |||
15 | (0,5 - 12,11) | 500. | 25.0 | 100. | |||
16 | (0,5 - 06,11) | 500. | 25.0 | 100. | |||
17 | (0,5 - 24,11) | 500. | 35.0 | 100. | |||
18 | (0,5 - 18,11) | 500. | 35.0 | 100. | |||
19 | (0,5 - 12,11) | 500. | 35.0 | 100. | |||
20 | (0,5 - 06,11) | 500. | 35.0 | 100. | |||
21 | (0,5 - 24,11) | 500. | 48.5 | 100. | |||
22 | (0,5 - 18,11) | 500. | 48.5 | 100. | |||
23 | (0,5 - 12,11) | 500. | 48.5 | 100. | |||
24 | (0,5 - 06,11) | 500. | 48.5 | 100. | |||
25 | (0,5 - 5,2 - 24,11) | 500. | 45.0 | 100. | |||
26 | (0,5 - 5,2 - 18,11) | 500. | 45.0 | 100. | |||
27 | (0,5 - 5,2 - 12,11) | 500. | 45.0 | 100. | |||
28 | (0,5 - 5,2 - 06,11) | 500. | 45.0 | 100. |
Provide the percent error formula used as well as the % error in each case according to the Table.
Conclusion:
State your conclusions of the experiment.
Discussion:
Provide a discussion if necessary.