CONSERVATION OF ENERGY

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1 Pre-Lab Assignment CONSERVATION OF ENERGY 1. Read the lab instructions. 2. A hanging mass of 1500 grams compresses a spring 2.0 cm. Find the spring constant in N/m. 3. The spring is compressed a total of 3.0 cm, and used to set a 500 gram cart into motion. Find the speed of the cart at the instant it is released, assuming all the elastic potential energy is converted to kinetic energy. 4. Suppose the cart is on a track that is elevated at one end. Find the maximum height h above the plane of the tabletop that the cart will rise to before coming to a stop; assume that all the kinetic energy is converted to gravitational potential energy. Apparatus Pasco 750 Interface 1.2 meter dynamics track, and a spirit level From track accessories: 1 adjustable end-stop, and 1 track support Three-hole pulley on a rod Universal clamp Dynamic cart (with plunger and weight) Photogate head with mounting bracket and five-pattern picket fence Meter-stick, table clamp O-Haus triple beam mass balance 1 inch by ¼ inch bolt, nut, and washers to attach to adjustable end-stop Theory If a mechanical process takes place without losses to friction or other dissipative force, the energy of the system is conserved. This means that the energy can be transformed without loss from one form of mechanical energy to another, such as elastic potential energy to kinetic energy, or kinetic energy to gravitational potential energy, for example. If friction is present, some or all of the energy is converted into heat, which is another form of energy. Three forms of mechanical energy are involved in this experiment: (1) Kinetic Energy = K = ½ m v 2 (2) Gravitational Potential Energy = U gr = mgh (3) Elastic Potential Energy = U el = ½ kx 2 In the last expression k is the spring constant, which applies to a spring that obeys Hooke s Law, F = -kx, where a force F causes the spring to be compressed or elongated a distance x away from its equilibrium position. If a compressed spring is used to accelerate an object and all of the elastic potential energy is transferred to the object, then ½ kx 2 = ½ mv 2. If the object rises until it stops, in the absence of friction, the kinetic energy will all be converted to gravitational potential energy, so ½ mv 2 = mgh, where h is the change in height of the object. 1/11/2009 1

2 Description of Experiment, and Objectives In this experiment you will: (A) measure the spring constant of a spring, and measure the elastic potential energy stored in this spring when it is compressed, (B) measure the kinetic energy of a cart that is released from the compressed spring, (C) measure the maximum height that the cart achieves, and (D) compare the results obtained in parts (A) and (B), parts (A) and (C), and parts (B) and (C). The experiment uses a dynamic cart with a spring-loaded plunger. First the spring constant is found by taking data on the spring force and compression distance. The spring is then used to fire the cart along a horizontal track. The velocity of the cart is measured by using a photogate and picket fence. The kinetic energy of the cart will be compared with the elastic potential energy stored in the spring. This comparison will be used to determined whether all the elastic potential energy was converted to kinetic energy. Finally, the spring fires the cart up an inclined track, and the change of height is measured. This tests whether the kinetic energy is all converted to gravitational potential energy. Procedure Figure 1. Equipment set up to measure the spring constant. A. Measurement of Spring Constant for Determination of the Potential Energy of a Compressed Spring I. Set up the apparatus as shown in Figure Place the track at the end of the lab table. Using the spirit level, adjust the bolt in the permanent end-stop to level the track. 2. DO NOT CLAMP THE TRACK TO THE TABLE. Attach the pulley tothe end of the track using the clamp. 3. Attach an adjustable end-stop as shown, with a 1 inch by ¼ inch bolt, nut, and washers for the spring-loaded plunger to push against. 1/11/2009 2

3 4. Place the dynamic cart on the track. Tie a string from the small hole above the plunger and lead it over the pulley to a 50 gram mass holder. 5. Add 150 more grams to the mass holder, making a total of 200 grams. Read the centimeter scale reading at the left end of the cart and record it. 6. Add 200 grams to the load and again read and record the scale at the left end of the cart. 7. Continue to add 200 grams at a time and record the scale readings. If you notice that friction causes a spread of possible readings for any load, choose the reading at the center of that spread. Continue until the total load is 1800 grams. II. Calculate the spring constant as follows (see example below) 1. Convert the load values to kilograms and multiply by g = 9.8 m/s 2 to obtain the weight in Newtons. This weight is equal to the magnitude of the spring force. 2. Convert the scale readings to meters. 3. Using Excel, plot the values of spring force versus the scale readings; the scale readings must be on the x-axis (see the example given below). Use chart wizard to plot the graph and to draw a linear trendline. Display the equation of the best-fit straight line on the graph, and print the graph. The coefficient of x in the graph is the spring constant. Record the spring constant. Figure 2. Track set up to measure the velocity of the cart after spring release. B. Measurement of Cart Velocity after Spring Release for Determination of the Kinetic Energy of the Cart In this part of the experiment the elastic potential energy stored in the spring will be converted to the kinetic energy of the cart (or cart plus mass). We will measure the velocity of the cart, determine the kinetic energy of the cart, and compare this with the elastic potential energy initially stored in the spring. Thus, in addition to the speed and mass of the cart, you will need to compute and record the initial potential energy stored in the spring. I. Set up the apparatus as shown in Figure 2 to measure the velocity of the cart after spring release. 1/11/2009 3

4 1. Place the track on the table so that the permanent end-stop is away from the computer. Level the track with the spirit level, using the bolt in the endstop. 2. DO NOT CLAMP THE TRACK TO THE TABLE. Just attach the heavy-duty table clamp to the table end near the computer, and push the track up against it. 3. Attach the adjustable end-stop about 10 centimeters from the end of the track near the computer. This end-stop should still have the 1 inch by ¼ inch bolt with nut and washers attached for the plunger to push on. 4. Place the dynamics cart on the track as shown with the plunger touching only the bolt described in step 3. Put the five-pattern picket fence in the cart, with the closest interval pattern on top. (In that pattern each light and dark band is.5 centimeters wide.) 5. Attach the photogate to the track using its mounting bracket, about 5 cm beyond the cart. Adjust its height so that the light beam passes through the close interval pattern of the picket fence. Plug its cable into digital channel 1 of the interface. 6. Weigh the dynamics cart (with picket fence on it.) Also weigh the black rectangular metal bar to be used as a load in the cart. 7. With the plunger not compressed, place the cart on the track with the plunger touching the bolt, and read the position of the end of the cart. Repeat this with the plunger locked in the most compressed position. Subtract to find the distance (x) the spring is compressed. II. Set up the interface and computer as follows: 1. Login to the computer, and open Data Studio. 2. Click on channel 1. From the list of sensors, choose Photogate and Picket Fence. 3. Under measurement, de-select acceleration and velocity so that only position is selected. Click on constant, and change the band spacing to 0.01m. 4. Set up a graph. III. To take a velocity measurement proceed as follows (Steps 1 to 10). 1. Move the track so that it pushes up against the table clamp. 2. Compress the plunger spring on the dynamic cart all the way in and latch it in the innermost position. 3. Place the cart on the track with the plunger touching the bolt on the endstop. 4. Click on start." 5. Release the spring by lightly tapping the spring release with a rigid object. 6. After the cart passes the photogate, click on Stop. 7. Select 'scale to fit.' 1/11/2009 4

5 8. Choose a linear fit. The slope is equal to the velocity. Record the velocity into an excel spreadsheet. IV. Measure the velocity at least five times (by doing steps 1 to 10 above.) Average the three highest values. V. Place the rectangular metal weight in the cart, and measure the velocity after release eight times. Average the three highest values. Figure 3. Equipment set up to measure the height the cart rises after spring release. C. Measurement of Cart Elevation After Spring Release to Determine the Gravitational Potential Energy In this part of the experiment, the elastic potential energy of the spring is converted into the kinetic energy of the cart and any mass the cart carries, which is then converted into gravitational potential energy. You will measure the change in the height of the cart, compute the change in gravitational potential energy, and compare this with the initial elastic potential energy, and the kinetic energy of the cart when it moved on a horizontal surface. I. Set up the apparatus as shown in Figure 3. II. 1. Leave the table clamp in place from part B of this experiment. 2. Attach a Pasco track support at the 1 centimeter mark and adjust its two bolts so that the bottom of the track is a distance d above the lab table; measure and record d. Use basic geometry to determine the angle θ the track makes with the tabletop. 3. Attach the adjustable stop that the plunger pushes against to the lower end of the track. To measure the distance traveled uphill by the cart. 1. Compress the plunger all the way in and latch it in the innermost position. 2. Place the cart on the track with the plunger resting against the bolt. 3. Record the initial scale reading at the high end of the cart. 4. Tap the spring release lightly with a metal bar. 5. As the cart rolls up the track, record the highest scale reading reached by the front of the cart. III. Measurements to be taken: 1/11/2009 5

6 Calculations 1. Remove the load block, but with the cart still holding the picket fence, follow steps 1 to 5 above to measure the highest scale reading five times. Average the three highest values, and record the average. Subtract the initial scale reading to find the distance L traveled along the track. Use this length L, and the angle θ determined in part I.2. to find the change in the height of the cart, h. 2. Repeat step 1, but with the mass load in the cart. 1. Calculate the elastic potential energy in the spring when the spring is latched in the most compressed position. Use the spring constant found in part A, and the maximum compression distance found in parts B. 2. Calculate the kinetic energy of the cart (and picket fence) after it is fired by the spring. Use the mass and velocity values found in B. Compare this with the elastic potential energy in the spring. What percent of elastic energy was not converted to kinetic energy? 3. Repeat calculation 2 for the cart containing the mass load (and picket fence.) 4. Calculate the gain of gravitational potential energy for the cart (and picket fence), using the change in elevation from part C. Compare this with the kinetic energy from calculation 2, and with the elastic potential energy from calculation Repeat calculation 4 for the cart containing the mass load (and picket fence.) 1/11/2009 6

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