Potential and Kinetic Energy

Transcription

1 LAB VI. ENERGY 95 Lab VI Potential and Kinetic Energy 1 Introduction This is a lab about the interplay between kinetic and potential energy. While we can calculate forces and accelerations of an object as it moves along a complicated path, the use of energy gives us a way to bypass these complex calculations, and understand motion in a simpler way. 1.1 Forms of Energy The kinetic energy T of an object can be written as: T = 1 2 mv2 (VI.1) where m is the mass of the object and v is its velocity. The potential energy that we will be dealing with is gravitational kinetic energy U g, and can be written as: U g = mgh (VI.2) where, again, m is the mass of the object, g is the acceleration of gravity (approximately 9.8 m/s 2 on earth) and h is the object s height above some defined position. We can select where h = 0 for our own convenience, because only changes in potential energy are significant. The total energy of our object is then given by: E = T + U g. (VI.3)

2 96 LAB VI. ENERGY 1.2 Energy conservation If our total energy is conserved, then E doesn t change and we can write E = 0. This implies that: T = U g (VI.4) So in the motion between points 1 and 2 : T 2 T 1 = (U g2 U g1 ) = mg (h 1 h 2 ) (VI.5) or 1 2 m ( ) v2 2 v1 2 = mg (h1 h 2 ). (VI.6) If an object starts from rest, v 1 = 0 and T 1 = 0. So: v 2 2 = (2g)h 1 (2gh 2 ) (VI.7) and if we plot v 2 2 vs. h 1, we should get a slope of 2g and an intercept of 2gh Energy losses Among the ways that an object can lose energy, friction and drag are the most important. In both cases, the object does work W: W = F d (VI.8) when it exerts a force F over distance d. While we will be working to minimize the effects of friction and drag on our measurements, they will always be present to some degree. The force from kinetic friction is approximately constant for a given object, so the work done (and energy lost) by friction will therefore be proportional to the distance traveled. The force of friction F f is related to the normal force F n (the force of the car on the track) by F f = µ k F n (VI.9) where µ k is the kinetic coefficient of friction. The normal force comes (in part) from the weight of the car, but a car traveling on a track that curves upward (for example) has additional normal force coming from the car being accelerated upward as it follows the curve. Drag forces are proportional to v 2 (or v, for small or slow objects); to see the effect of drag forces will be more complicated.

3 LAB VI. ENERGY 97 We can now update our expression for the energy of an object: E = T + U g W f (VI.10) where W f is the work done by the object in the form of friction or drag. Note that while total energy is conserved, friction and drag are energy lost by our object because the work done has generated heat or other forms of energy.

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5 LAB VI. ENERGY 99 Name: Sec/Group: Date: 2 Pre-Lab A car is traveling horizontally at 4 m/s, after which its track slopes upward. At what height will the car stop, neglecting friction? 2. A 250 g car traveling on a horizontal track at 1 m/s comes to a stop in 5 m because of friction. What is the frictional force on the car? What is the coefficient of kinetic friction? 3. A car on a loop-the-loop track of radius R = 10 cm needs to travel at 1 m/s when it is at the top of the loop (point a ) to keep from falling off the track. What is the velocity of the car when it is at the bottom of the loop (point b ), and from what height h should the car be released in order to stay on the track through the loop? c a h R b

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7 LAB VI. ENERGY Equipment 3.1 Track The track may already be set up before you start. Should you need to set up or change the track configuration, please follow these guidelines. There are three boards for holding track, assemble them with the smaller board in the middle, with the plastic support bases located at the joint between the boards. You may also have binder clips to hold the board together at the top. There are colored dots on the upper corners of the boards: match the dots on the corners that go together. The track is supported by plastic pegs, which come in two varieties: photogate pegs, which have a screw fitting for holding a photogate, and normal pegs that have plain end. Figure VI.1: Track support pegs; photogate support peg is on top. You must first attach a peg-holder inside the hole in the board, before you can insert a peg. The peg-holder is a round insert with a nut that tightens on the back of the board. Please note: there is a groove inside the peg-holder that fits a pin that protrudes from the photogate pegs to keep them from twisting. Insert the photogate pegs carefully so that they have their pin in the groove. The track is attached to the pegs by clips. You put the clips in the bottom of the track by turning them so that the flat tab part of the clip attaches to the grooves on the bottom of the track. Then attach the other end of clip to the peg. You may need to slide the clip along the track slightly to get it in the right position. One of your lab group should be stationed at the end of the track as a catcher to stop the car before it falls off the end of the track. The catcher should get their hands in place before the car is released!

8 102 LAB VI. ENERGY Figure VI.2: Track clip. Figure VI.3: Car with photogate flag.

9 LAB VI. ENERGY 103 Figure VI.4: Photogate for time and velocity measurement. 3.2 Cars The cars are quite simple, with three wheels that fit in grooves on the top of the track. Make sure the wheels are in the right place, or the car will have extra friction or fall off the track! There are slots on the sides of the car for fitting a small vertical flag (see Fig. VI.3). This flag is used with the photogates to measure the velocity of the cars. Put only one flag on a car. Weights can be attached to the underside of the cars, using a screw that threads into the brass weight. Drag has less of an effect when the mass of the car is increased, so it s a good idea to use the added weight. 3.3 Photogates Photogates are attached to the special photogate pegs and used for timing and velocity measurements (see Fig. VI.4). There are small holes (one of which is visible in the figure) for the an infrared beam that is blocked by the flag on the cars. Adjust the photogates so that the body of the photogate is perpendicular to the track, and the height of the infrared beam above the track is the right height so that the cars flag (and not the car body) blocks the beam. You can put the timer in test mode to check if the flags are correctly blocking the infrared beam.

10 104 LAB VI. ENERGY Figure VI.5: Timer module used for time and velocity measurement. 3.4 Timer Plug the photogate into the timer (see Fig. VI.5), using the input labeled 1 on the right side of the timer. Connect the timer s power supply on the left side of the timer, and switch it on. Using the control buttons on the timer, select Velocity, and One photogate. Before each measurement it is necessary to press the START button on the timer. The timer gives velocities in cm/s. 4 Procedure 4.1 Consistency check First we must check the consistency of our measurements. Set up the track with a downward slope leading to a long level segment of track. This is track configuration with the peg positions marked on the board with red dots (See Fig. VI.6). Put the photogate peg and the photogate in the position marked with two red dots. We want to start the car from rest at the same place on the track, so pick one of the pegs in the downward slope as your starting point. Attach a small binder clip to the track to mark the starting position of the back of the car. The main point is that a car released from the starting point will have a consistent velocity along the level part of the track. You should make sure that when the car is at the starting point, the car is against the binder clip, and that when the car is released it is neither pushed nor held back.

11 LAB VI. ENERGY 105 Figure VI.6: Track configuration #1, marked with red dots Connect the photogate to input 1 of the timer and set the timer to measure velocity. Press START on the timer and release the car from the starting point. Record the velocity of the car in Table VI.1. NOTE: the timer gives you velocities in cm/s, convert to m/s when you enter the velocity measurements in the data tables. Repeat the measurement five times, entering the table in Table VI.1. Calculate the average of your measurements and enter it in the Table. 4.2 Potential to Kinetic Energy 1 Keep the track configuration unchanged for this part. Weigh your car and enter the weight in Table VI.2. Then take the meter stick and measure the height from the tabletop to the track at the photogate, and enter the value as h 2 in Table VI.2. NOTE: that part of the track may be on top of the motion table cover, rather than the lab bench. Measure the height of the cover, and add to the height from the top of the cover to your starting position. Select five starting points for your car, Your starting points should be spread out along the downward sloping part of the track to get a full range of heights. Measure the approximate horizontal distance between your starting points (take the middle starting point) and the photogate, and record it as the d in Table VI.2. At each starting point measure the height of the track, then center the car above the starting point. Release the car and take a velocity measurement. Enter the height (h 1 ) and velocity (v 2 ) in Table VI.2. Calculate the car s kinetic energy from its velocity in Table VI.2. Note that the timer gives velocities in cm/s, and you must convert to m/s when calculating the kinetic energy. Calculate the change in potential energy U between the starting point and the photogate and enter it in the table. Make a graph of the car s velocity squared (v 2 2 on the y-axis) vs. h 1 (x-axis) and

12 106 LAB VI. ENERGY Figure VI.7: Track configuration #2, marked with yellow dots. fit with a straight line. Enter the slope and intercept in Table VI.2, calculate g from your fit, and get a value for h 2 from the intercept of your graph (see Eqn. VI.7). Attach your graph to the data you hand in. 4.3 Potential and Kinetic Energy 2 Now adjust your track configuration to the one marked with yellow dots. As before, put the photogate in the spot marked with two yellow dots (See Fig. VI.7). Now repeat the previous series of measurements, entering your data in Table VI.3, and plotting and fitting the data the same way as you did previously. Try to get your starting points close to the heights used for your measurements using the first track configuration.

13 LAB VI. ENERGY 107 Name: Sec/Group: Date: 5 Data Table VI.1: Consistency check measurement # average velocity (m/s) Table VI.2: Potential and Kinetic Energy 1 h 2 : m m car : kg d: m h 1 [m] v 2 [m/s] T 2 [J] U [J] Fit results: slope: m/s 2 intercept: m 2 /s 2 g: m/s 2 h 2 m

14 108 LAB VI. ENERGY Name: Sec/Group: Date: Table VI.3: Potential and Kinetic Energy 2 h 1 [m] v 2 [m/s] T 2 [J] U [J] Fit results: slope: m/s 2 intercept: m 2 /s 2 g: m/s 2 h 2 m

15 LAB VI. ENERGY 109 Name: Sec/Group: Date: 6 Conclusions 1. The two track configurations differ in their shape; compare the speeds measured for the two tracks when the starting heights are close to the same. How do you explain what you observed? 2. Calculate the potential energy difference for a car going between one starting point (pick one from your data) and the photogate, and compare to the final kinetic energy you measured. Can you explain why there might be a difference between the energies? 3. How good a value for g do you get from your data? What systematic errors might be present? Suggest some techniques for reducing these errors. 4. If the frictional force on the car was 0.1 N, estimate how much energy would be lost by the car during its travel. Now work out how much frictional force is needed to account for any difference you measured between initial potential energy and final kinetic energy.