ΣE before ± W = ΣE after

Transcription

1 The Law of Conservation of Energy The Law of Conservation of Energy states: Energy is never created nor destroyed just transformed into other forms of energy. OR ΣE before = ΣE after Yet if energy is added to or removed from the system the total amount of energy has changed. This can only be accomplished by external forces, which is work. If energy is added: positive work was done on the object. If energy is removed: negative work was done on the object. Therefore, work must be accounted for in our equation. Law of Conservation of Energy ΣE before ± W = ΣE after Solving Conservation of Energy Problems Using Conservation of Energy, many complicated problems can be solved simply. Before 2 kg v = 0 m/s After 2 kg v =? Example 1: A 2 kg object compresses a spring 0.5 m. If k = 72 N/m, how fast is the object going after the spring is released? Step 1: Identify the energies before and after. Step 2: Decide if energy was added (+W), removed ( W), or just transformed (W = 0). Step 3: Put the information from steps 1 and 2 into the Conservation of Energy formula. Step 4: Put in the formulas for the different kinds of energy or for work. Step 5: Put in all of the given information and solve. Step 1: E before = PE el (a compressed spring) E after = E k (it is moving) Step 2: No forces energy is just transformed. W = 0 (no work added or removed) Step 3: ΣE before ± W = ΣE after PE el + 0 = E k Step 4: (½)kx 2 = (½)mv 2 Step 5: Example 2 : A 3 kg mass at rest on the ground is pushed by an 8 N force for 4 m. How fast is the mass going afterwards? Before m = 3 kg +W 8 N 4 m After m = 3 kg Step 1: E before = 0 (at rest on the ground) E after = E k (it is moving) Step 2: Energy is added: +W Step 3: ΣE before ± W = ΣE after 0 + W = E k Step 4: 0 + Fd = (½)mv 2 Step 5: v =? Example 3 : An object is thrown into the air going 60 m/s. How high up does it go? NOTE: Follow the steps even though you are not given the mass of the object. Step 1: E before = E k (above the ground) E after = E p (it is falling) Step 2: No work, energy is transferred, so, W = 0 Step 3: ΣE before ± W = ΣE after E k + 0 = E p Step 4: (½)mv 2 = mgh Step 5: How high? NOTE: Often you don t need the mass. It may cancel.

2 1. Potential Energy (E p ), Kinetic Energy (E k ), Potential Elastic Energy (PE el ), or Work (W)? A. A car is moving 22 m/s. B. When a spring is compressed. C. When an object is sitting on a desk. D. After a falling object hits the ground. E. The ground stopping a falling object. F. While a rock is flying thru the air. G. When a force pushes on an object. H. Decreases as an object goes downhill. 3.5 m 40kg C. Where does this energy come from? 2. Slim Jim lifts a 40 kg object up 3.5m. A. How much energy does it have before it is lifted? B. What kind of energy will it gain? I. What is gained as an object goes downhill. J. Friction acting on an object. K. A box on the ground at rest. L. Decreases as an object slows down. M. Provided by the engine of car. N. Pushing an object up a ramp. O. For a dropped object, as it is falling. P. Decreases as a spring is released. 950 kg 3. If the car has 30,400 J of kinetic energy, how fast is it moving? D. Calculate the box s energy after it is lifted. 35 N 10 kg 4. Slim Jim pulls on a box for 6.5m. A. How much work did Jim do? 10 m 30º 6kg 5. A 6 kg object is slides down a frictionless 10 m ramp. A. What kind of energy does it have at the top? B. h must always be vertical. Calculate h at the top of the ramp. C. Calculate the energy at the top. B. What kind of energy does the box gain? C. If there is no friction, how much energy does the box gain? k =? 3 kg 6. A 3 kg object compresses a spring 0.6m, giving it 8.1 J of energy. A. What kind of energy does it have? B. Calculate the spring constant for this spring. D. What is this energy transforming into as it slides down the ramp? C. What kind of energy will the object have when the spring is released? I 2kg 7. A. If the ball has 120J of energy at v i = 3 m/s v f = 3 m/s 8. Slim Jim carries position I, calculate the height at I. 12 kg 12 kg a box for 6m at constant speed. Save me, Joe! II III B. What kind of energy does it have at II? C. Where does the energy go at III? d = 6 m A. How much energy does the box have? B. How much work does Jim do? Two identical balls roll down opposite sides of a frictionless platform, which one will be going faster at the bottom?

3

4 The Law of Conservation of Energy The Law of Conservation of Energy states: Energy is never created nor destroyed just transformed into other forms of energy. OR ΣE before = ΣE after Yet if energy is added to or removed from the system the total amount of energy has changed. This can only be accomplished by external forces, which is work. If energy is added: positive work was done on the object. If energy is removed: negative work was done on the object. Therefore, work must be accounted for in our equation. Law of Conservation of Energy ΣE before ± W = ΣE after Solving Conservation of Energy Problems Using Conservation of Energy, many complicated problems can be solved simply. Before 2 kg v = 0 m/s After 2 kg v =? Example 1: A 2 kg object compresses a spring 0.5 m. If k = 72 N/m, how fast is the object going after the spring is released? Step 1: Identify the energies before and after. Step 2: Decide if energy was added (+W), removed ( W), or just transformed (W = 0). Step 3: Put the information from steps 1 and 2 into the Conservation of Energy formula. Step 4: Put in the formulas for the different kinds of energy or for work. Step 5: Put in all of the given information and solve. Step 1: E before = PE el (a compressed spring) E after = E k (it is moving) Step 2: No forces energy is just transformed. W = 0 (no work added or removed) Step 3: ΣE before ± W = ΣE after PE el + 0 = E k Step 4: (½)kx 2 = (½)mv 2 Step 5: Example 2 : A 3 kg mass at rest on the ground is pushed by an 8 N force for 4 m. How fast is the mass going afterwards? Before m = 3 kg +W 8 N 4 m After m = 3 kg Step 1: E before = 0 (at rest on the ground) E after = E k (it is moving) Step 2: Energy is added: +W Step 3: ΣE before ± W = ΣE after 0 + W = E k Step 4: 0 + Fd = (½)mv 2 Step 5: v =? Example 3 : An object is thrown into the air going 60 m/s. How high up does it go? NOTE: Follow the steps even though you are not given the mass of the object. Step 1: E before = E k (above the ground) E after = E p (it is falling) Step 2: No work, energy is transferred, so, W = 0 Step 3: ΣE before ± W = ΣE after E k + 0 = E p Step 4: (½)mv 2 = mgh Step 5: How high? NOTE: Often you don t need the mass. It may cancel.

5 1. Ep, Ek, PEel, W, or No Energy? Compressing a spring Friction acting on an object. An object at rest on the ground. Pushing an object. An object above the ground. An object moving. A compressed spring. An object as it is falling. 2. Is Energy Added (+W), Removed (-W), or Transferred (T) Slowing down an object. Lifting an object into the air. Lowering an object to the ground slowly. An object falling. Law of Conservation of Energy p2 Speeding up an object. A moving object compressing a spring. A force compressing a spring. An object slides up a frictionless ramp. 3. For each of the following, develop the Conservation of Energy Equation A. A moving object speeds up. E before = Ek Work? = +W E after = Ek Conservation of Energy Equation: Ek + W = Ek. B. An object is dropped. There is air friction. E before = Work? = E after = Conservation of Energy Equation: C. A moving object compresses a spring. E before = Work? = E after = Conservation of Energy Equation: E. A relaxed spring is compressed. E before = Work? = E after = Conservation of Energy Equation: F. A spring causes an object to move. E before = Work? = E after = Conservation of Energy Equation: G. An object slides down a frictionless ramp. E before = Work? = E after = Conservation of Energy Equation: D. An object is thrown up, going 2 m/s. How high does it go? E before = Work? = E after = Conservation of Energy Equation: 4. A 5 kg mass at rest on the ground is raised up to 15 m. Find the work that was done on the object. A. E before = Work? = E after = B. Conservation of Energy equation: C. Solve. H. An object is dropped. How fast is it going part way down? E before = Work? = E after = Conservation of Energy Equation: 5. A 8 kg mass going 2 m/s compresses a spring 0.5 meters. Find the spring constant of the spring. A. E before = Work? = E after = B. Conservation of Energy equation: C. Solve. 6. A 6 kg mass going 4 m/s is slowed to 3 m/s by a 2 N force. For how much distance did the force act? A. E before = Work? = E after = B. Conservation of Energy equation: C. Solve. 7. A mass at rest is dropped from 12 m in the air. How fast is it going 2 m above the ground? A. E before = Work? = E after = B. Conservation of Energy equation: C. Solve.

6

7 Conservation of Energy Practice 1. Define and give the units for the variables given a the right. Variable What it is Units m mass kg v P k g Variable What it is Units x F PEel d h 2. A 2 kg mass compresses a spring 1.2m. If the object has 160 J of energy, calculate the spring constant of the spring. 3. A 4 kg object has 180 J of kinetic energy. How fast is it moving? 70 kg 1.4 m 4. Slim Jim, continually maintaining his svelte body, lifts a 70kg barbell 1.4m above the ground. A. How much energy did the barbell have when it was on the ground at rest? B. What kind of energy does the barbell have in its current position? C. Where did the energy come from? D. Calculate the energy it has at its current position. E. How much work did Jim do to lift the object? F. If he lifted it in 1.5 seconds, how much power did he use? 5. On Slim Jim s last cave adventure he accidentally dropped his lantern while studying a formation of stalactites. The lantern was dropped from 35m up. How fast was it going when it smashed into the cave floor? A. What color is his lantern (of course)? B. E before = Work? = E after = C. Conservation of Energy equation: D. Substitute the formulas for each type of energy and solve. stalactites 9 m? m/s 6. Slim Tony Hawk Jim starts at rest at the top of a 9m long ramp that is tilted at 30. How fast is he going at the bottom? A. Calculate his height at the top of the ramp. B. E before = Work? = E after = 30 C. Conservation of Energy equation: D. Substitute the formulas for each type of energy and solve.

8 20kg 7. Slim Jim s dog Bim is on a ledge 2.5m above a spring board. If he compresses it 0.45m, what is the spring constant for the board? A. E before = Work? = E after = 2.5 m B. Conservation of Energy equation: C. Solve. 8. Slim Jim finds a giant spring that has a 350 N/m spring constant. Jim figures out how to compress it 1.5 m. If Jim and his spring propelled cart s mass is 85kg, how fast is he going after he releases the lever? A. E before = Work? = E after = B. Conservation of Energy equation: C. Solve. 350N/m 85kg 9. Slim Jim runs 5m/s at full speed. If he grabs a rope at the ground, how high off the ground will he swing? A. E before = Work? = E after = B. Conservation of Energy equation: C. Solve. 60 kg 5 m/s 10. A 4 kg object is moving 15m/s. If it stops due to friction in 3.5m, what is the magnitude of the force of friction? (Calculate Fk.) A. E before = Work? = E after = B. Conservation of Energy equation: C. Solve. Before 4 kg 3.5m 15 m/s 2 sec After 4 kg 0 m/s D. How much energy did it lose? E. Since this energy was lost in 2 seconds, how much power did friction dissipate (use up)?

9

10

11 Advanced Conservation of Energy 1. Using a pulley system a person pulls 48cm and with 60N to lift an object 12cm. A. Convert all numbers to standard units on the diagram. 48 cm B. What kind of energy is the object gaining? C. Where does it come from? D. Write the Conservation of Energy equation: E. Solve for the mass of the object. 60 N 12 cm M F. If it was lifted in 5.5 seconds, how much power was used? before P 2. A mass is dropped from 10m. At what height above the ground will it be going 6m/s? A. At letter Q what kind of energy does it have? B. Write the Conservation of Energy equations: 10 m Q 6m/s h =? C. Solve for the height at Q. R 3m C 3. Slim Jim is standing on a 6m tall ledge and holds a 1.5kg ball 3m above his head. He drops the ball to the ground below. This time there is air friction. A. How much total energy does the ball have when he holds it above his head? 6m D E The ball is moving 11m/s just before it hits the ground, due to air friction. B. How far does it fall? C. How far does air friction act on the ball? D. E before = Work? = E after = E. Conservation of Energy equation: F. Solve for the force of air friction. 4. Always the consummate outdoorsman, Slim Jim is skiing 6m/s when he begins to ski up a ramp that is tilted at 30º. A. Using Conservation of Energy, calculate how high he goes. 6 m/s 30º B. Now that you have his vertical distance, calculate how far up the ramp he goes.

12

13 Efficiency In the real world, energy transfers are not perfect and energy is lost to friction. The more energy that is lost, the less efficient the transfer of energy. 40 kg 4 m Efficiency (in %) Energy gained by the object (in J) W Eff = W out in x100 Energy you tried to give the object (in J) V = 0 m/s E k = 0 J F = 25 N; d = 4 m W in = 100 J Energy added V = 2 m/s E k = 80 J Energy gained If all of the energy was transferred, 100 J of work would become 100 J of kinetic energy. Yet friction took 20 J and converted it into thermal energy. The transfer was 80% efficient. Understanding percent: part % = x100 whole You know that on a test 95 problems out of 100 is 95%. 95 is the part of the whole 100 points (the total). So 42 points out of 60 total points is 42/60 =.70 or 70%. Work In How much energy you tried to give to the object thru an energy transfer or work. Work Out How much energy is actually gained by the object (how much it got out). Here work is done on the object, pulling it up the ramp. This is the total energy that you tried to give the object. W = Fd = in 30(8) = 240 J Before 10 kg F in = 30 N Work tried to put in 240 J. D = 8m The object only got out 200 J. After 10 kg 2 m W out = = = Ep mgh 10(10)2 = 100(2) = 200J gained E ff E ff = = = = W W E W o u t p in x1 0 0 x J x J.8 3 x = 8 3 % Efficiency in Energy Transfers W in = E p = mgh = 1(10)4 = 40 J W in is the total energy that could be given to the object. For a dropped object, the available energy is potential energy. W in = 40 J In the real world no energy transfer is really 100% efficient, due to friction. In an energy transfer, the W in is the energy it starts with. 4 m 1 kg 8 m/s Air friction slows the object, removing energy. W out is the energy actually gained by the object. For a dropped object it gained kinetic energy. W out = 32 J W out = E k = (½)mv 2 = (½)1(8) 2 = 32 J Wout Ek Eff = x100 = x100 W E = = in 32 J x J.80 x 100 = 80% So 20% was lost to air friction. p Consumer Efficiency There are a lot of steps consumers can take to increase efficiency, whether with their cars (idle less), their houses (add insulation), or buying more efficient appliances. More efficient means less money spent on energy and less pollution for the environment. Compact fluorescent light bulbs give the same amount of light, but produce less heat. Thus, they are more efficient and cheaper to use. Fluorescent: uses only 15 W. Very little energy is lost to heat. Both of these light bulbs give the same amount of light. Incandescent: uses 60 W. At least 45 W is lost as heat.

14 Efficiency p2 1. Identify W in and W out for each of the following situations. A. An object is pushed up a ramp. There is friction on the ramp. D. An object is at the top of a ramp. It slides down the ramp. There is friction. W in =. W out = W in = W out = B. An object is against a compressed spring. The spring is released and pushes the object, but the object rubs against the ground. E. An object is accelerated by a force, but because of friction it is not moving as fast afterwards as it should. W in = W out = W in = W out = C. An object is launched into the air with an initial velocity. It goes to the top. There is air friction. F. An object is dropped. There is air friction. W in = W out = W in = W out = 2. A 6 kg object is at rest on a table. It is pushed for 20 m with a 3 N force. It is moving 4 m/s afterwards. A. W in = B. W out = C. Calculate efficiency. 3. A 2 kg object is moving 6 m/s. It compresses a spring 1 meter that has a spring constant of 8 N/m. A. W in = B. W out = C. Calculate efficiency. D. How much energy was lost to friction? 4. A 3 kg object falls off an 8 m tall ledge. Due to friction it is only going 11 m/s at the ground. A. W in = B. W out = C. Calculate efficiency. D. Where did the energy go? 5. A 2 kg object is pushed up a 12 m long ramp by a 7 N force to get the object to the top of a 3 m tall table. A. W in = B. W out = C. Calculate efficiency. D. How much energy was lost to friction? 6. A 6 kg object is at the top of a 2 m tall ramp. Due to friction it is only moving 5 m/s at the bottom. A. W in = B. W out = C. Calculate efficiency. D. How much thermal energy was created? 7. A 6 kg object is moving 1 m/s. It is pushed by a 4 N force for 20 m. It is going 5 m/s afterwards. A. W in = B. W out = C. Calculate efficiency. D. How much thermal energy was created? D. How much thermal energy was created?

15

16 Work and Energy In Class Review 1. Work 2. Power 3. Kinetic Energy 4. Potential Energy 5. Potential Elastic Energy A. Rate of doing work; how fast you transfer energy. B. Energy of position or height. C. Applied energy; can create energy. D. Energy of something that can be compressed. E. Energy due to motion and inertia. 6. Law of Conservation of Energy 7. Rate 8. Work-Kinetic Energy Theorem 9. Energy 10. Perpetual motion A. How fast something is done. B. An object that moves forever without added energy. C. A change in kinetic energy comes from work. D. Energy can be transformed, but not created nor destroyed. E. Stored work; ability to create forces or cause motion. 11. Chemical 12. Nuclear 13. Mechanical 14. Thermal 15. Electrical 16. Radiant A. Energy stored in the atom. B. Energy stored in molecular bonds. C. Caused by friction. Heat. D. Due to moving electrons. E. From light. F. Any kinetic or potential energy. 18. A person pulls down with 50 N to lift an object up 1 m. A) What is the MA of the pulley system? B) How much rope will you pull out? C) What is W in? D) What is W out? 17. Which of the following shows positions from highest to lowest kinetic energy? E) Calculate efficiency. 50 N 1 m i. E, G, F ii. E, F, A iii. A, F, D F) If the pulley was 100% efficient, how much force would you have needed? 19. A more powerful motor does more work. True or false? 20. In the same amount of time a more powerful motor: 23. A. Which of the 3 forces does no work on the object? B. Find the total work done on the 6 kg mass. 5 N 6 N 60 o 8 N 2 m 6 kg 6 kg 21. How much energy does a 60 W light bulb use in 2 minutes? (Be sure to use seconds.) C. If there is no friction, how much energy does it gain? 22. A 70 kg person climbs up 2 meters in 2.8 seconds. A) How much E p did they gain? B) How much power did they use? 24. You hold onto a book for an hour. A. Does your body get tired? B. Does your body use energy? C. Do you do any work on the object? D. Why? 25. How do all simple machines multiply force? 26. With a simple machine (like the ramp below), do you do more or less work if there is no friction? 27. With a simple machine, do you do more or less work if there is friction? 28. With a simple machine, do you use less or more force? 29. With a simple machine, do you use less or more time? 30. With a simple machine, do you use less or more power? F = 10N

17 Energy In Class Review p2 31. Can a simple machine ever have an efficiency greater than 100%? Why or why not? A frictionless ramp is inclined at 20 o. An object going 6 m/s slides up. A) Find the final height of the object. B) How far up the ramp does it roll? 33. A person pushes down on a lever 3.2 meters to lift a 850 N object 0.25 meters up. The person pushes down on the lever with 70 N of force. Find the efficiency of the lever. 70 N 850N 34. A 1.2 kg rock is dropped from 20 meters. The rock is going only 15 m/s just before it hits the ground because of air friction. A) How far does friction act on the rock? B) How far does the rock drop? C) Does all of the Ep turn into Ek? D) Does friction add or subtract energy? E) Find the force of air friction on the rock. 35. A 6 kg object going 2 m/s speeds up to 7 m/s due to a 4 N force. A) How many meters does the force act? B) What is the acceleration of the object? 36. A 4 kg mass going 6 m/s stops by compressing a spring 1.3 meters. Find the spring constant of the spring. (VEO) (VEO variables and equation only. Give equations, put in numbers, and do not solve.) 37. A 5 kg object is dropped from 30 meters up. How fast is it going 10 meters above the ground? (VEO) 38. A 3 kg object is originally at rest is pushed on by the force shown on the graph at the right. A) Find the work done on the object in the first 10 m. B) Find the final velocity of the object. Force (N) Force vs. Distance Distance (m)

18

19

20 PreAP Energy Notes - Ep W E k PEel P Due to height ONLY (path doesn t matter. Only the final position.) conservative (path doesn t matter). mg (weight) = Force, so F w times h = E p. Must be relative to a point (Can be negative) h is always vertical Double m, Ep doubles; etc. Graphing (E p vs m is linear; E p vs h is also linear). Force times distance. F must be in direction of motion (use sin or cos). W = E. Friction always removes energy so friction = -W. F could = mg, when lifting an object. Either F = mg OR W = E p No work done if object doesn t move. Units = joules. [in simplier units = Fd = mad = kg(m/s 2 )m = kgm 2 /s 2 ] Graphed (area under F vs D graph = work done. Can be negative area.) Energy of motion. Doesn t matter if it is above the ground or not. Can have other kinds of energy at the same time. Double m, E k doubles; Double v, E k quadruples. Always positive (since v is squared). Graphed (E k vs m is linear; E k vs v is quadratic [an upward parabola]) k is spring constant bigger for stronger spring X is distance from equilibrium position (at rest position) Always positive (since x is squared) Double k, doubles PE; Doubling x, quadriples PE. Graphed (E k vs m is linear; E k vs v is quadratic [an upward parabola]) rate of doing work or giving energy (how FAST you do work or change energy) Slope of W vs. time graph is power. P = W/t OR P = Fd/t OR P = E/t. You don t have to know the work to find P. You could know the amount of energy you gave the object. Units = watts. (Like joules, it can be expressed as simpler units.) Conservation of Energy If no work is done, the total amount of mechanical E remains constant. If there is work (in [gained] or out [lost, like friction]), then E total increases or decreases. Pendulums and roller coaster are good examples of systems where E p turns to E k and back. In the absence of friction, E total doesn t change. If there is friction, then E total decreases over time. Mechanical Energy any kind of E p or E k. Mechanical energy is organized energy (as opposed to thermal energy which is random) and easy to utilize easy to transfer to another kind. Energy is always conserved, but mechanical energy can be lost to thermal energy.