SPH4U: Energy and Momentum Course Website:

Transcription

1 SPH4U: Energy and Momentum Course Website: This syllabus contains a list of all classes, topics and homework in the Gr. 12 Kinematics Unit. You are strongly encouraged to explore the simulations and videos listed for each lesson they are optional but quite interesting! Day Topics Homework Extras 1 Oomph Read: pg. 232, Handbook: Oomph parts C and D, pg. 243 #5 2 The Idea of Conservation 2 Types of Collisions CGPS 4 Car Crash! Conservation of Momentum in 2D Read: pg Problems: pg. 237 #8, 1, pg. 248 #6 Read: pg Problems: pg. 252 #12, 13, 14 Read: pg Handbook: 2D Momentum Problem Solving Problems: pg. 257 #3 Active Physics: Save an astronaut Video: Intro to Momentum Lesson: Conservation of Momentum Active Physics: Momentum and Energy Lesson: Isolated Systems Video: Slow Motion Collision Video: Impulse Lesson: Impulse Active Physics: Elasticity Video: Elastic and Inelastic Collisions Video: 1968 Crash Test 5 2D Collisions Problem: pg. 257 #5, pg. 258 #3 Active Physics: P and E Conservation Simulation: Collision Lab 6 Work and Kinetic Energy Problems: pg. 181 #7, pg. 183 #7, pg. 186 #4 Active Physics: Work Simulation: The Ramp 7 Collisions Quiz, Energy and Coordinate Systems Problems: pg. 197 #6, pg. 21 #5, pg. 227 #17 Video: Roller Coasters Lesson: Work 8 Energy Transfers Problems: pg. 2 #13, pg. 22 #8, 1 Handbook: Transfers of Energy Active Physics: Energy Bar Charts Simulation: Energy Skate Park 9 The Ballistics Problem: pg. 27 #2 Active Physics: Pendulum Bashes Box Pendulum 1 Spring Force and Energy Read: pg Problems: pg. 26 #5 Video: Hooke s Law Video: Energy in Springs 11 Spring Force and Energy, continued Read: pg Problems: pg. 211 #1, 12a, 13 Active Physics: Inverse Bungee Simulation: Masses and Springs 12 CGPS Problems: pg. 229 #38 Review: Momentum Review: Work and Energy Note: There are many questions in the above links that do not apply to what we have discussed. If it looks unfamiliar.skip it and don t worry about it. 13 Test Review: pg. 37 #4, 5, 1, 14, 18, 2, 26, 48 1

2 SPH4U: Oomph When you catch a heavy object you feel a lot of Oomph. What is this mysterious quantity that we all kind of know? Let s find out. Recorder: Manager: Speaker: A: Figuring Out the Formula for Oomph! The more oomph something has, the harder it is to stop, and the more ability it has to knock other things over. Let s figure out the formula for oomph. 1. Reason. A small pebble and a larger rock are thrown at the same speed. (a) Which one has more oomph? Why? (b) The rock is twice as massive as the pebble. Intuitively, how does the rock s oomph compare to the pebble s oomph? Is it twice as big? Half as big? Three times as big? 2. Reason. Picture two identical bowling balls, one of which is rolling faster than the other. (a) Which ball, the faster or slower one, has more oomph? Why? (b) The faster ball is exactly 7 times as fast as the slower one. Intuitively, how does the faster ball s oomph compare to the slower ball s oomph? 3. Find a Relationship. The physics concept corresponding to oomph is momentum. Building on your above answers, figure out a formula for momentum (oomph) in terms of mass and velocity. Explain how the formula expresses your intuitions from parts A and B above. (For nutty historical reasons, physicists use the letter p for momentum.). ** check with your teacher at this point ** B: Testing Our Momentous Intuitions (together as a simulation: explorelearning.com) You will need a dynamics track and two carts. In the previous section, your intuitions about oomph led to a formula for momentum. Now let s see if your ideas hold true for collisions. Cart A (1 kg) is rolling with negligible friction at 3 m/s and collides with and sticks to cart B (identical to cart A). So, after colliding, the carts roll together as a single unit. 1 kg 1 kg 1. Predict. Don t do it yet! Using your intuitions, guess the postcollision speed of the two carts. Briefly explain your reasoning. Moment 2? 3 m/s Moment 1 2 C. Meyer, 214, adapted from University of Maryland Physics Education Research Group, Fall 24

3 2. Test. Use the carts and dynamics track to test your prediction. Roughly speaking, do your observations agree with your prediction? Did the first cart gain or lose momentum? What about the second cart? What about the system of two carts, did it gain or lose momentum? To help represent the momenta of a system before and after a process like a collision we can construct an impulsemomentum bar chart. In our notation, we use letters to denote different objects and numbers to indicate different moments in time. We draw a bar that represents the size and direction of the momentum of each object in the system. The exact heights are not important, but the bars must clearly show the correct ideas. The change in the momentum of the system is called the impulse and is represented by the symbol J. 3. Represent and Explain. Complete the momentumimpulse bar chart for the system of the two carts before and after the collision described above. For convenience, you can think of the cart A as having 3 1 = 3 units of momentum. Once complete, explain two ways in which the graph visually represents the fact that the impulse on the system was zero. p A1 p B1 J p A2 p B2 Here is another situation to consider: in a similar experiment, cart A collides with cart B magnetically. The two carts don t actually touch the magnets act like a perfect spring between the two carts. After the collision, cart A is at rest. 4. Predict. Again using intuitions, predict the postcollision speed of cart B. 3 m/s Moment 1 1 kg 1 kg Moment 2 stopped? 5. Test. Use the carts and dynamics track to test your prediction. Roughly speaking do your observations agree with your prediction? Did the first cart gain or lose momentum? What about the second cart? What about the system of two carts, did it gain or lose momentum? Complete the momentumimpulse bar chart. p A1 p B1 J p A2 p B2 6. Summarize. Based on your work above, state a general rule about how the total momentum of a system changes during a collision. Here is one last situation to try out. The two carts are initially moving at 3 m/s in opposite directions. They collide and stick using Velcro. 7. Predict and Test. Intuitively, after the collision, how fast do the carts move and in what direction? Test your prediction. 3 m/s BEFORE 3 m/s 1 kg 1 kg AFTER? 3

4 8. Reason. In all cart collisions explored above, momentum was conserved; it was the same before and after the collision. Because conserved quantities are useful in problemsolving, it would be cool if we could define momentum in such a way that it s always conserved in collisions (between objects that are free to move). Is there some way to modify or clarify the momentum formula you figured previously so that momentum is conserved in the headon collision between the two carts? Explain. (Hint: Maybe oomph cares about direction.) 9. Represent. Complete a momentumimpulse bar chart for this collision. Explain how the idea of direction is visually represented in the chart. p A1 p B1 J p A2 p B2 ** check with your teacher ** C: The Conservation of Momentum Conservation of momentum is a fundamental physical law. Among other things, it says that when two objects collide, the total momentum of the system immediately after the collision equals the total momentum of the system immediately before the collision: Conservation of momentum: mava 1 mbvb 1 mava2 mbvb2 Since p mv, and since velocity cares about direction, so does momentum. So, a negative oomph (momentum) can partially or fully cancel a positive oomph, as the Velcro carts demonstrated. Problem. Let s practice using momentum conservation. A 5 g cart moves east at 1. m/s and collides with a 1 g cart that is at rest. The carts bounce magnetically off each other. After the bounce, the 1 g car is moving east at.6 m/s. 1. Represent. A good, first step in any conservation problem is sketching the initial and final states of the process. These are two key events. Complete Part A below. A: Pictorial Representation Sketch showing before and after, coordinate system, label givens and unknowns with symbols, conversions, describe events 2. Predict. Without doing calculations, make a quick guess for the final direction of motion of the 5 g car. Briefly explain your reasoning. 3. Reason. Are there any important interactions between the system objects and the external environment? Will these affect the momentum? Explain. 4

5 4. Represent. Complete part B for the process. Draw the diagrams for the two carts during the collision. B: Physics Representation momentum bar chart, interaction diagram, force diagram p A1 p B1 J = p A2 p B2 5. Calculate. Now calculate the 5 g car s speed and direction of motion after the collision. Use the bar chart to help construct your momentum equation (leave out any terms you know are zero). D: Mathematical Representation Complete equations, describe steps, algebraic work, substitutions with units, final statement 6. Evaluate. Is your final answer reasonable (size, magnitude and direction)? Does it agree with your prediction? E: Evaluation Answer has reasonable size, direction and units? Why? 7. Test. Observe the simulation. Does our momentum law work? D: A Further Look 1. Represent. Some of you are probably wondering about impulse. We didn t see any examples where there was an impulse. Try this: p A1 J A = p A2 complete a momentumimpulse bar chart for the system of cart A and for the system of cart B in the first collision example (B#1). Draw a FD for each cart below the bar chart. 2. Explain. What does each chart and FD tells us about the impulse experienced by the individual carts? P B1 J B = p B2 FD FD 5

6 SPH4U: The Idea of Conservation There are some situations where momentum seems to appear or disappear. Let s study one of these situations carefully. A: The Slowing Block A 1. kg block initially sliding at 1.5 m/s along a rough surface comes to a stop. Recorder: Manager: Speaker: p 1 J = p 2 1. Represent. Complete a momentumimpulse chart for the system of the block. How much momentum did the system lose? 2. Reason. Are there any important interactions between the system objects and the external environment? How does this help to explain the loss of momentum? The Momentum Conservation Condition: The momentum of a system is conserved when the net force the system experiences is zero. The total momentum of a system can change if the system experiences a net force from its environment (objects outside the system). This change, also known as the impulse, is related to the net force and the amount of time the force acts: p J F t. When F net (or J) = the system s momentum is conserved. Use this as the criteria to decide system whether to use momentum to solve a problem. net 3. Represent. Draw an ID and FD for the system of the block. Write an expression for the net force. ID FD 4. Speculate. Consider the force responsible for slowing the block. What is the other force in a 3 rd law pair with that force? Use that other force to help you guess where the block s momentum went. Make a guess and move on! B: The Block on a Track (demonstration) The block is moving at 1.5 m/s, just like before, and is gently lowered on to a level track that is supported on wheels and is free to move (no friction). The track has a mass of 2.3 kg and is initially at rest. 1. Predict. What will happen after the block is released? 2. Test and Observe. Use the equipment at the front of the class to test our two predictions. Describe how your observations help to confirm your predictions. 6

7 3. Represent and Reason. Complete a momentumimpulse bar chart for the blocktrack system. What is the net force on the blocktrack system? Explain. p A1 p B1 J = p A2 p B2 4. Calculate. What is the final velocity of the track? 5. Reason. Imagine the mass of the track was increased enormously to equal that of the earth. Describe what would be different. How does this relate to the situation of part A? C: The Process of a Collision Collisions often occur very quickly and we don t usually notice what is actually happening during a collision. In this example, cart A (1. kg) collided with a smaller cart B (.5 kg) using an uncompressed spring. The velocity of each cart was recorded at 9 moments in time and used to calculate the momentum and kinetic energy. A third line on each graph represents the total momentum and total kinetic energy of the system of two carts. Initial Final Cart A Cart B Cart A Cart B 1. Represent. Draw a vertical line on the graph labelled A to indicate the moment in time when the collision begins and one labelled C to indicate when the collision ends. What is the duration of the collision? 2. Reason. What would we observe about the spring at moments A and C? 3. Represent. Draw a single ID and two FDs for each cart during the collision. ID FD Cart A FD Cart B 7

8 4. Reason. Based on the momentum graph, is there any evidence for a small net force acting on this system? Explain. 5. Find a Pattern. Calculate the impulse and average net force experienced by each cart during the collision. How do the impulses and net forces compare (try to disregard the small variations due to friction). Impulse Cart A Cart B F net The Idea of Conservation: A conserved quantity is one whose total for a system remains the same at every moment in time. 6. Reason. Carefully study the graphs showing the total momentum and total kinetic energy of the system. Ignoring the small losses due to friction, which of these quantities is a conserved quantity? Explain. 7. Reason. Notice how the total kinetic energy dips down during the collision. This indicates a transfer of energy. Where has been transferred? 8. Represent. Draw a vertical line on the kinetic energy graph labelled B to indicate the moment in time when the spring was at its maximum compression. Approximately how much energy was stored in the spring at this moment? D: The Buggy Challenge 1. Predict. Your teacher will turn on a physics buggy, gentle lower it on to the sliding track used earlier today, and release it. The mass of the buggy is less than the track. How will the buggy and track move? Use appropriate physics diagrams to support your prediction. 8

9 SPH4U: Types of Collisions A collision may roughly fall in to three categories based on how the system s total kinetic energy (E k = ½mv 2 ) before and after the collision compares. If the total kinetic energy decreases due to the collision, it is called an inelastic collision. If the total kinetic energy remains is the same, the collision is called elastic. If the total kinetic energy increases, the collision is called superelastic. Our goal today is to study collisions that fall in to these three categories. You will observe the interaction of two carts. We will not be making precise measurements. We will simply estimate speeds or changes in speeds and try to making some decisions based on that. A: The Sticky Collision (Together as a simulation) 1. Observe and Represent. For the collision below, sketch the before and after situations. 2. Reason. Is the total momentum of the system constant during this collision? Explain. Recorder: Manager: Speaker: Represent. Complete a momentumimpulse bar chart for the two cart system. Use your understanding of momentum conservation to help estimate the velocities just before or after the collision (for example, v A1 = 1 unit). Situation: Sticky Type of Collision: Before Moment 1 After Moment 2 Momentum Bar Chart Energy Bar Chart v A1 = 1 unit v B1 = p A1 p B1 J = p A2 p B2 E ka1 E kb1 W ext E ka2 E kb2 E int A workenergy bar chart shows the amount of energy stored in different mechanisms of a system (motion, gravitational field, etc.) at two moments in time. The shaded column represents the flow of energy in or out of the system due to external forces, also known as the external work. A new column has been included to account for the internal energy: energy stored in other mechanisms of this system (for example heat, vibrations and many more). The heights of the bars in the graph do not need to be exact we typically draw these charts before we make any calculations. What is important is that bars illustrate the correct ideas. 4. Interpret. What does this bar chart tell us about the amount of kinetic energy in the system before and after the collision? What about the total energy? Which quantity is conserved? 5. Reason. What type of collision is this an example of: elastic, inelastic, or superelastic? Record this above the chart. A sticky collision is not the only kind of inelastic collision. Technically, any collision that loses kinetic energy is an inelastic collision. A sticky collision is special because is produces the greatest loss of kinetic energy for the given starting conditions. For this reason it is also called a completely inelastic collision. 9

10 B: The Bouncy Collision (Together as a simulation) 1. Represent and Reason. For the collision below, sketch the before and after situations. Estimate the speeds involved. Complete a momentumimpulse bar chart and energy bar chart. Does the system gain or lose kinetic energy? Decide what type of collision it is. Situation: Bouncy Type of Collision: Before Moment 1 After Moment 2 Momentum Bar Chart Energy Bar Chart p A1 p B1 J = p A2 p B2 E ka1 E kb1 W ext E ka2 E kb2 E int v A1 = 1 unit v B1 = Carts collide magnetically C: The Explosion (Together as a simulation) 1. Represent and Reason. Sketch the before and after situations. Estimate the speeds involved (for example, v A1 = 1 unit). Complete a momentumimpulse bar chart and a new kinetic energy bar chart. Does the system gain or lose kinetic energy? Decide what type of collision it is. Situation: Explosion Type of Collision: Before Moment 1 After Moment 2 MomentumImpulse Bar Chart Energy Bar Chart v A1 = v B1 = p A1 p B1 J = p A2 p B2 E ka1 E kb1 E e1 W ext E ka2 E kb2 E e2 Compressed spring is released. Get ready to catch! In this final example energy was stored in the spring (elastic energy, E e ). This could have been counted as another storage mechanism for internal energy, but since we will soon be able to measure and describe energy stored in springs carefully, we gave spring energy its own bar. 2. Reason. Isaac drew his bar chart for the previous example with E ka2 as a positive bar and E kb2 as an equalsized positive bar. Albert drew E kb2 as an equalsized negative bar. Who do you agree with? Explain. 3. Reason. Is energy a vector or scalar quantity? Use the bar charts you drew in this investigation to help explain. 1

11 SPH4U: 2D Momentum Homework Name: A typical problem involving the conservation of momentum in 2D is often challenging, but usually due to lack of organization and careless mistakes. Follow the solution format and these following suggestions carefully! Note that parts B and C have been omitted here, but continue to do them whenever you can. Problem Two hover pucks glide towards each other, collide and then glide away. Puck A (5. kg) was initially travelling at 2. m/s [E 25 o N]. Puck B (3. kg) was initially travelling at 4. m/s [E 3 o S]. After the collision, puck A travelled at 1.6 m/s [E 3 o S]. Determine the velocity of Puck B after the collision. A: Pictorial Representation Sketch showing before and after, coordinate system, label given information, conversions, unknowns, key events v A1x = v A1y = v B1x = v B1y = v A2x = v A2y = v B2x = v B2y = D: Mathematical Representation Complete equations, describe steps, algebraic work, substitutions with units, final statement Use the conservation of momentum to find the velocity of puck B after the collision: E: Evaluation Answer has reasonable size, direction and units? 11

12 SPH4U: Momentum in 2D Momentum is a vector quantity and the Law of Conservation of Momentum is a vector equation. An object s momentum can be broken up into components and so can the law, meaning that momentum is conserved in each component direction. Recorder: Manager: Speaker: A: The Collision Consider an example where a small, fast moving hover puck (m a = 2. kg) collides with a large stationary hover puck (m b = 4. kg). Friction is small enough to be negligible. v a2 = 2. m/s N y E x v a1x = v a1y = 6 o v b1x = v b1y = m a v a1 = 6. m/s m b v a2x = v a2y = v b1 = 1. Predict. Use your intuition to predict the direction m b will travel after the collision. Draw a vector arrow in the diagram to show this. Briefly explain your reasoning. Do this quickly and move on! 2. Solve. For simplicity we choose a coordinate system that lines up with v a1. Determine the x and ycomponents of all the known velocities. Record these in the table above. Indicate which components are unknown. Show your work and be sure to use the sign convention! B: The xmomentum 1. Solve. Find the xcomponents of all the three momentum vectors we have velocities for. Show your work. p a1x = p b1x = p a2x = 2. Represent. Draw a bar in the momentum bar chart for the three xmomenta you calculated above. 3. Reason. Are there any external forces acting on the system of two pucks? Does the system experience a net force from its environment? What does this imply about J x? p a1x p b1x J x = p a2x p b2x 4. Predict. Draw a fourth bar in the momentum bar chart that represents the xmomentum of puck B after the collision. Explain your prediction. 5. Represent. Write an equation showing the relationship between the four momentum values in the bar chart. Use the symbols in the bar chart. 12

13 Since there is no net force in the xdirection, the xmomentum is conserved and: p a1x p b1x = p a2x p b2x 6. Solve. Use your new equation to explicitly solve for p b2x. 7. Reason. Isaac remarks that the magnitude of p b2x is quite large compared with the other components. It must be going really fast. Do you agree or disagree? Explain. C: The ymomentum 1. Solve. Find the ycomponents of all the three momentum vectors we have velocities for. Show your work. p a1y = p b1y = p a2y = 2. Represent. Draw a bar in the momentum bar chart for the three ymomenta you calculated above. 3. Predict. Draw a fourth bar in the momentum bar chart that represents the y momentum of puck B after the collision. Explain your prediction without math. p a1y p b1y J y = p a2y p b2y Since there is no net force in the ydirection, the ymomentum is conserved and: p a1y p b1y = p a2y p b2y 3. Solve. Use the conservation of momentum in the ydirection to explicitly solve for p b2y. D: The Final Result 1. Explain. Describe how we can use the results found so far to find the missing momentum b2 p. 2. Solve. Find the momentum vector p b2. Be sure to draw the component triangle. 3. Solve and Test. Find v b2. Compare this with your prediction and the simulation, if possible. 4. Summarize. What does it mean to say, Momentum is conserved in two dimensions? 13

14 SPH4U: 2D Collisions The collision on the next page has had the path of one object removed! Your challenge is to reconstruct the path. Examine the collision and read the details describing it, then follow the steps below. Recorder: Manager: Speaker: Observe. Label the points where the collision appears to begin and end. Explain how you can tell. 2. Reason. Marie says, Let s choose a coordinate system that lines up with the page that will be the most helpful. Emmy says, I think we should choose one that lines up with the initial velocity of puck A. That will be easiest. Who do you agree with? Explain. 3. Observe and Reason. Find v a1x on the opposite page and record the result on your whiteboard. Isaac says, Let s use units of cm/s for our result that will be convenient. Albert says, Hmm maybe we should use m/s, even though the numbers will be very small they are S.I. units. Who do you agree with? Explain. 4. Observe and Calculate. Write an equation for the conservation of momentum in the xdirection where v b2x is the unknown. Make the necessary measurements from the first page (show these) and record your calculated results in the chart. Solve the equation for v b2x. Note: the equation will simplify. Explain why. 5. Observe and Calculate. Write an equation for the conservation of momentum in the ydirection where v b2y is the unknown. Make the necessary measurements from the first page and record these in the chart. Solve the equation for v b2y. 6. Calculate and Test. Determine the vector v b2. Carefully draw the vector beginning at the circle under the B (Bonus: draw the circles too!) ** Check this with your teacher! ** 7. Calculate and Reason. Use kinetic energy calculations to help explain what type of collision this is. 14

15 va1x = pa1x = Eka1 = va1y = pa1y = va2x = pa2x = Eka2 = Vb1x = Pb1x = Ekb1 = va2y = pa2y = Vb1y = Pb1y = Vb2x = Pb2x = Ekb2 = pt1x = pt1y = Ekt1 = Vb2y = Pb2y = pt2x = pt2y = Ekt2 = A Puck A Puck A Before the Collision Puck B After the Collision Puck B Totals Totals On this page is pictured part of a collision between two dryice pucks A and B (ma = mb = 3 g). The collision takes place using magnets and is captured using a strobe light that flashes 1 times a second. The centre of each puck is indicated using a small circle and the path of puck B has been erased. Puck B was initially at rest and was positioned at the small circle near the letter B. Show your measurements and results on this page and your work on the following page. B 15

16 SPH4U: Work and Kinetic Energy How can the energy of a system change? When an external force acts on a system while a system object moves through a displacement, energy flows into or out of the system. This process is called work and the amount of work done (or energy that has flowed) due to the force can be found from the equation: W = F d cosθ, where θ is the angle between the force and displacement vectors. A: The Flow of Energy Your teacher has a track with two carts set up. A large cart, cart A (1. kg) is moving right and a small cart, cart B (5 g) is at rest. They collide magnetically and travel on a frictionless track. 1. Predict. (individually). Describe how each cart will move after the collision. Recorder: Manager: Speaker: Observe (as a class). Describe the motion of each cart after the collision. An energyflow diagram is similar to an interaction diagram, but with two differences: we only draw the lines connecting objects if there is a flow of energy between them and if there is an energy flow, we add an arrow showing the direction of the flow. 3. Reason. Which object has gained energy? Which has lost energy? Is energy lost to the environment? 4. Represent. Draw an energy flow diagram for the system of the two carts. 5. Represent. Moments 1 and 2 occur at the beginning and end of the collision. Follow the instructions and complete the chart below for the system of cart A and then for the system of cart B. (a) Draw a vector arrow for the displacement of the cart and the force acting on the cart during the collision. (b) Complete a motion diagram for the cart between moments 1 and 2. Describe its motion: is the cart speeding up, slowing down, or moving with a constant speed? (c) What is happening to the energy of the system of the cart? Draw an energy flow diagram for the system of the cart. (d) According to the equation for work, is the work done by this force positive, negative or zero? Give a sample calculation for the sign. (e) Complete the workenergy bar chart for the system of the cart. System = Cart A Displacement of cart Motion Diagram EnergyFlow Diagram WorkEnergy Bar Chart E k1 W ext E k2 Sign of Work Force of B on A Description of Motion 16

17 System = Cart B Displacement of cart Motion Diagram EnergyFlow Diagram WorkEnergy Bar Chart Sign of Work Force of A on B Description of Motion E k1 W ext E k2 6. Reason. We have been assuming (consciously or not) that the gravity and the normal force do not cause energy to flow into or out of the system. Use the equation for work to help explain why this was correct. (Hint: ) 7. Reason. Does the result of the work equation depend on your choice of sign convention? Is work a scalar or vector quantity? Explain. B: Work from Multiple Forces 1. Represent. In a new experiment, two hands push horizontally on the same cart. Hand 1 pushes to the left on the cart and Hand 2 pushes to the right. The cart is moving to the right and is now speeding up. (a) Draw a motion diagram for the cart. (b) Draw a complete force diagram for the system of the cart. (c) Decide whether the work done by each force is positive, negative, or zero. Indicate this by writing a,, or along side the force vector. (d) Draw an energyflow diagram for the cart (e) Is the net work on the system above positive, negative or zero? Motion Diagram Force Diagram EnergyFlow Diagram Net Work When a system has multiple interactions with its environment it may gain or lose energy due to a number of forces. The total change in energy of the system is the net work. The net work can be found in two different ways: by adding up the work from the individual forces or by finding the work due to the net force. W net = W= W 1 W 2 or W net = F net d cos A system that experiences a net force will accelerate and gain or lose kinetic energy. This idea is called the net work kinetic energy theorem, W net = E k. This theorem is closely related to Newton s 2 nd Law. 2. Represent. Two hands continue to push on opposite ends of the cart, but with different magnitudes than before. For each situation, complete the chart below like in question #1 above. Description Motion Diagram Force Diagram EnergyFlow Diagram Net Work The cart is moving to the left and is slowing down. 3. Summarize. What is the physical significance of positive, negative and zero net work? What happens to the system? 17

18 SPH4U: Energy and Coordinate Systems Gravitational energy (E g = mgy) is a special kind of energy that depends on an object s vertical position. But what happens if we choose coordinate systems that have different vertical origins for our measurements? Let s find out! Recorder: Manager: Speaker: On a hot day, you look out a third story window and hold in your hands a 2. kg water balloon (a big one!). Your friend walks below the window, not noticing you 1 meters above her. You release the water balloon. We will analyze what happens next from the frame of reference of you and also of your friend. Flow 1. Represent. Draw an energy flow diagram for the Earthballoon system while the balloon falls. A: Coordinate System One You set the vertical origin at your hand. Calculate the energies involved. Use a kinematics calculation to help find v 2. Add up the total energies at each moment. Complete the workenergy bar chart for the earthballoon system. Sketch y Gravitational Energy E g1 = E k1 = Kinetic Energy Total Energy E T1 = WorkEnergy Bar Chart E k1 E g1 W ext E k2 E g2 y 1 = v 1 = E g2 = E T2 = y 2 = 1. m v 2 < E k2 = B: Coordinate System Two Now you set the vertical origin at your friend s head. Calculate the energies involved. Use a kinematics calculation to help find v 2. Add up the total energies at each moment. Complete the workenergy bar chart. Sketch y E g1 = Potential Energy E k1 = Kinetic Energy Total Energy E T1 = WorkEnergy Bar Chart E k1 E g1 W ext E k2 E g2 y 1 = 1. m v 1 = E g2 = E T2 = y 2 = v 2 < E k2 = C: Thinking About Energy 1. Reason. How does E g, E k and E t change as measured in each coordinate system? 2. Reason. The two observers do not agree on the total energies, but this is not a problem. What is important in physics are energy changes. What does each observer conclude about the amount of energy that transfers from gravitational to kinetic within the system? Explain. 18 Adapted from: Laws, P. Workshop Physics Activity Guide. Module II. John Wiley, 24

19 SPH4U: Transfers of Energy Homework Name: 1. Represent. Each row in the chart below gives multiple representations of one situation. Complete the missing parts for each row come up with new and interesting situations where appropriate. Word Description Sketch of Initial and Final States WorkEnergy Bar Chart and Equation (i) A stunt car has a springpowered ejector seat. When the spring is released, the seat with its passenger is launched out of the car and reaches a maximum height y 2 above its starting position. (The elastic energy stored in the compression x of a spring is E e ) (ii) An elevator is initially moving downwards at speed v 1. It approaches the ground floor and slows to a stop in a distance h. y y y 1 = v 1 = x > y 2 > v 2 = x = y 1 > v 1 < System: Equation: y 2 = v 2 = System: Equation: (iii) E k1 W ext = E k2 System: (iv) Equation: W ext = E g2 E th2 System: Equation: 19

20 SPH4U: Transfers of Energy Nobody really knows what energy is, but we do know how it behaves. Energy is a quantity that can be transferred between systems and which can be used to predict whether some event may occur. By carefully keeping track of where the energy is located, or stored, we can construct equations to help with our predictions. A: An Energetic Example Your teacher has a ramp set up at the front of the class. A block is on top of a cart that will be released and glide down an inclined track. The cart will collide with a barrier and stop. The block slides across another track until it comes to a stop. Your challenge is to predict how far your block will slide along the bottom track surface. 1. Reason. Which objects interact with the block? Which interactions involve a transfer of energy? Recorder: Manager: Speaker: y 2. Reason. There are three important events in this situation. Describe them here and label them on the diagram. To understand the role of energy in a problem, we need to choose a system, or a collection of objects, whose energies we will track. We have a freedom to include any objects in our system and, if we do our work properly, everyone should always agree on the final answer. If an interaction can easily be described using energy (e.g. gravity), include those objects in your system. If an interaction is more easily described using work (e.g. applied, tension), make one object external. Friction effects involving transfers to thermal energy are best described as internal energy that is shared between system objects (they each get warmer). If this is the case, we will keep objects like the ground surface, roads, and tracks inside the system. 3. Reason. Based on the advice above, chose your system. Draw an energyflow diagram for the system between moments 1 and 2. Do the same for moments 2 and System = The law of the conservation of energy states that the total energy of a system remains constant unless energy flows into or out of the system. This flow of energy can be due to the work done by an external force. The total energy of the system and its changes may be expressed by a workenergy equation: E T1 W ext = E T2. 4. Reason. What is a convenient position for the vertical origin in the situation? Explain how you chose it and label it in the diagram above. E T1 W ext = E T2 E T2 W ext = E T3 E k1 E g1 W ext = E k2 E g2 E k2 E g2 W ext E k3 E g3 E int 5. Represent. Complete a workenergy bar chart for moments 1 and 2. Complete a workenergy bar chart for moments 2 and

21 7. Represent. The bar charts help you to write the workenergy equations for the system comparing the two moments in time. Do this using the same symbols as in the chart. 8. Explain. When you practice problems like this in your homework, part C asks for a word explanation. Let s practice: describe the motion of the system and use energy transfers and work to explain why it moves the way it does. To find the amount of thermal energy (which we are calling E int ) due to the sliding block, we need to use the definition of work to create a new equation. W = F dcos. When an object is sliding, the angle between F f and d is 18 o. This gives W=F f d. The amount of thermal energy should be a positive number so we create the equation, E th = F f d. 9. Measure. Choose a block for your group. Make the measurements you think will be necessary to find all the energies. Record these in your diagram above. 1. Predict. Use your workenergy equations to predict the sliding distance of the block. Write this up as you would part D of your homework solutions. (Hint: only substitute in your final step!) D: Mathematical Representation Describe steps, complete equations, algebraically isolate, substitutions with units, final statement 11. Evaluate. Is your answer reasonable? Explain why you think so before you test it! 12. Test. Ask your teacher to watch you test your prediction. Note: there are very large uncertainties in this test! Try it a few times to see how it compares with your prediction. If you are within about 2 cm it is a good result! 21

22 SPH4U: The Ballistic Pendulum Here s a problem for you: how can you determine the speed of a bullet using only measurements of mass and distance? The answer was found in 1742 by the English mathematician Benjamin Robbins using his invention, the ballistic pendulum. Now it s your turn to repeat his clever work. Recorder: Manager: Speaker: A: Pictorial Representation Sketch with before and after, coordinate system, label givens & unknowns, conversions, events Measurements: Do these later. Hint: There are three important events in this problem (not including the launch). You may assume that its velocity does not change much while traveling through the air. B: Physics Representation Energy / momentum bar chart, flow diagram, events p A1 p B1 J = p A2 p B2 E k1 E g1 W ext = E k2 E g2 E int E k2 E g2 W ext = E k3 E g C: Word Representation Describe motion, energy / momentum transfers, assumptions Moments 12: Moments 23: 22

23 D: Mathematical Representation Describe steps, complete equations, algebraically isolate, substitutions with units, final statement Now try out the pendulum and make your measurements. Record these values in your sketch for part A. Hint: Make sure you consider the geometry of the swing carefully! E: Evaluation Answer has reasonable size, direction and units? 23

24 SPH4U: Spring Force and Energy How does the force exerted by a spring change as we stretch it? How much energy is stored in the spring? In this investigation we will find out. Recorder: Manager: Speaker: A: Force and Springs A few things about springs: Don t overstretch springs they can be permanently deformed and damaged. The position of the end of a spring experiencing no other forces, whether oriented horizontally or vertically, is called the equilibrium position. Physics springs have no mass. 1. Explain. There is an important different between the length of a spring and the amount of stretch. Explain this difference. Which one are you being asked to use? 2. Explain. Describe an experiment to determine how the size of the spring force is affected by the amount of stretch in a spring. A good experiment will feature many data points to reduce uncertainties and use consistent intervals to help find patterns. 3. Represent. Draw a sketch of your experiment showing the measurements involved. 4. Observe. Conduct your experiment and make your measurements. 5. Represent. Plot a graph that shows the relationship between your variables. 6. Analyze. Use appropriate techniques to analyze the relationship in the graph. (Always show units!) 7. Represent. Construct an equation using the symbols F s (the force the spring exerts), Δx (the stretch of the spring), and k the spring constant (your slope) based on your graphical analysis. 24

25 8. Predict and Test. You will hang a 1. kg object from your spring. Predict the amount of stretch that will result. Test your prediction. 9. Interpret. What does the value of the spring constant, k, tell us about the spring itself? What would a large value indicate? What would a small value indicate? The relationship found above was first discovered by Robert Hooke around 166. Within a certain range of stretch or compression most springs behave according to Hooke s Law. Note that the direction of the force the spring exerts is opposite to the direction of the stretch it experiences. Official version: Magnitude only: B: Energy and Springs 1. Reason. Why do we believe there is a flow of energy when you stretched the spring during your experiment? Represent this flow with an energy flow diagram. 2. Reason. Emmy is discussing how to measure the energy transferred while she stretched the spring. I measured a force of 2.1 N when the spring had 2 cm of stretch and a force of 3.3 N when it had 3 cm of stretch. I moved it by 1 cm so the work I did is W = (2.1 N)(.1 m) =.21 J. Marie says Emmy s result is too low. Who do you agree with? Explain Reason. Albert joins in the discussion. But the force gets bigger as you stretch it, so we have to use the bigger value. The work will be (3.3 N)(.1m) =.33 J. Isaac says Albert s result will be too high. Who do you agree with? Explain Reason. Both Albert and Emmy were wrong! The force is changing in a steady way when the spring is stretched from 2 to 3 cm. What single value of force would be best to use in our work equation? How much work did Emmy actually do? ** check your answer with your teacher ** 25

26 5. Calculate. Complete the chart below. Calculate the work done during each increment of stretch based on your force data from earlier. The Increment of Stretch is the extra distance the spring was stretched between one force measurement and the next. If you re not sure, try two rows and call your teacher over. Displacement (stretch) from equilibrium (m) Force (N) Increment of stretch (m) Average force during increment (N) Work done during increment (J) 6. Calculate. Now you are ready to calculate the total work done to stretch a spring from its equilibrium position to each displacement. To do this, consider that the work to displace the spring to.2 m is equal the work to go from to.1 m plus the work to go from.1 to.2 m. Complete the chart to the right and plot a graph of total work vs. displacement. Displacement from equilibrium (m) Total Work (J) Work can be found by calculating the area under a forcedisplacement graph. This is especially helpful for forces that change magnitude. 7. Calculate. Find the work needed to stretch the spring to a displacement of.3 m by computing the area under the F s vs. Δx graph you created earlier. Does this agree with the result in your chart above? 8. Speculate. Create an equation that relates W and Δx. Hint: Consider how you computed the area to get the work. Official version: 26 ** check your result with your teacher **

27 C: The Test Launch Your teacher has a dynamics cart set up on a track at the front of the class. It has a spring launching mechanism. Your challenge is to make a few measurements and predict the launch speed of the cart. When thinking about this process, there are three important events: (1) the cart is at rest and the spring is uncompressed, (2) the spring is now fully compressed due to your hand, and (3) the cart has been launched and the spring is uncompressed. 1. Represent. Draw an energy flow diagram for the system of the cart between moments 1 and 2 and between moments 2 and 3. Draw an energy bar chart for each moment in time. E k1 E e1 W ext = E k2 E e E k2 E e2 W ext = E k3 E e3 2. Plan. What steps will you take in your solution in order to find the launch speed? 3. Measure. Make any appropriate measurements to determine the quantities you will need to find the launch speed. (Hint: to find k, simply place a weight vertically on top of the cart s spring.) 4. Calculate and Predict. Use your measurements to predict the launch speed of the dynamics cart. D: Mathematical Representation Describe steps, complete equations, algebraically isolate, substitutions with units, final statement with uncertainty 5. Test. Use the motion detector. You probably noticed that the speed was a bit slower than you predicted. Explain why. 27