* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Unit 4 Packet (Labs)
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Specific impulse wikipedia , lookup
Fictitious force wikipedia , lookup
Eigenstate thermalization hypothesis wikipedia , lookup
Hooke's law wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Classical mechanics wikipedia , lookup
Centrifugal force wikipedia , lookup
Nuclear force wikipedia , lookup
Mass versus weight wikipedia , lookup
Equations of motion wikipedia , lookup
Electromagnetism wikipedia , lookup
Work (thermodynamics) wikipedia , lookup
Hunting oscillation wikipedia , lookup
Seismometer wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Centripetal force wikipedia , lookup
under-inflated billiard balls colliding 100% basketball bouncing 90% 30% ELASTIC 0% INELASTIC inflated proton colliding with another proton basketball bouncing Conservation Laws AP Physics 1 Mr. Kuffer Motion Detector Force Sensor Elastic cord Check Your Understanding Express your understanding of the concept and mathematics of momentum by answering the following questions. 1 NOTES: I. (6.1)Work, measured in Joules (J), is done only in the direction of the force causing the displacement. Work = Force x Displacement W = F x d… well, W = F x s… (always in direction of force) W = (F Cos Θ) s ??(‘s’ is the ‘customary’ symbol for displacement when discussing work)?? In the words of Mike Tomlin… “It is what it is” II. (6.9) Work done by variable force, Simple Harmonic Motion (SHM), & Elasticity (ex: Drawing back a bowstring… compressing or stretching a spring) - The work done by a variable force in moving an object is equal to the area under the curve of ‘ FCosΘ vs s ‘ graph. III. (10.1-10.2) The ‘Ideal Spring’ & Hooke’s Law – and ideal spring behaves according to Hooke’s Law as follows: Fx = -kx Where k is the spring constant and x is the displacement of the spring 2 IV. (10.3) Energy and Simple Harmonic Motion (SHM) - Elastic Potential Energy – energy stored when a ‘spring’ is compressed or stretched. B/c the spring is ideal… Fx = -kx However, we know the force is variable… that is it changes in time… B/c the spring force on x is linear (‘ideal’ spring), the magnitude of the average force is just ½ the sum of the initial and final values… Fx = ½ (kxi + kxf) So… Welastic = (Fx Cos Θ) x = ½ (kxi + kxf) (Cos 0°) (xi-xf) Or Welastic = ½ kxi2 - ½ kxf2 V. (6.2) Work-Energy Theorem When Work is done… there is a change in Kinetic Energy (ΔKE) W = KEf - KEi = ½ mvf2 – ½ mvi2 3 VI. (6.3) Gravitational Potential Energy (Work done by gravity) Wgravity = (mg Cos0°) (Δh) = mg (Δh) Wgravity = mghf - mghi VII. (6.4) Conservative vs Nonconservative Forces - - - A force is conservative when the work it does on a moving object is independent of the path between the object’s initial and final positions. (Gravitational force is a conservative force) A force is conservative when it does no net work on an object moving around a closed path, starting and ending at the same point… (ex: riding a roller coaster in a complete loop) A force is nonconservative if the work it does on an object moving between two points depends on the path of the motion between the points. (Kinetic friction force is one example of a nonconservative force… air resistance being another nonconservative force) o The concept of potential energy is not defined for a nonconservative force. Wnc = ΔPE + ΔKE 4 VIII. (6.5) The Conservation of Mechanical Energy ME = KE + PE or Wnc = Ef –Ei Note: if there are no nonconservative forces then… Ef = Ei Energy of a Horizontally Launched Projectile… IX. (6.6) Nonconservative Forces and the Work-Energy Theorem (most moving objects, in reality, experience nonconservative forces… So,) Wnc = Ef –Ei & Wnc = (½ mvf2 + mghf ) – (½ mvi2 + mghi) X. (6.7) Power – Simply the rate at which Work is done. P=W/t (Measured in Joules/s… 1 J/s = 1 Watt) 5 XI. (7.1) Impulse-Momentum Theorem FΔt = mΔv XII. (7.2) Conservation of Linear Momentum - Internal forces – forces that the objects within the system exert on each other - External forces – Forces exerted on the objects by agents external to the system - Remember F12 = -F21 (action-reaction pair) ρ1 + ρ2 = ρ1' + ρ2' m1v1 + m2v2 = m1v1' + m2v2' ** It is important to realize that the total linear momentum may be conserved even when the kinetic energies of the individual parts of a system change (example below) XIII. (7.3) Collisions in One Dimension - Elastic Collision – Total KE before equals total KE after collision - Inelastic Collision - Total KE before does NOT equal total KE after collision. If the objects stick together, the collision is said to be completely elastic under-inflated billiard balls colliding 100% basketball bouncing 90% ELASTIC 30% inflated proton colliding with another proton basketball bouncing 0% INELASTIC 6 XIV. (7.4) Collisions in Two Dimensions XV. (7.5) Center of Mass (Average location of total mass… abbreviated ‘cm’) 7 YOU ARE POWERFUL! Lab Data Sheet PART I: Stairwell Lab Step height __________ # of steps __________ Vertical displacement (m) _________ My Mass (kg) ____________ My Weight (lbs) ____________ My Force (N) ____________ (1 lb = 4.45N) Directions: Walk or run up the stairs from the first to third floor three times. Increase your speed with each trial. Carry a stop watch with you to measure the time. SAFETY FIRST!!! If you feel faint, sick, or weak, STOP!! My Data Trial 1 Trial 2 Trial 3 Average Time (s) Work (J) Power (W) Group Member Mass (kg)* Height of stairs (m) Avg Time (s) Avg Work (J) Avg Power (W) (you) 8 1. Calculate the work for both you and your group members. Show sample calcs for your first trial in your lab notebook. Trial 1 data. 2. Calculate the power for you and your group members. Show work for your first trial in your lab notebook. Trial 1 data. Analysis: (lab notebook) 3. Which of you did more work? Why? 4. Without changing the height of the stairs, could you have done something so that the person that did the least amount of work did more work? Explain. 5. Which of you had more power? Why? 6. Without changing the height of the stairs, could you have done something so that the person who had the least amount of power had more power? Explain. 7. (a) Calculate your power in kilowatts. (b) Calculate your work done in calories. (1000 calories = 4186 J) Applications: 8. Your local electric company charges approximately $0.103 for one kilowatt-hour. Suppose you could climb the stairs continuously for one hour. How much would this climb be worth? (1 J = 2.78x10-7 kWh) 9 PART II: Build-a- Meal Instructions: Each student will construct a meal and then analyze how much of a certain activity must be done in order to “burn off” the energy that was consumed in the meal. The meal must represent a meal you would actually order. Additional side orders are permitted. Use your Nutritional Value Chart to find the number of Calories in each item. Fill in the chart below, then show your calculations (lab notebook). You should show ALL work, use units throughout the problem and use proper problem skills. Useful Information: 1 Cal = 4,184 J 1,000 cal = 1 Cal Your Restaurant: _______________ Total Calories: _________________ Your Meal: Quantity Item Total # Calories Total # Joules 10 PART III: Nutristrategy.com In this section you are now going to use your energy to work. Remember: work = energy and energy = work. First, you are going to do push-ups. You will need to measure your arm’s length and change your weight in lbs to your mass in kg. How many push-ups can you do? How many would you have to do to ‘burn’ off ALL of your caloric intake (a.k.a. your meal)? W=Fxd W = (mass x g)(arm’s length in meters) Show Calculations here: Number of push-ups: Now, choose at least THREE activities from the website http://www.nutristrategy.com/activitylist4.htm. You must calculate how long each activity is done so that at the end of your last activity there are ZERO Calories left from the meal you consumed. Activity #1_________________ How long?______________ Activity #2__________________How long?_______________ Activity #3__________________ How long?_______________ 11 Reference: Figure B A 1.0 kg box, initially at rest and a height of 30.0 m, begins sliding down a frictionless surface. When it reaches the bottom it starts sliding across the horizontal surface, then has a totally inelastic collision with a 2.0 kg box. They stick together and begin to slide off together. After leaving this surface, the system is projected horizontally, from a vertical height of 5.0m. A) Calculate the speed of the 1.0 kg box at the bottom of the hill (before striking M2). B) Calculate the speed of the two boxes after they collide. C) Calculate the speed that the system has when it leaves the surface without becomes a projectile. D) Calculate the horizontal range of this projectile. friction and Figure B: M1 h = 30m M1 = 1 kg M2 = 2 kg M2 y = 5m x = ? m 12 Projectiles & Energy Conservation Pre-Lab Experimental (Method 1) Projectile Method Theoretical (Method 2) Conservation of Energy Method 1. Launch car - measure dx, dy, and h as discussed in class. 2. Determine ‘t’ using projectile method (using dy = Vyi t + ½ at2) 3. Using ‘t’, solve for Vx… Vx = dx / t 4. Solve for Vy… Vyf2 = Vyi2 + 2ad 5. Solve for Vtot, using vector analysis… Pythagorean Theorum 6. Using Vtot, and the KE equation determine the EXPERIMENTAL KE. 7. KE = the energy of the system 1. Determine total energy of the system using the total mass, gravitational acceleration, and initial height of the projectile. 2. PE = mgh, where ‘h’ is the total height of the projectile before the launch (htot = h + Δy as shown below). 3. Notice that the PE determined above is converted into the KE the projectile experiences throughout the “trip”. (PE = KE) 4. THEORETICAL PE at the top is ideally equal to KE at the bottom! h htot dy After solving for the energy of dx the system using both method 1 & 2, determine the “energy loss” of the system! (theoretical minus experimental). Name _______________________ Period _______ Date ___________ 13 CONSERVATION OF ENERGY LABORATORY Purpose: The purpose of this lab is to apply the law of conservation of energy and the principle governing projectile motion to determine the amount of energy lost to friction as a matchbox car rolls down a ramp. Background and Theory: There are several ways that the motion of objects can be analyzed. One way is by using kinematic equations and the idea of independence of horizontal and vertical components of the motion. Another way is by using the Law of Conservation of Mechanical Energy. In this laboratory, both methods will be used in determining the mechanical energy “lost” due to the dissipative force of friction. It is important to remember that the energy is not actually gone; rather it is converted into some other form. When using the Law of Conservation of Mechanical Energy, conditions are assumed to be ideal, with no dissipative forces involved. Realistically, however, as a car rolls down a ramp, the force of friction does lead to a loss of mechanical energy. In calculating the mechanical energy at the bottom of the ramp, the value will be higher than it should be, since the force of friction has not been considered. By finding the actual velocity at the bottom of the ramp using projectile motion methods, a more accurate value of the energy can be obtained. The difference in the energy values of each method gives a good indication of how much energy was “lost” to friction. Materials: matchbox car, ramp, meter stick, tape Procedure: 1. Set up the apparatus as demonstrated. 2. Measure the height (h) to the top of the track in reference to the table top. 3. Measure the height of the table (Δdy). 4. Set your matchbox car at the top of the ramp, launch it, and note its approximate landing position on the floor. 5. Place a piece of carbon paper on top of a piece of notebook paper on the floor where the car landed. 6. Measure the horizontal distance (Δx) from the point of launch at the end of the ramp to the landing location of the car. Data and Calculations: h htot h dy dx2 14 Table 1: Raw Data Trial dy (m) h (m) htot (m) 1 2 3 “ “ “ “ “ “ Table 2: Calculated data Method 1 (projectile motion) Trial 1 2 3 Ave m (kg) “ “ Method 2 (PE = mgh = KE) IN PRACTICE Vx (m/s) dx (m) IN THEORY Vy (m/s) “ “ “ Vtot (m/s) KE (1) (J) PE (J) KE (J) “ “ 0 0 0 0 Energy Loss (J) *Show one sample calculation for each variable of table 2 in lab notebook. Analysis: 1. In which method of calculating kinetic energy should you reach a result that is closer to the “actual” kinetic energy of the car? Why? 2. In which method of calculating kinetic energy will the answer be true only if there was no friction or outside forces acting on the system? 3. In this lab, you calculated an energy loss. Where did this energy go? In what form was it lost? 4. Explain the main idea behind this lab. Specifically: Why are we able to subtract the kinetic energies from each method to find the energy lost to friction? 15 Energy of a Tossed Ball When a juggler tosses a bean ball straight upward, the ball slows down until it reaches the top of its path and then speeds up on its way back down. In terms of energy, when the ball is released it has kinetic energy, KE. As it rises during its free-fall phase it slows down, loses kinetic energy, and gains gravitational potential energy, PE. As it starts down, still in free fall, the stored gravitational potential energy is converted back into kinetic energy as the object falls. If there is no work done by frictional forces, the total energy will remain constant. In this experiment, we will see if this works out for the toss of a ball. Motion Detector No basket necessary, do not let the ball hit the motion detector In this experiment, we will study these energy changes using a Motion Detector. OBJECTIVES Measure the change in the kinetic and potential energies as a ball moves in free fall. See how the total energy of the ball changes during free fall. MATERIALS computer Vernier computer interface Logger Pro Vernier Motion Detector volleyball, basketball, or other similar, fairly heavy ball 16 PRELIMINARY QUESTIONS For each question, consider the free-fall portion of the motion of a ball tossed straight upward, starting just as the ball is released to just before it is caught. Assume that there is very little air resistance. 1. What form or forms of energy does the ball have while momentarily at rest at the top of the path? 2. What form or forms of energy does the ball have while in motion near the bottom of the path? 3. Sketch a graph of velocity vs. time for the ball. 4. Sketch a graph of kinetic energy vs. time for the ball. 5. Sketch a graph of potential energy vs. time for the ball. 6. If there are no frictional forces acting on the ball, how is the change in the ball’s potential energy related to the change in kinetic energy? PROCEDURE 1. Measure and record the mass of the ball you plan to use in this experiment. 2. Connect the Motion Detector to the DIG/SONIC 1 channel of the interface. Place the Motion Detector on the floor and protect it by placing a wire basket over it. 3. Open the file “16 Energy of a Tossed Ball” from the Physics with Vernier folder. 4. Hold the ball directly above and about 1.0 m from the Motion Detector. In this step, you will toss the ball straight upward above the Motion Detector and let it fall back toward the Motion Detector. Have your partner click to begin data collection. Toss the ball straight up after you hear the Motion Detector begin to click. Use two hands. Be sure to pull your hands away from the ball after it starts moving so they are not picked up by the Motion Detector. Throw the ball so it reaches maximum height of about 1.5 m above the Motion Detector. Verify that the position vs. time graph corresponding to the free-fall motion is parabolic in shape, without spikes or flat regions, before you continue. This step may require some practice. If necessary, repeat the toss, until you get a good graph. When you have good data on the screen, proceed to the Analysis section. 17 DATA TABLE Mass of the ball Position Time (s) Height (m) (kg) Velocity (m/s) PE (J) KE (J) TE (J) After release Top of path Before catch ANALYSIS 1. Click on the Examine tool, , and move the mouse across the position or velocity graphs of the motion of the ball to answer these questions. a. Identify the portion of each graph where the ball had just left your hands and was in free fall. Determine the height and velocity of the ball at this time. Enter your values in your data table. b. Identify the point on each graph where the ball was at the top of its path. Determine the time, height, and velocity of the ball at this point. Enter your values in your data table. c. Find a time where the ball was moving downward, but a short time before it was caught. Measure and record the height and velocity of the ball at that time. d. For each of the three points in your data table, calculate the Potential Energy (PE), Kinetic Energy (KE), and Total Energy (TE). Use the position of the Motion Detector as the zero of your gravitational potential energy. 2. How well does this part of the experiment show conservation of energy? Explain. 3. Logger Pro can graph the ball’s kinetic energy according to KE = ½ mv2 if you supply the ball’s mass. To do this, choose Column Options Kinetic Energy from the Data menu. Click the Column Definition tab.You will see a dialog box containing an approximate formula for calculating the KE of the ball. Edit the formula to reflect the mass of the ball and click . 4. Logger Pro can also calculate the ball’s potential energy according to PE = mgh. Here m is the mass of the ball, g the free-fall acceleration, and h is the vertical height of the ball measured from the position of the Motion Detector. As before, you will need to supply the mass of the ball. To do this, choose Column Options Potential Energy from the Data menu. Click the Column Definition tab. You will see a dialog box containing an approximate formula for calculating the PE of the ball. Edit the formula to reflect the mass of the ball and click . 5. Go to the next page by clicking on the Next Page button, . 18 6. Inspect your kinetic energy vs. time graph for the toss of the ball. Explain its shape. 7. Inspect your potential energy vs. time graph for the free-fall flight of the ball. Explain its shape. 8. Record the two energy graphs by printing or sketching. 9. Compare your energy graphs predictions (from the Preliminary Questions) to the real data for the ball toss. 10. Logger Pro will also calculate Total Energy, the sum of KE and PE, for plotting. Record the graph by printing or sketching. 11. What do you conclude from this graph about the total energy of the ball as it moved up and down in free fall? Does the total energy remain constant? 12. Should the total energy remain constant? Why? 13. If it does not, what sources of extra energy are there or where could the missing energy have gone? THOUGHT EXPERIMENTS (DO NOT PERFORM, JUST PREDICT) 1. What would change in this experiment if you used a very light ball, like a beach ball? 2. What would happen to your experimental results if you entered the wrong mass for the ball in this experiment? 3. Predict what would happen in a similar experiment using a bouncing ball. To do this, you would mount the Motion Detector high and pointed downward so it can follow the ball through several bounces. 19 Work and Energy Work is a measure of energy transfer. In the absence of friction, when positive work is done on an object, there will be an increase in its kinetic or potential energy. In order to do work on an object, it is necessary to apply a force along or against the direction of the object’s motion. If the force is constant and parallel to the object’s path, work can be calculated using W F d where F is the constant force and d the displacement of the object. If the force is not constant, we can still calculate the work using a graphical technique. If we divide the overall displacement into short segments, s, the force is nearly constant during each segment. The work done during that segment can be calculated using the previous expression. The total work for the overall displacement is the sum of the work done over each individual segment: W F (d )d This sum can be determined graphically as the area under the plot of force vs. position.1 These equations for work can be easily evaluated using a Force Sensor and a Motion Detector. In either case, the work-energy theorem relates the work done to the change in energy as W = PE + KE where W is the work done, PE is the change in potential energy, and KE the change in kinetic energy. In this experiment you will investigate the relationship between work, potential energy, and kinetic energy. OBJECTIVES Use a Motion Detector and a Force Sensor to measure the position and force on a hanging mass, a spring, and a dynamics cart. Determine the work done on an object using a force vs. position graph. Use the Motion Detector to measure velocity and calculate kinetic energy. Compare the work done on a cart to its change of mechanical energy. MATERIALS computer Vernier computer interface Logger Pro Vernier Motion Detector Vernier Force Sensor 1 masses (200 g and 500 g) spring with a low spring constant (10 N/m) masking tape wire basket (to protect Motion Detector) rubber band, dynamics cart s final If you know calculus you may recognize this sum as leading to the integral W F ( s ) ds . sinitial 20 PRELIMINARY QUESTIONS 1. Lift a book from the floor to the table. Did you do work? To answer this question, consider whether you applied a force parallel to the displacement of the book. 2. What was the average force acting on the book as it was lifted? Could you lift the book with a constant force? Ignore the very beginning and end of the motion in answering the question. 3. Holding one end still, stretch a rubber band. Did you do work on the rubber band? To answer this question, consider whether you applied a force parallel to the displacement of the moving end of the rubber band. 4. Is the force you apply constant when you stretch the rubber band? If not, at what point in the stretch is the force the least. At what point is the force the greatest? PROCEDURE Part I Work When The Force Is Constant In this part you will measure the work needed to lift an object straight upward at constant speed. The force you apply will balance the weight of the object, and so is constant. The work can be calculated using the displacement and the average force, and also by finding the area under the force vs. position graph. 1. Connect the Motion Detector to DIG/SONIC 1 of the interface. Connect the Vernier Force Sensor to Channel 1 of the interface. Set the range switch to 10 N. 2. Open the file “18a Work and Energy” from the Physics with Computers folder. Three graphs will appear on the screen: position vs. time, force vs. time, and force vs. position. Data will be collected for 5 s. 3. You may choose to calibrate the Force Sensor, or you can skip this step. a. Choose Calibrate CH1: Dual Range Force from the Experiment menu. Click . b. Remove all force from the Force Sensor. Enter a 0 (zero) in the Value 1 field. Hold the sensor vertically with the hook downward and wait for the reading shown for Reading 1 to stabilize. Click . This defines the zero force condition. c. Hang the 500 g mass from the Force Sensor. This applies a force of 4.9 N. Enter 4.9 in the Value 2 field, and after the reading shown for Reading 2 is stable, click . Click to close the calibration dialog. 4. Hold the Force Sensor with the hook pointing downward, but with no mass hanging from it. Click , select only the Force Sensor from the list, and click to set the Force Sensor to zero. 21 5. Hang a 200 g mass from the Force Sensor. 6. Place the Motion Detector on the floor, away from table legs and other obstacles. Place a wire basket over it as protection from falling weights. Dual-Range Force Sensor 7. Hold the Force Sensor and mass about 0.5 m above the Motion Detector. Click to begin data collection. Wait about 1.0 s after the clicking sound starts, and then slowly raise the Force Sensor and mass about 0.5 m straight upward. Then hold the sensor and mass still until the data collection stops at 5 s. 8. Examine the position vs. time and force vs. time graphs by clicking the Examine button, . Identify when the weight started to move upward at a constant speed. Record this starting time and height in the data table. 9. Examine the position vs. time and force vs. time graphs and identify when the weight stopped moving upward. Record this stopping time and height in the data table. Figure 1 10. Determine the average force exerted while you were lifting the mass. Do this by selecting the portion of the force vs. time graph corresponding to the time you were lifting (refer to the position graph to determine this time interval). Do not include the brief periods when the up motion was starting and stopping. Click the Statistics button, , to calculate the average force. Record the value in your data table. 11. On the force vs. position graph select the region corresponding to the upward motion of the weight. (Click and hold the mouse button at the starting position, then drag the mouse to the stopping position and release the button.) Click the Integrate button, , to determine the area under the force vs. position curve during the lift. Record this area in the data table. 12. Save the graphs (optional). Part II Work Done To Stretch A Spring In Part II you will measure the work needed to stretch a spring. Unlike the force needed to lift a mass, the force done in stretching a spring is not a constant. The work can still be calculated using the area under the force vs. position graph. 13. Open the file “18b Work Done Spring” from the Physics with Computers folder Three graphs will appear on the screen: position vs. time, force vs. time, and force vs. position. Data will be collected for 5 seconds. 14. Attach one end of the spring to a rigid support. Attach the Force Sensor hook to the other end. Rest the Force Sensor on the table with the spring extended but relaxed, so that the spring applies no force to the Force Sensor. 22 15. Place the Motion Detector about one meter from the Force Sensor, along the line of the spring. Be sure there are no nearby objects to interfere with the position measurement. Motion Detector Force Sensor Dual-Range Force Sensor Figure 2 16. Using tape, mark the position of the leading edge of the Force Sensor on the table. The starting point is when the spring is in a relaxed state. Hold the end of the Force Sensor that is nearest the Motion Detector as shown in Figure 3. The Motion Detector will measure the distance to your hand, not the Force Sensor. With the rest of your arm out of the way of the Motion Detector beam, click . On the dialog box that appears, make sure that both sensors are highlighted, and click . Logger Pro will now use a coordinate system which is positive towards the Motion Detector with the origin at the Force Sensor. Force Sensor Motion Detector Figure 3 17. Click to begin data collection. Within the limits of the spring, move the Force Sensor and slowly stretch the spring about 30 to 50 cm over several seconds. Hold the sensor still until data collection stops. Do not get any closer than 40 cm to the Motion Detector. 18. Examine the position vs. time and force vs. time graphs by clicking the Examine button, . Identify the time when you started to pull on the spring. Record this starting time and position in the data table. 19. Examine the position vs. time and force vs. time graphs and identify the time when you stopped pulling on the spring. Record this stopping time and position in the data table. 20. Click the force vs. position graph, then click the Linear Fit button, , to determine the slope of the force vs. position graph. The slope is the spring constant, k. Record the slope and intercept in the data table. 21. The area under the force vs. position graph is the work done to stretch the spring. How does the work depend on the amount of stretch? On the force vs. position graph select the region corresponding to the first 10 cm stretch of the spring. (Click and hold the mouse button at the starting position, then drag the mouse to 10 cm and release the button.) Click the Integrate button, , to determine the area under the force vs. position curve during the stretch. Record this area in the data table. 23 22. Now select the portion of the graph corresponding to the first 20 cm of stretch (twice the stretch). Find the work done to stretch the spring 20 cm. Record the value in the data table. 23. Select the portion of the graph corresponding to the maximum stretch you achieved. Find the work done to stretch the spring this far. Record the value in the data table. 24. Save the graphs (optional). Part III Work Done To Accelerate A Cart In Part III you will push on the cart with the Force Sensor, causing the cart to accelerate. The Motion Detector allows you to measure the initial and final velocities; along with the Force Sensor, you can measure the work you do on the cart to accelerate it. 25. Open the file “18c Work Done Cart”. Three graphs will appear on the screen: position vs. time, force vs. time, and force vs. position. Data will be collected for 5 seconds. 26. Remove the spring and support. Determine the mass of the cart. Record in the data table. 27. Place the cart at rest about 1.5 m from the Motion Detector, ready to roll toward the detector. 28. Click . Check to see that both sensors are highlighted in the Zero Sensors Calibration box and click . Logger Pro will now use a coordinate system which is positive towards the Motion Detector with the origin at the cart, and a push on the Force Sensor is positive. 29. Prepare to gently push the cart toward the Motion Detector using the Force Sensor. Hold the Force Sensor so the force it applies to the cart is parallel to the sensitive axis of the sensor. 30. Click to begin data collection. When you hear the Motion Detector begin clicking, gently push the cart toward the detector using only the hook of the Force Sensor. The push should last about half a second. Let the cart roll toward the Motion Detector, but catch it before it strikes the detector. 31. Examine the position vs. time and force vs. time graphs by clicking the Examine button, . Identify when you started to push the cart. Record this time and position in the data table. 32. Examine the position vs. time and force vs. time graphs and identify when you stopped pushing the cart. Record this time and position in the data table. 33. Determine the velocity of the cart after the push. Use the slope of the position vs. time graph, which should be a straight line after the push is complete. Record the slope in the data table. 34. From the force vs. position graph, determine the work you did to accelerate the cart. To do this, select the region corresponding to the push (but no more). Click the Integrate button, , to measure the area under the curve. Record the value in the data table. 35. Save the graphs (optional). 24 DATA TABLE Part I Time (s) Position (m) Start Moving Stop Moving Average force(N) Work done (J) Integral (during lift): force vs. position (N•m) PE (J) Part II Time (s) Position (m) Start Pulling Stop Pulling Spring Constant (N/m) Stretch 10 cm 20 cm Maximum Integral (during pull) (N•m) PE (J) Part III Time (s) Position (m) Start Pushing Stop Pushing Mass (kg) Final velocity (m/s) Integral during push (N•m) KE of cart (J) 25 ANALYSIS 1. In Part I, the work you did lifting the mass did not change its kinetic energy. The work then had to change the potential energy of the mass. Calculate the increase in gravitational potential energy using the following equation. Compare this to the average work for Part I, and to the area under the force vs. position graph: PE = mgh where h is the distance the mass was raised. Record your values in the data table. Does the work done on the mass correspond to the change in gravitational potential energy? Should it? 2. In Part II you did work to stretch the spring. The graph of force vs. position depends on the particular spring you used, but for most springs will be a straight line. This corresponds to Hooke’s law, or F = – kx, where F is the force applied by the spring when it is stretched a distance x. k is the spring constant, measured in N/m. What is the spring constant of the spring? From your graph, does the spring follow Hooke’s law? Do you think that it would always follow Hooke’s law, no matter how far you stretched it? Why is the slope of your graph positive, while Hooke’s law has a minus sign? 3. The elastic potential energy stored by a spring is given by PE = ½ kx2, where x is the distance. Compare the work you measured to stretch the spring to 10 cm, 20 cm, and the maximum stretch to the stored potential energy predicted by this expression. Should they be similar? Note: Use consistent units. Record your values in the data table. 4. In Part III you did work to accelerate the cart. In this case the work went to changing the kinetic energy. Since no spring was involved and the cart moved along a level surface, there is no change in potential energy. How does the work you did compare2 to the change in kinetic energy? Here, since the initial velocity is zero, KE = ½ mv where m is the total mass of the cart and any added weights, and v is the final velocity. Record your values in the data table. EXTENSIONS 1. Show that one N•m is equal to one J. 2. Start with a stretched spring and let the spring do work on the cart by accelerating it toward the fixed point. Use the Motion Detector to determine the speed of the cart when the spring reaches the relaxed position. Calculate the kinetic energy of the cart at this point and compare it to the work measured in Part II. Discuss the results. 3. Repeat Part I, but vary the speed of your hand as you lift the mass. The force vs. time graph should be irregular. Will the force vs. position graph change? Or will it continue to correspond to mgh? 4. Repeat Part III, but start with the cart moving away from the detector. Pushing only with the tip of the Force Sensor, gently stop the cart and send it back toward the detector. Compare the work done on the cart to the change in kinetic energy, taking into account the initial velocity of the cart. 26 Egg Drop Lab Theory and Inquiry I. Explain the theory behind the egg drop lab impulse = Ft = mv = Explain where the equation above came from and how it helps in constructing the egg drop apparatus. II. Choose an equation from your equation sheet that will allow you to solve for the final velocity of your egg. Calculate the final velocity of your egg. You have to measure a variable in order to solve for the final velocity. What is the variable? Scoring: 1. Complete all questions in lab notebook 2. Egg drop apparatus is constructed 3. Egg survives drop from 1st flight of bleachers 4. Egg survives drop from top of bleachers Total: 10 pts 5 pts 5 pts +5 pts 20 pts must be able to easily place and remove egg in apparatus may only use cardboard, drinking straws, paper, and non-padded tape may not use any parachute apparatus apparatus may not exceed an 8 inch cube or sphere in size 27 Momentum Lab Preliminary Questions: 1. Momentum is “__________________________________________” 2. List three examples of objects with a large amount of momentum Instructions: 1. Set up the lab as indicated (and shown below) 2. Set the carts into motion by activating the spring mechanism 3. Calculate their momenta at the two points on the track. 4. IN YOUR LAB NOTEBOOK… fill in all of the data table 5. REMEMBER… Trials 6-9 are done on a slightly inclined track Photogates Flag Calculations: Since you cut a flag for the cart that is 2.54 cm (0.0254 m) wide, the velocity of the cart can be calculated in the following manner… V = ∆d/t * Where ∆d is 0.0254 m and the time is given by the photogate Analysis Questions: 1. The momentum should not be the same at both positions… Why? 2. What would you find if a third photogate was placed along the track? 28 Data at Position #1 Trial # Horizontal track Incline in track Mass of Cart (kg) Extra mass (kg) Total Mass (kg) Time at Position #1 (s) Velocity at Position #1 (m/s) Momentum at Position #1 (kg m/s) 1 2 3 4 5 6 7 8 9 Data at Position #2 Trial # Horizontal track Incline in track Time at Position #2 (s) Velocity at Position #2 (m/s) Momentum at Position #2 (kg m/s) “Lost” / Gained Momentum (kg m/s) 1 2 3 4 5 6 7 8 9 29 Explosion Lab Investigating the Law of Conservation of Momentum NAME:_____________________ m1V1 + m2V2 = m1V1' + m2V2' Cart #1 (Cart moving to the left after explosion) Trial # Mass of Cart (kg) 1 2 3 4 5 6 7 8 9 .5 .5 .5 .5 .5 .5 .5 .5 .5 Additional Mass in Cart (kg) 0 .1 .2 .3 .4 .5 .6 .7 .8 Total Mass (kg) Time Velocity Momentum (s) (m/s) (kg m/s) Cart #2 (Cart moving to the right after explosion) Trial # 1 2 3 4 5 6 7 8 9 Mass of Cart (kg) .5 .5 .5 .5 .5 .5 .5 .5 .5 Additional Mass in Cart (kg) Total Mass (kg) Time Velocity Momentum Difference in (s) (m/s) (kg m/s) Momentum of Two Carts (kg m/s) 0 0 .1 .2 .3 .4 .5 .6 .7 * Was there a difference in the momentum of the two carts? If there was, why? If there wasn’t, why not? 30 Momentum and Collisions The collision of two carts on a track can be described in terms of momentum conservation. If there is no net external force experienced by the system of two carts, then we expect the total momentum of the system to be conserved. This is true regardless of the force acting between the carts. Collisions are classified as elastic (kinetic energy is conserved), inelastic (kinetic energy is lost) or completely inelastic (the objects stick together after collision). In this experiment you can observe most of these types of collisions and test for the conservation of momentum and energy in each case. OBJECTIVES Observe collisions between two carts, testing for the conservation of momentum. Classify collisions as elastic, inelastic, or partially elastic. MATERIALS computers Vernier computer interface Logger Pro two Vernier Motion Detectors dynamics cart track two low-friction dynamics carts with magnetic and Velcro™ bumpers PRELIMINARY QUESTIONS 1. Consider a head-on collision between two billiard balls. One is initially at rest and the other moves toward it. Sketch a position vs. time graph for each ball, starting with time before the collision and ending a short time afterward. 2. As you have drawn the graph, is momentum conserved in this collision? PROCEDURE 1. Measure the masses of your carts and record them in your data table. Label the carts as cart 1 and cart 2. 2. Set up the track so that it is horizontal. Test this by releasing a cart on the track from rest. The cart should not move. 3. Practice creating gentle collisions by placing cart 2 at rest in the middle of the track, and release cart 1 so it rolls toward the first cart, magnetic bumper toward magnetic bumper. The carts should smoothly repel one another without physically touching. 4. Place a Motion Detector at each end of the track, allowing for the 0.4 m minimum distance between detector and cart. Connect the Motion Detectors to the DIG/SONIC 1 and DIG/SONIC 2 channels of the interface. 5. Open the file “18 Momentum Coll” from the Physics with Vernier folder. 31 6. Click to begin taking data. Repeat the collision you practiced above and use the position graphs to verify that the Motion Detectors can track each cart properly throughout the entire range of motion. You may need to adjust the position of one or both of the Motion Detectors. 7. Place the two carts at rest in the middle of the track, with their Velcro bumpers toward one another and in contact. Keep your hands clear of the carts and click . Select both sensors and click . This procedure will establish the same coordinate system for both Motion Detectors. Verify that the zeroing was successful by clicking and allowing the still-linked carts to roll slowly across the track. The graphs for each Motion Detector should be nearly the same. If not, repeat the zeroing process. Part I: Magnetic Bumpers 8. Reposition the carts so the magnetic bumpers are facing one another. Click to begin taking data and repeat the collision you practiced in Step 3. Make sure you keep your hands out of the way of the Motion Detectors after you push the cart. 9. From the velocity graphs you can determine an average velocity before and after the collision for each cart. To measure the average velocity during a time interval, drag the cursor across the interval. Click the Statistics button to read the average value. Measure the average velocity for each cart, before and after collision, and enter the four values in the data table. Delete the statistics box. 10. Repeat Step 9 as a second run with the magnetic bumpers, recording the velocities in the data table. Part II: Velcro Bumpers 11. Change the collision by turning the carts so the Velcro bumpers face one another. The carts should stick together after collision. Practice making the new collision, again starting with cart 2 at rest. 12. Click to begin taking data and repeat the new collision. Using the procedure in Step 9, measure and record the cart velocities in your data table. 13. Repeat the previous step as a second run with the Velcro bumpers. Part III: Velcro to Magnetic Bumpers 14. Face the Velcro bumper on one cart to the magnetic bumper on the other. The carts will not stick, but they will not smoothly bounce apart either. Practice this collision, again starting with cart 2 at rest. 15. Click to begin data collection and repeat the new collision. Using the procedure in Step 9, measure and record the cart velocities in your data table. 16. Repeat the previous step as a second run with the Velcro to magnetic bumpers. 32 DATA TABLE Mass of cart 1 (kg) Run number Run number Mass of cart 2 (kg) Velocity of cart 1 before collision Velocity of cart 2 before collision Velocity of cart 1 after collision Velocity of cart 2 after collision (m/s) (m/s) (m/s) (m/s) 1 0 2 0 3 0 4 0 5 0 6 0 Momentum of cart 1 before collision (kg•m/s) Momentum of cart 2 before collision (kg•m/s) 1 0 2 0 3 0 4 0 5 0 6 0 Momentum of cart 1 after collision (kg•m/s) 90% ELASTIC proton colliding with another proton Total momentum before collision (kg•m/s) Total momentum after collision (kg•m/s) Ratio of total momentum after/before under-inflated basketball bouncing billiard balls colliding 100% Momentum of cart 2 after collision (kg•m/s) 30% inflated basketball bouncing 0% INELASTIC 33 In this lab, we will collide two collision carts together to determine if momentum is conserved in all types of collisions. To do this, we will obviously need to measure the velocity and mass of each cart both before and after the collision. The above diagram might help you keep things straight. ANALYSIS 1. Determine the momentum (mv) of each cart before the collision, after the collision, and the total momentum before and after the collision. Calculate the ratio of the total momentum after the collision to the total momentum before the collision. Enter the values in your data table. 2. If the total momentum for a system is the same before and after the collision, we say that momentum is conserved. If momentum were conserved, what would be the ratio of the total momentum after the collision to the total momentum before the collision? 3. For your six runs, inspect the momentum ratios. Even if momentum is conserved for a given collision, the measured values may not be exactly the same before and after due to measurement uncertainty. The ratio should be close to one, however. Is momentum conserved in your collisions? 4. Classify the three collision types as elastic, inelastic, or completely inelastic. EXTENSIONS 1. Using a collision cart with a spring plunger, create a super-elastic collision; that is, a collision where kinetic energy increases. The plunger spring should be compressed and locked before the collision, but then released during the collision. Measure momentum before and after the collision. Is momentum conserved in this case? 2. Using the magnetic bumpers, consider other combinations of cart mass by adding weight to one cart. Is momentum conserved in these collisions? 3. Using the magnetic bumpers, consider other combinations of initial velocities. Begin with having both carts moving toward one another initially. Is momentum conserved in these collisions? * Answer in Lab Notebook!!! 34 Impulse Momentum Theorem The impulse-momentum theorem relates impulse, the average force applied to an object times the length of time the force is applied, and the change in momentum of the object: F t mv f mvi Here we will only consider motion and forces along a single line. The average force, F , is the net force on the object, but in the case where one force dominates all others it is sufficient to use only the large force in calculations and analysis. For this experiment, a dynamics cart will roll along a level track. Its momentum will change as it reaches the end of an initially slack elastic tether cord, much like a horizontal bungee jump. The tether will stretch and apply an increasing force until the cart stops. The cart then changes direction and the tether will soon go slack. The force applied by the cord is measured by a Force Sensor. The cart velocity throughout the motion is measured with a Motion Detector. Using Logger Pro to find the average force during a time interval, you can test the impulse-momentum theorem. Motion Detector Force Sensor Elastic cord OBJECTIVES * Measure a cart’s momentum change and compare to the impulse it receives. * Compare average and peak forces in impulses. MATERIALS computer Vernier computer Logger Pro Vernier Motion Detector Vernier Force Sensor dynamics cart and track clamp elastic cord string 500 g mass 35 PRELIMINARY QUESTIONS 1. In a car collision, the driver’s body must change speed from a high value to zero. This is true whether or not an airbag is used, so why use an airbag? How does it reduce injuries? 2. You want to close an open door by throwing either a 400 g lump of clay or a 400 g rubber ball toward it. You can throw either object with the same speed, but they are different in that the rubber ball bounces off the door while the clay just sticks to the door. Which projectile will apply the larger impulse to the door and be more likely to close it? PROCEDURE 1. Measure the mass of your dynamics cart and record the value in the data table. 2. Connect the Motion Detector to DIG/SONIC 1 of the interface. Connect the Force Sensor to Channel 1 of the interface. If your Force Sensor has a range switch, set it to 10 N. 3. Open the file “20 Impulse and Momentum” in the Physics with Computers folder. Logger Pro will plot the cart’s position and velocity vs. time, as well as the force applied by the Force Sensor vs. time. 4. Optional: Calibrate the Force Sensor. a. Choose Calibrate CH1: Dual Range Force from the Experiment menu. Click . b. Remove all force from the Force Sensor. Enter a 0 (zero) in the Reading 1 field. Hold the sensor vertically with the hook downward and wait for the reading shown for CH1 to stabilize. Click . This defines the zero force condition. c. Hang the 500 g mass from the sensor. This applies a force of 4.9 N. Enter 4.9 in the Reading 2 field, and after the reading shown for CH1 is stable, click . Click to close the calibration dialog. 5. Place the track on a level surface. Confirm that the track is level by placing the lowfriction cart on the track and releasing it from rest. It should not roll. If necessary, adjust the track. 6. Attach the elastic cord to the cart and then the cord to the string. Tie the string to the Force Sensor a short distance away. Choose a string length so that the cart can roll freely with the cord slack for most of the track length, but be stopped by the cord before it reaches the end of the track. Clamp the Force Sensor so that the string and cord, when taut, are horizontal and in line with the cart’s motion. 7. Place the Motion Detector beyond the other end of the track so that the detector has a clear view of the cart’s motion along the entire track length. When the cord is stretched to maximum extension the cart should not be closer than 0.4 m to the detector. 36 8. Click Sensor. , select Force Sensor from the list, and click to zero the Force 9. Practice releasing the cart so it rolls toward the Motion Detector, bounces gently, and returns to your hand. The Force Sensor must not shift and the cart must stay on the track. Arrange the cord and string so that when they are slack they do not interfere with the cart motion. You may need to guide the string by hand, but be sure that you do not apply any force to the cart or Force Sensor. Keep your hands away from between the cart and the Motion Detector. 10. Click to take data; roll the cart and confirm that the Motion Detector detects the cart throughout its travel. Inspect the force data. If the peak exceeds 10 N, then the applied force is too large. Roll the cart with a lower initial speed. If the velocity graph has a flat area when it crosses the time-axis, the Motion Detector was too close and the run should be repeated. 11. Once you have made a run with good position, velocity, and force graphs, analyze your data. To test the impulse-momentum theorem, you need the velocity before and after the impulse. Choose an interval corresponding to a time when the elastic was initially relaxed, and the cart was moving at approximately constant speed away from the Force Sensor. Drag the mouse pointer across this interval. Click the Statistics button, , and read the average velocity. Record the value for the initial velocity in your data table. In the same manner, choose an interval corresponding to a time when the elastic was again relaxed, and the cart was moving at approximately constant speed toward the Force Sensor. Drag the mouse pointer across this interval. Click the statistics button and read the average velocity. Record the value for the final velocity in your data table. 12. Now record the time interval of the impulse. There are two ways to do this. Use the first method if you have studied calculus and the second if you have not. * Method 1: Calculus tells us that the expression for the impulse is equivalent to the integral of the force vs. time graph, or t final F t F (t )dt tinitial On the force vs. time graph, drag across the impulse, capturing the entire period when the force was non-zero. Find the area under the force vs. time graph by clicking the integral button, . Record the value of the integral in the impulse column of your data table. * Method 2: On the force vs. time graph, drag across the impulse, capturing the entire period when the force was non-zero. Find the average value of the force by clicking the Statistics button, , and also read the length of the time interval over which your average force is calculated. The number of points used in the average divided by the data rate of 50 Hz gives the time interval t. Record the values in your data table. 37 13. Perform a second trial by repeating Steps 10 – 12, record the information in your data table. 14. Change the elastic material attached to the cart. Use a new material, or attach two elastic bands side by side. 15. Repeat Steps 10 – 13, record the information in your data table. DATA TABLE Mass of cart kg Trial Final Velocity vf Initial Velocity vi Average Force F Duration of Impulse t Impulse Elastic 1 (m/s) (m/s) (N) (s) (Ns) 1 2 Elastic 2 1 2 Trial Elastic 1 Impulse Ft (Ns) Change in momentum (kgm /s) or (Ns) % difference between Impulse and Change in momentum (Ns) 1 2 Elastic 2 1 2 38 ANALYSIS 1. From the mass of the cart and change in velocity, determine the change in momentum as a result of the impulse. Make this calculation for each trial and enter the values in the second data table. 2. If you used the average force (non-calculus) method, determine the impulse for each trial from the average force and time interval values. Record these values in your data table. 3. If the impulse-momentum theorem is correct, the change in momentum will equal the impulse for each trial. Experimental measurement errors, along with friction and shifting of the track or Force Sensor, will keep the two from being exactly the same. One way to compare the two is to find their percentage difference. Divide the difference between the two values by the average of the two, then multiply by 100%. How close are your values, percentage-wise? Do your data support the impulsemomentum theorem? 4. Look at the shape of the last force vs. time graph. Is the peak value of the force significantly different from the average force? Is there a way you could deliver the same impulse with a much smaller force? 5. Revisit your answers to the Preliminary Questions in light of your work with the impulse-momentum theorem. 6. When you use different elastic materials, what changes occurred in the shapes of the graphs? Is there a correlation between the type of material and the shape? 7. When you used a stiffer or tighter elastic material, what effect did this have on the duration of the impulse? What affect did this have on the maximum size of the force? Can you develop a general rule from these observations? EXTENSIONS 1. Try other elastic materials, doing the same experiment. 39 Momentum Weblab Using the following link, complete the following questions; http://media.pearsoncmg.com/bc/aw_young_physics_11/pt1a/Media/Momentum/Collisio nsElasticity/Main.html 40 NOTES 41