Lab 4: Gauss Gun Conservation of Energy

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1 Lab 4: Gauss Gun Conservation of Energy Before coming to Lab Read the lab handout Complete the pre-lab assignment and hand in at the beginning of your lab section. The pre-lab is written into this weeks lab. See "pre-lab" below. Introduction In this lab you will explore the Gauss gun. This will give you the opportunity to explore the notions of work and energy regarding this interesting system. Although, the physics laws responsible for the interaction of the magnet and balls is unknown to us at this point in the course, we can tell that there is energy associated with that interaction. It appears that the magnetic interaction can be a source of kinetic energy, so there must be a potential energy that gets transformed into kinetic energy. Even though we don't have a way of measuring energy directly, we can use our understanding of work to get at the potential energy by measuring the magnetic force at different distances. Once we know the energies, we can explain why the kinetic energy changes so much in the operation of the Gauss gun. Background Potential energy Potential energy is the stored energy of an interaction. The key identifying characteristic of potential energy is that it can be converted into kinetic energy by the force associated with the interaction. The most familiar example of potential energy is gravitational potential energy. An object of mass m in the gravitational field of the earth has gravitational potential energy equal to U grav = mgh, where h is its height, and g the gravitational constant on earth. This means that in order to raise an object by a height h, you need to supply it with an amount of energy equal to mgh; conversely, letting it fall a distance h releases mgh of potential energy. (In free-fall, this would all be converted into kinetic energy of the object.) Another example is elastic potential energy, the potential energy stored in a spring or other elastic material. If a spring has spring constant k, then the potential energy stored in the spring is equal to U elastic = 1/2 k x 2, where x is the distance by which the spring is stretched from its relaxed (neutral) length. (x is negative if the spring is compressed.) If you stretch a spring, the spring has elastic energy, which can be converted into some other form of energy (e.g. kinetic) through the action of the spring force: the spring will try to return to its relaxed length, and will exert a pulling force on whatever is attached to it in order to do so. One important fact about potential energy is that there is no absolute measurement of potential energy. Instead, only differences of potential energy between two configurations can be measured. This means that any time you express potential energy as a function of position (for example, U grav (h) = mgh or U elastic (x) = 1/2 kx 2 ), you could add an arbitrary constant to the potential energy without affecting the physical meaning of the equation. For example, if you are dealing with the gravitational potential energy of a 1-kg mass, you could say that U=0 when the mass is on the ground. If you then lift the mass onto a table 1 m high its new gravitational potential energy is m=1 kg, times g=10 m/s 2, times h=1 m, which gives U=10 joules. However, you could

2 also say that U= -10 J when the mass is on the ground, and 0 J when it is on the table. (This amounts to defining your h=0 to be at the height of the table.) Figure: Three equivalent ways of defining gravitational potential energy You could even say that U=300 J at ground level and U=310 J at table level. It doesn't matter. The only relevant physical quantity is the difference in potential energy between the floor and the table: U(B) - U(A) = amount of energy required to move the object from A to B. If this quantity is positive, it means that the potential energy is higher at B, and therefore you need to add energy to the system in order to go from A to B. (As we will see shortly, the way you would accomplish this is by doing work on the system.) If it is negative, it means that you would get extra energy by letting the object go from A to B. Work and force A force that acts on a moving object does work on that object. In general, work is defined as the dot product of the (vector) force and (vector) displacement of the object. But if the force and motion are confined to one dimension, we can use a simpler expression: The work done by a force F is equal to the (scalar) force F multiplied by the (scalar) displacement dx. If the force and displacement are in the same direction, the work done by F is positive; if they are in opposite directions, the work done by F is negative. If the force is not constant during the object's motion, we can use calculus to get the work anyway. As usual, we divide up the motion into very short segments, and approximate the force as being constant over each segment. Then we just add up all the little bits of work, F dx for every little dx. The result is an integral: the work done by F as an object moves from x=a to x=b is equal to If there is more than one force acting on the object, then the net work is equal to the algebraic sum of the work done by each force. Work and energy: 3 important results In lecture, we saw the work-kinetic energy theorem, which states that the net work done on an object (that is, the sum of the work done by every force acting on it) is equal to the change in that object's kinetic energy. If a force F is conservative, then there is a potential energy function U(x) associated with it, such that

3 Plugging this into the equation for the work done by a force, we get In words, the work done by a conservative force in moving from one point to another is equal to minus the change in potential energy. This is an important relationship between work and potential energy. If there are both conservative and non-conservative forces, we can combine the two results above. The total work is equal to the work done by conservative forces, plus the work done by other forces. This should be equal to the change in the object's kinetic energy; but using the equation for work done by a conservative force, we can get the work-energy theorem: The change in an object's total mechanical energy (kinetic plus potential) is equal to the work done on it by non-conservative forces.

4 Pre-lab Answer the following questions on a separate sheet of paper and hand in at your lab section. You have been given a little gauss gun in class to explore. Careful: The magnet is quite strong. Keep it away from credit cards, your Crimson card etc.!!! Take a moment to play with this and explore how it works. Be careful the ball bearings are easily lost. The basic idea is to arrange two the magnet (M) and two balls (2,3) in the following way, and let another ball (1) approach the magnet from the opposite side: Make sure the balls are well aligned. If you got the balls in a tube you can try arranging them inside the tube. Try it out! Prelab questions: 1 M M 2 3 Ball (3) flies away with large kinetic energy. In the lab we will explore how the magnet can accomplish this. 1. To understand the gauss gun, let s first consider a simpler system: Imagine ball a is moving and hits ball b which is initially at rest. Assume that it is a fully elastic collision (no plastic deformation, i.e. non conservative forces), a central collision (all movement takes place in one dimension), and let s neglect rolling and friction. What is the state of motion of ball a and b after the collision? a b 2. Let s now assume we have one additional ball in between. What is the state of motion of ball a, b and c after the collision? a b c x 3. Let s come back to the Gauss gun: Ball (1) gets attracted by the magnet (M). Describe in words the motion of ball (1) and of the magnet with the two balls attached (M,2,3) as ball (1) approaches the magnet. Is this in keeping with Newton s third law? 4. Can we understand why ball (3) flies away, even though the magnet is just attracting the balls? Ball (1) is really fast right before it hits the magnet. Let s call the speed of ball (1) at the moment right before collision v 1. If the magnet and ball (2) is just transferring the momentum like in question 2, how large is the speed of ball (3) (v 3 ) right after the collision event (in terms of v 1 )? 5. Design an experiment with which you could measure the launch velocity of the outgoing ball bearing. Draw a sketch and show how you would calculate this initial velocity. Hint: Think of the bonus question in lab 1, you just need a table and a ruler. Roughly measure the launch velocity of your Gauss gun. 6. Show how to calculate the recoil velocity of the magnet and two remaining ball bearings. Hint: Think about conservation of total momentum.

5 Lab Names: Material: Force sensor, the same as in the last lab. A large spring and mass stand with hook. The spring is a coil spring. It can be stretched, but not compressed. Lab Jack: This is a simple piece of equipment that enables you to adjust the height of a small platform. Magnet and steel ball bearings A low friction track on a ruler Tubing clamp The tubing clamp consists of a clear plastic tube open at both ends, and a clamp to tighten around the tube. This one shown has the magnet on top and a ball below for the ballmagnet-ball set-up. Dial caliper Shim stock Gauss gun Let s first use the provided balls and magnets and the glass rod track to build a gauss gun. To get reproducible results, let ball 3 approach with small velocity, so that you can neglect this velocity compared to the launch velocity. 1. Use the method you found in the prelab to measure the launch velocity of ball (1). Average between 5 trials. 2. Calculate the recoil velocity of the rebounding system (3,M,2)

6 3. Calculate the total kinetic energy of the outgoing and recoiling balls Next we will try to quantitatively explore where this kinetic energy comes from. As the ball approaches the magnet, it gets accelerated towards this magnet. Let s use a force sensor to measure the force that leads to this acceleration. Work on a spring This quick activity will help us to understand the force sensor and how to calculate the work done on a spring. We can apply these concepts later to the Gauss Gun. Open the file Lab4.cmbl in Logger Pro. Set up the force sensor so that the hook hangs downward about cm higher than the table, and flip the range switch to the ±10 N setting. If Logger Pro complains about this change, click "Use the sensor setting" instead of "OK." Hang the spring from the hook and then zero the force sensor by clicking the zero button. Place 500 g on the mass stand and put it on top of the lab jack. Adjust the height of the jack so that you can hang the hook of the mass stand from the spring while the stand is still resting on the jack. The spring should be slightly stretched, as shown: Measure the height of the lab jack above the table, and measure the spring force using the force sensor. The meter on the Logger Pro screen will probably jump around a little bit, but you can easily estimate it to the nearest 0.1 N. Lower the lab jack (and everything on it) by 1 cm, and measure the force again, but make sure that the mass stand is still resting easily on the lab jack. (If the mass is hanging, raise the lab jack by 5 or 10 cm and start over.) Repeat until you have a total of five force measurements at 1 cm intervals. Enter the data into the table on page 1 of your Logger Pro file. Distance = 0 corresponds to whichever height you started at, which is not the same thing as the relaxed position of the spring. 4. Make a graph of the force exerted by the spring versus distance and paste it. (Scale the graph so that the relevant part of it is showing.) Graphically calculate the following quantities. Click the integral fit button in the Logger Pro tool menu. Adjust the cursors to highlight the appropriate section of the graph. Don't worry about uncertainties. 5. The work required to stretch the spring from 0 to 2 cm 6. The work required to stretch the spring from 2 cm to 4 cm 7. Are these equal to each other? Why or why not? 8. What is the total work done by the spring in stretching from 0 to 4 cm. 9. Use the graph to find the spring constant. 10. Consider the potential energy U spring. What is the difference in U spring between the initial (less stretched) position of the spring and its final (more stretched) position? Save your work.

7 How does the Gauss gun work? We observed that a large amount of kinetic energy is produced in the Gauss gun collision. How is it possible that the ball acquires such large velocity? Where does all the kinetic energy come from? Somehow, it must have to do with the nature of the magnetic force and, although we have not studied magnetic forces yet, you must have noticed that the force appears to diminish with the distance. In the Gauss gun collision, we go from the initial configuration, to the configuration, with a third ball bearing far away in both cases. 11. Which configuration do you think has a higher potential energy? 12. Can you make an hypothesis using concepts of energy for what happens during the operation of the gauss gun and why does the second ball comes out so fast? Discuss it with your TF. Let s investigate your hypothesis and determine the potential energy in each configuration by measuring the work done by the magnetic force on the ball bearing 2. Measure the magnetic force versus distance Open the Logger Pro file Lab4-work.cmbl (inside the Lab4 folder on the Desktop). Save the file giving it a new name including your initials. Read the instructions on page 1 and move to page 2 for you lab. You should find two tables and three graphs already setup. The graph Force vs Time is where you will be taking your data. You should use the two tables to record the data for the two configurations: ball-ballmagnet and ball-magnet-ball. The two other graphs will build-up automatically from the data in the tables. You will be using the force sensor with a ball bearing hanging by a yellow string. Make sure to zero the sensor by pressing the zero key. Configuration: ball-ball-magnet Setup the magnet and ball in the tube and clamp them tightly so that they don t move (only a small part of the ball needs to protrude from the end of the tube). Touch the tubing clamp ball to the hanging ball and pull it straight down. Collect some data and observe the Force vs Time graph. The maximum force reached before release is equal to the magnetic force on the hanging ball. Refer to the first page of the Logger Pro file for instructions on how to make an estimation of the magnetic force. Make measurements of the magnetic force for different distances using the shims provided. Start without any shim and move on to the thicker and thicker ones. You don t need to use all of them as long as you get a smooth graph and measure a magnetic force down to close to zero. Do 5 trials each and estimate the error by eye based on range of peak value variations. Attention: make sure the shim is always in contact with both balls. Also, when using rigid shims, make sure you don t push the ball upwards with the shim

8 13. Create a graph of force versus distance. Paste the resulting graph and save your file. Configuration: ball-magnet-ball Change the configuration in the clamp and repeat the measurement as above. You will likely need to use thicker shims before the force drops to zero and you can forego some of the thinner shims. Why is that? Fill-out the data for this configuration in the second table.. Again, do 5 trials each and estimate the error by eye based on range of peak value variations. You will need to change to the +/- 50 N setting on the force meter. 14. Paste the resulting graph and save your file Work done by the magnetic force 15. Use the graphs obtained above to determine the work done by the magnetic force on ball 2 as the separation decreases from very far away to zero, for the two configurations 16. What is the magnetic potential energy for each configuration, if we define that the potential energy is zero when the ball 2 is very far away? (be careful about signs.) 17. Calculate the difference in magnetic potential energy ΔUmag between the first and second configuration. 18. How does this difference compare with the difference in kinetic energy in the gauss gun collision? Is the reduction in potential energy enough to explain the kinetic energy of the launched ball? How might some of the energy get dissipated? Print one copy of the relevant section for the whole group and hand this into the lab TF. Don t forget your names. Please fill out the survey. Have fun!

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