Energy Work Kinetic Energy Potential Energy

Transcription

1 Energy Work Kinetic Energy Potential Energy Lana Sheridan De Anza College Oct 25, 2017

2 Last time energy work

3 Overview Work as an integral Kinetic energy Work-Kinetic energy theorem Potential energy

4 N through the rtant. WorkMoving ving it 2 cm if system) undert force of magt. r orce on the itude Dr of Figure 7.2 An object undergoes For a constant applied force Work is defined as: u, where u is W = F r (7.1) Work done by a Work is the amount constant of energy force transferred to a system by an fined interaction terms with the environment. xplore Units: how Joules, to J. 1 J = 1 Nm t of application u S F S a displacement D r S under the action of a constant force F S.

5 Solutions Manual/Study Guide Work example Enhanced WebAssign Page 204, #1 Section 7.2 Work Done by a Constant Force 1. A shopper in a supermarket pushes a cart with a Q/C force of 35.0 N directed at an angle of below the horizontal. The force is just sufficient to balance various friction forces, so the cart moves at constant speed. (a) Find the work done by the shopper on the cart as she moves down a 50.0-m-long aisle. (b) The shopper goes down the next aisle, pushing horizontally and maintaining the same speed as before. If the friction force doesn t change, would the shopper s applied force be larger, smaller, or the same? (c) What about the work done on the cart by the shopper? 2. A raindrop of mass kg falls vertically at W constant speed under the influence of gravity and 3. 4.

6 e of the force. nt Work: represents more a general environment. definition N through What ifthe F is not constant? rtant. Moving S ving it 2 cm if F system) undert force of magt. represents the influence from the r orce on the itude Work, Dr of W Figure 7.2 An object undergoes a displacement D S r under the u, where For an u is applied force, action F(r): of a constant force S F. W = F(r) dr (7.1) Work done by a constant force Work is the amount of energy transferred to a system by an fined in terms interaction with the environment. xplore how to u S

7 If the size of the s The total work done for the We could break up our displacement distancefrom into x terms in the sum in little i to x f slices is and ask what force is applied in each slice. approximately The workequal for each to the little sum slice nite is value equal to of the areas of all the rectangles. W = F (x) x Work: more general definition F x Area = F x x Therefore, we can moves from x i to x f F x x i x a Then add them together: W total = x F x x The work done by the component As the length of the little slices goes to zero: F x of the varying force as the particle moves from x i to x f is exactly lim F x x = F (x)dx equal x 0 to the area under the curve. x x f x This equation redu constant. If more than on the total work don express the net for the particle moves

8 x x i x f This equation red x constant. a If more than o the total work do The work done by the component express the net fo F x of the W varying = F(r) force as drthe particle moves from x i to x f is exactly the particle moves Work done is theequal areato under the area a force-displacement under the curve. curve. Work: more general definition F x b x i Work Figure 7.7 (a) The work done on a particle by the force component 1 Figures from Serway F for & the Jewett. small displacement Dx is x f x For the general ca vary, we use the sc where the integra The subscript ex agent on the syste differentiate this w If the system c deformable), we c

9 Question te the What is the work The done net by work the done force indicated by this force in the graph as the particle movesis from the x area = 0under to x = 6the m? curve. force reases could ecause not fit ticle is otion, F x (N) x (m) Figure 7.8 (Example 7.4) The force acting on a particle is constant for the first 4.0 m of motion and then

10 Question te the force reases could ecause not fit ticle is otion, What is the work The done net by work the done force indicated by this force in the graph as the particle movesis from the x area = 0under to x = 6the m? curve. W = 25 J. F x (N) x (m) Figure 7.8 (Example 7.4) The force acting on a particle is constant for the first 4.0 m of motion and then

11 Springs and Work The force exerted by many types of springs is governed by Hooke s Law. F spring = kx where x is the amount of displacement of one end of a spring from its natural length. (The amount of compression or extension. k is the force constant or spring constant. 1 Figure from CCRMA Stanford Univ.

12 have large k values, and soft springs have small k values. As can be seen from Equation 7.9, the units of k are N/m. Spring Force a x 0 S F s x x When x is positive (stretched spring), the spring force is directed to the left. b x When x is zero (natural length of the spring), the spring force is zero. c S F s x x When x is negative (compressed spring), the spring force is directed to the right. kx max d 0 where x can take a positive or negative sign. F s kx x max F s F s = kxi W = F(x) dx x The work done by the spring force on the block as it moves from x max to 0 is the area of the shaded triangle, 1 2 kxmax 2. Fig by a the the (a) is n x fo

13 x Work done by a spring on a block b c d kx max S F s x x max F s 0 F s kx x x x When x is zero (natural length of the spring), the spring force is zero. The work done by the spring on the block as the spring moves the block from ( x max ) 0: When x is negative (compressed spring), the spring force is 0 directed to the right. W s = 0 The work done by the = spring force on the block as it moves from x max to 0 is the area 0 of the shaded = triangle, 1 2 kxmax 2. x max F(x) dx Figure 7.9 The fo by a spring on a bloc the block s position the equilibrium pos (a) x is positive. (b) is negative. (d) Grap x for the block sprin x max ( kx)i dx x max k x dx = 1 2 k(x max) 2

14 e for this disnt Work must done push by an block applied is carried force out very on slowly, block / spring If the process of moving the S n is opposite then F app is equal in magnitude S force Why as was the the spring compressed to begin with? Suppose it is an and opposite in direction to F s applied force. Suppose at all Ftimes. app does not allow the block to accelerate: block s v is constant. system by the (7.13) S F app S F s shing the ssed a disared with as much x i x max x f 0 Figure 7.10 A block moves from 0 x i 5 2x max to x f 5 0 on a friction- surface as a force S app = (kx)dx FW app app is = W s Wless x applied to max the block.

15 Question Quick Quiz A dart is inserted into a spring-loaded dart gun by pushing the spring in by a distance x. For the next loading, the spring is compressed a distance 2x. How much work is required to load the second dart compared with that required to load the first? (A) four times as much (B) two times as much (C) the same (D) half as much 2 Serway & Jewett.

16 Question Quick Quiz A dart is inserted into a spring-loaded dart gun by pushing the spring in by a distance x. For the next loading, the spring is compressed a distance 2x. How much work is required to load the second dart compared with that required to load the first? (A) four times as much (B) two times as much (C) the same (D) half as much 2 Serway & Jewett.

17 Work and Kinetic Energy Kinetic energy, K the energy that a system has as a result of its motion, or the motion of its constituent parts. K = 1 2 mv 2 but where did this expression come from? Why is it an energy?

18 Work and Kinetic Energy The net work done on a (particle) system is the total energy that is transferred to the system from the environment. Let the system s mass be m. How much work do we do in accelerating the system from rest to velocity v?

19 Work and Kinetic Energy The net work done on a (particle) system is the total energy that is transferred to the system from the environment. Let the system s mass be m. How much work do we do in accelerating the system from rest to velocity v? W net = = x 0 x 0 F net dx = m dv dt dx x 0 ma dx

20 Work and Kinetic Energy The net work done on a (particle) system is the total energy that is transferred to the system from the environment. Let the system s mass be m. How much work do we do in accelerating the system from rest to velocity v? W net = = = = x 0 x 0 x 0 v 0 F net dx = m dv dt dx m dx dt = 1 2 mv 2 mv dv x 0 dv dx dx ma dx (chain rule)

21 Work-Kinetic Energy Theorem For accelerating from v i to v f So, W net = vf v i mv dv = 1 2 mv 2 f 1 2 mv 2 i = K f K i W net = K This is the Work-Kinetic Energy Theorem, which could also be stated as: When the environment does work on a system and the only change in a system is in its speed, the net work done on the system equals the change in kinetic energy of the system.

22 Question Quick Quiz A dart is inserted into a spring-loaded dart gun by pushing the spring in by a distance x. For the next loading, the spring is compressed a distance 2x. How much faster does the second dart leave the gun compared with the first? (A) four times as fast (B) two times as fast (C) the same (D) half as fast

23 Question Quick Quiz A dart is inserted into a spring-loaded dart gun by pushing the spring in by a distance x. For the next loading, the spring is compressed a distance 2x. How much faster does the second dart leave the gun compared with the first? (A) four times as fast (B) two times as fast (C) the same (D) half as fast

24 F r 2F onceptual argument. Work and a simple machine Ramp Can Lessen we dothe lesswork work by Required? using a ramp to lift a refrigerator? k using claims ck if the valid? a hand ase, for k, DK 5 system oint of Because h Figure 7.14 (Conceptual Example 7.7) A refrigerator attached to (Assume the axle of the wheels on the cart is frictionless...) a frictionless, wheeled hand truck is moved up a ramp at constant speed. L u 5 W by man 1 W by gravity 5 0

25 Dxr 5 F F r Dx 5 F 2F Dx Dx is shorter as suggested by our conceptual argument. Work and a simple machine xample 7.7 load a refrigerator onto a truck using u as shown in Figure He claims uld be required to load the truck if the amp were increased. Is his claim valid? Does the Ramp Lessen the Work Required? e refrigerator is wheeled on a hand mp at constant speed. In this case, for refrigerator and the hand truck, DK 5 orce exerted by the ramp on the system 08 to the displacement of its point of so does no work on the system. Because k kinetic energy theorem gives work on the W ext 5fridge W by man 1 Wagainst by gravity 5 0 gravity: by the gravitational force equals the product of the weight mg of the system, the distance L through erator is displaced, and cos (u 1 908). Therefore, W by man 52W by gravity 521mg21L23cos 1u mgl sin u 5mgh u is the height of the ramp. Therefore, the man must do the same amount of work mgh on the system length of the ramp. The work depends only on the height of the ramp. Although less force is required mp, the point of application of that force moves through a greater displacement. h Figure 7.14 (Conceptual Example 7.7) A refrigerator attached to a frictionless, wheeled hand truck is moved up a ramp at constant speed. If we just lifted the fridge directly, we would require F = mg j to do L u tial Energy of a System apter, we have defined a system in general, but have focused our

26 Dxr 5 F F r Dx 5 F 2F Dx Dx is shorter as suggested by our conceptual argument. Work and a simple machine xample 7.7 load a refrigerator onto a truck using u as shown in Figure He claims uld be required to load the truck if the amp were increased. Is his claim valid? Does the Ramp Lessen the Work Required? e refrigerator is wheeled on a hand mp at constant speed. In this case, for refrigerator and the hand truck, DK 5 orce exerted by the ramp on the system 08 to the displacement of its point of so does no work on the system. Because k kinetic energy theorem gives work on the W ext 5fridge W by man 1 Wagainst by gravity 5 0 gravity: by the gravitational force equals the product of the weight mg of the system, the distance L through erator is displaced, and cos (u 1 908). Therefore, W by man 52W by gravity 521mg21L23cos 1u mgl sin u 5mgh u is the height of the ramp. Therefore, the man must do the same amount of work mgh on the system length of the ramp. The work depends only on the height of the = ramp. mgh Although less force is required mp, the point of application of that force moves through a greater displacement. h Figure 7.14 (Conceptual Example 7.7) A refrigerator attached to a frictionless, wheeled hand truck is moved up a ramp at constant speed. If we just lifted the fridge directly, we would require F = mg j to do W = F r = (mg j) (h j) L u tial Energy of a System apter, we have defined a system in general, but have focused our

27 Dxr 5 Dx 5 Dx 5 1 F r 2F 2 Dx is shorter as suggested by our conceptual argument. Work and a simple machine xample 7.7 load a refrigerator onto a truck using u as shown in Figure He claims uld be required to load the truck if the amp were increased. Is his claim valid? Does the Ramp Lessen the Work Required? e refrigerator is wheeled on a hand mp at constant speed. In this case, for refrigerator and the hand truck, DK 5 orce exerted by the ramp on the system 08 to the displacement of its point of so does no work on the system. Because k kinetic energy theorem gives F = mg sin W ext 5θ Wi by man (i 1 Wis by gravity up5along 0 the slope): by the gravitational force equals the product of the weight mg of the system, the distance L through erator is displaced, and cos (u 1 908). Therefore, W by man 52W by gravity 521mg21L23cos 1u mgl sin u 5mgh u is the height of the ramp. Therefore, the man must do the same amount of work mgh on the system length of the ramp. The work depends only on the height of the ramp. Although less force is required mp, the point of application of that force moves through a greater displacement. h Figure 7.14 (Conceptual Example 7.7) A refrigerator attached to a frictionless, wheeled hand truck is moved up a ramp at constant speed. If we pushed the fridge up a ramp, we would require W = F r = (mg sin θ i ) (Li ) L u tial Energy of a System apter, we have defined a system in general, but have focused our

28 Dxr 5 Dx 5 Dx 5 1 F r 2F 2 Dx is shorter as suggested by our conceptual argument. Work and a simple machine xample 7.7 load a refrigerator onto a truck using u as shown in Figure He claims uld be required to load the truck if the amp were increased. Is his claim valid? Does the Ramp Lessen the Work Required? e refrigerator is wheeled on a hand mp at constant speed. In this case, for refrigerator and the hand truck, DK 5 orce exerted by the ramp on the system 08 to the displacement of its point of so does no work on the system. Because k kinetic energy theorem gives F = mg sin W ext 5θ Wi by man (i 1 Wis by gravity up5along 0 the slope): by the gravitational force equals the product of the weight mg of the system, the distance L through erator is displaced, and cos (u 1 908). Therefore, W by man 52W by gravity 521mg21L23cos 1u mgl sin u 5mgh u is the height of the ramp. Therefore, the man must do the same amount of work mgh on the system length of the ramp. The work depends only on the height = of the mgl ramp. Although sin θless force is required mp, the point of application of that force moves through a greater displacement. tial Energy of a System apter, we have defined a system in general, but have focused our h Figure 7.14 (Conceptual Example 7.7) A refrigerator attached to a frictionless, wheeled hand truck is moved up a ramp at constant speed. If we pushed the fridge up a ramp, we would require W = F r = (mg sin θ i ) (Li ) = mgh L u

29 Work and a simple machine Implication: the ramp allows us to use less force, but we still must do the same amount of work.

30 is shorter as suggested by our conceptual argument. Potential Energy xample 7.7 load a refrigerator onto a truck using u as shown in Figure He claims uld be required to load the truck if the amp were increased. Is his claim valid? Does the Ramp Lessen the Work Required? e refrigerator is wheeled on a hand mp at constant speed. In this case, for refrigerator and the hand truck, DK 5 orce exerted by the ramp on the system 08 to the displacement of its point of so does no work on the system. Because k kinetic energy theorem gives refrigerator W ext 5through W by man 1 W by a gravity height 5 0 h required work W = mgh regardless by the gravitational force equals the product of the weight mg of the system, the distance L through erator is displaced, of whether and cos (u 1 we 908). Therefore, lifted it directly or pulled it up an incline. W by man 52W by gravity 521mg21L23cos 1u Now, imagine 5 mgl sin uwe 5mghdrop it. What will its final velocity be? u is the height of the ramp. Therefore, the man must do the same amount of work mgh on the system length of the ramp. The work depends only on the height of the ramp. Although less force is required mp, the point of application of that force moves through a greater displacement. h Figure 7.14 (Conceptual Example 7.7) A refrigerator attached to a frictionless, wheeled hand truck is moved up a ramp at constant speed. Think again about the refrigerator lifting problem. To lift the L u tial Energy of a System apter, we have defined a system in general, but have focused our ily on single particles or objects under the influence of external w consider systems of two or more particles or objects interacting

31 is shorter as suggested by our conceptual argument. Potential Energy xample 7.7 load a refrigerator onto a truck using u as shown in Figure He claims uld be required to load the truck if the amp were increased. Is his claim valid? Does the Ramp Lessen the Work Required? e refrigerator is wheeled on a hand mp at constant speed. In this case, for refrigerator and the hand truck, DK 5 orce exerted by the ramp on the system 08 to the displacement of its point of so does no work on the system. Because k kinetic energy theorem gives refrigerator W ext 5through W by man 1 W by a gravity height 5 0 h required work W = mgh regardless by the gravitational force equals the product of the weight mg of the system, the distance L through erator is displaced, of whether and cos (u 1 we 908). Therefore, lifted it directly or pulled it up an incline. W by man 52W by gravity 521mg21L23cos 1u Now, imagine 5 mgl sin uwe 5mghdrop it. What will its final velocity be? u is the height of the ramp. Therefore, the man must do the same amount of work mgh on the system length of the ramp. The work depends only on the height of the ramp. Although less force is required mp, the point of application of that force moves through a greater displacement. tial Energy of a System apter, we have defined a system in general, but have focused our ily on single particles or objects under the influence of external w consider systems of two or more particles or objects interacting h Figure 7.14 (Conceptual Example 7.7) A refrigerator attached to a frictionless, wheeled hand truck is moved up a ramp at constant speed. Think again about the refrigerator lifting problem. To lift the vf 2 = vi 2 + 2ax vf 2 = 0 + 2gh v f = 2gh L u

32 Potential Energy v f = 2gh The final Kinetic Energy of the fridge is 1 2 mv f 2 = 1 2 m(2gh) = mgh

33 Potential Energy v f = 2gh The final Kinetic Energy of the fridge is 1 2 mv f 2 = 1 2 m(2gh) = mgh So an object of mass m at a height h has the potential to possess an amount of kinetic energy K = mgh.

34 Potential Energy v f = 2gh The final Kinetic Energy of the fridge is 1 2 mv f 2 = 1 2 m(2gh) = mgh So an object of mass m at a height h has the potential to possess an amount of kinetic energy K = mgh. We call this potential for kinetic energy potential energy, U.

35 Potential Energy Notice that the refrigerator has potential energy once it is lifted because of its configuration in space. It was a height h above the ground, and therefore could fall a distance h. Also notice that the amount of work done on the fridge was W = mgh.

36 Summary Kinetic Energy Work-Kinetic Energy Theorem Potential Energy 2nd Collected Homework! due Friday, Oct 27. 2nd Test Friday, Nov 3. (Uncollected) Homework Serway & Jewett, Read Chapter 7. Ch 7, onward from page 205. Probs: 25, 35, 39 (41, 43 - wait to do these)