Introduction to Mechanics Potential Energy Energy Conservation

Transcription

1 Introduction to Mechanics Potential Energy Energy Conservation Lana Sheridan De Anza College Nov 28, 2017

2 Last time power conservative and nonconservative forces friction

3 Overview conservative forces and potential energy potential energy diagrams mechanical energy energy conservation

4 distance is h, the work you do on the box is W = mgh. If you now release the box and allow it to drop back to the floor, gravity does the same work, in the process gives the box an equivalent amount of kinetic energy. Conservative Forces: Work Done Lifting W = mgh, a Box and Work done by person (applied force) W = Fd cos(0 ) = mgh. Work done by person = mgh Work done by gravity = mgh FIGURE 8 Lifting a bo stant speed box is releas work on the conservativ h Contrast Whenthis box with falls, the force this energy of kinetic becomes friction, which kinetic is nonconservative. energy. To slide a box of mass m across the floor with constant speed, as shown in Figure 8 2, you must Wexert net = a mgh force of = magnitude K. m k N = m k mg. After sliding the box a distance d, the work you have done is W = m k mgd. In this case, when you release the box it simply stays put friction does no work on it after you let go. Thus, the work

5 in the process gives the box an equivalent amount of kinetic energy. Conservative Forces: Potential Energy Work done by person = mgh Work done by gravity = mgh FIGURE 8 Lifting a bo stant speed box is releas work on the conservativ h Contrast this with the force of kinetic friction, which is nonconservative. To slide a box of mass m across the floor with constant speed, as shown in Figure 8 2, you must exert a force of magnitude m k N = m k mg. After sliding the box a distance energy. d, the work you have done is W = m k mgd. In this case, when you release the box it simply stays put friction does no work on it after you let go. Thus, the work done by a nonconservative force cannot be recovered later as kinetic energy; instead, it is converted to other forms of energy, such as a slight warming of the When the box is in the air, it has the potential to have kinetic The man put in work lifting it, as long as the box is held in the air, this energy is stored.

6 Conservative Forces: Potential Energy Any box that has been lifted a height h has had the same work done on it: mgh. The path the box took to get to that height doesn t matter. This is because gravity is a conservative force.

7 Conservative Forces: Potential Energy Any box that has been lifted a height h has had the same work done on it: mgh. The path the box took to get to that height doesn t matter. This is because gravity is a conservative force. For any conservative force acting on an object, we can say that the object has some amount of stored energy that depends on its configuration.

8 Conservative Forces: Potential Energy Any box that has been lifted a height h has had the same work done on it: mgh. The path the box took to get to that height doesn t matter. This is because gravity is a conservative force. For any conservative force acting on an object, we can say that the object has some amount of stored energy that depends on its configuration. Potential energy energy that system has as a result of its configuration. Is always the result of the effect of a conservative force.

9 Potential Energy Potential energy energy that system has as a result of its configuration. Is always the result of the effect of a conservative force. Only conservative forces can have associated potential energies! If a nonconservative force acts, any work done to displace the system (at constant velocity) leaves the system again as heat and sound. That energy isn t stored no potential energy.

10 Gravitational Potential Energy The change of potential energy when lifting an object of mass m near the Earth s surface: U = mg( h) If we choose the convention that U = 0 at the Earth s surface, then an object (mass m) at a height h has gravitational potential energy: U = mgh

11 Gravitational Potential Energy The change of potential energy when lifting an object of mass m near the Earth s surface: U = mg( h) One technical point: in order for a box to be at one height or another, we need the Earth (which creates the gravitational force on the box) to be part of our system description.

12 Gravitational Potential Energy The change of potential energy when lifting an object of mass m near the Earth s surface: U = mg( h) One technical point: in order for a box to be at one height or another, we need the Earth (which creates the gravitational force on the box) to be part of our system description. The configuration of the system refers to how close the box is to center of the Earth. To have a potential energy, we must include the Earth in the system and make the weight of the box an internal force.

13 tional potential energy of the stone Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential energy of the system from release to reaching the bottom of the well? Gravitational Potential Energy or on nt raec- e- w- ct rts so es. of nd arnd ur ge he 42. A 400-N child is in a swing that is attached to a pair W of ropes 2.00 m long. Find the gravitational potential energy of the child Earth system relative to the child s lowest position when (a) the ropes are horizontal, (b) the ropes make a angle with the vertical, and (c) the child is at the bottom of the circular arc. Section 7.7 Conservative and Nonconservative Forces 43. A 4.00-kg particle moves M from the origin to position, having coordi- Q/C nates x m and y m (Fig. P7.43). One force on the particle is the gravitational force y (m) er 1 Problem from Serway & Jewett, 9th ed, page 207. (5.00, 5.00)

14 tional potential energy of the stone Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential energy of the system from release to reaching the bottom of the well? Gravitational Potential Energy or on nt raec- e- w- ct rts so es. of nd arnd ur ge he er 42. A 400-N child is in a swing that is attached to a pair W of ropes 2.00 m long. Find the gravitational potential energy of the child Earth system relative to the child s lowest position when (a) the ropes are horizontal, (b) the ropes make a angle with the vertical, and (c) the child is at the bottom of the circular arc. (a) Section U = (mg)y 7.7 Conservative = (400 N)(2and m) Nonconservative = 800J Forces 43. A 4.00-kg particle moves y (m) M from the origin to position, having coordi- Q/C (5.00, 5.00) nates x m and y m (Fig. P7.43). One force on the particle is 1 Problem the gravitational from Serway & Jewett, force 9th ed, page 207.

15 tional potential energy of the stone Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential energy of the system from release to reaching the bottom of the well? Gravitational Potential Energy or on nt raec- e- w- ct rts so es. of nd arnd ur ge he er 42. A 400-N child is in a swing that is attached to a pair W of ropes 2.00 m long. Find the gravitational potential energy of the child Earth system relative to the child s lowest position when (a) the ropes are horizontal, (b) the ropes make a angle with the vertical, and (c) the child is at the bottom of the circular arc. (a) Section U = (mg)y 7.7 Conservative = (400 N)(2and m) Nonconservative = 800J Forces 43. A 4.00-kg particle moves y (m) (b) U = (mg)y = (400 N)(2 m)(1 cos 30 from the origin to position, having coordi- ) = 107J M Q/C (5.00, 5.00) nates x m and y m (Fig. P7.43). One force on the particle is 1 Problem the gravitational from Serway & Jewett, force 9th ed, page 207.

16 tional potential energy of the stone Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential energy of the system from release to reaching the bottom of the well? Gravitational Potential Energy or on nt raec- e- w- ct rts so es. of nd arnd ur ge he er 42. A 400-N child is in a swing that is attached to a pair W of ropes 2.00 m long. Find the gravitational potential energy of the child Earth system relative to the child s lowest position when (a) the ropes are horizontal, (b) the ropes make a angle with the vertical, and (c) the child is at the bottom of the circular arc. (a) Section U = (mg)y 7.7 Conservative = (400 N)(2and m) Nonconservative = 800J Forces 43. A 4.00-kg particle moves y (m) (b) U = (mg)y = (400 N)(2 m)(1 cos 30 from the origin to position = 0., having coordi- ) = 107J M (c) Q/CU (5.00, 5.00) nates x m and y m (Fig. P7.43). One force on the particle is 1 Problem the gravitational from Serway & Jewett, force 9th ed, page 207.

17 Spring Force: Another Conservative Force The spring force is also a conservative force. If we stretch a spring, we can say that the spring stores the energy. That energy is converted to kinetic energy when the end of the spring is released.

18 Spring Force: Another Conservative Force The spring force is also a conservative force. If we stretch a spring, we can say that the spring stores the energy. That energy is converted to kinetic energy when the end of the spring is released. There is also spring potential energy! Choosing U = 0 when the spring is at its natural length (relaxed): U = 1 2 kx 2 (The spring must be part of our system.)

19 Potential Energy Diagrams U 1 2 kx 2 s Potential energy can be plotted as a function of position. eg. potential energy of200 a spring: Chapter 7 Energy of a System U s a x max 0U s 1 2 a kx 2 U s The restoring force exerted by the spring always acts toward x 0, x the x position max of stable 0 equilibrium. E x max The restoring force x 0 exerted x max by the spring b always acts toward x 0, the Figure position 7.20 of stable equilibrium. (a) Potential energy as a function of x for the frictionless block spring system shown F in s S (b). For a given energy E of the sys- m E F sp = kx 1 Figure from Serway & Jewett, 9th ed. x max S F s x energy function for a block plotted versus x in Figure 7.2 exerted by the spring on the As we saw in Quick Quiz 7.8 tive energy of the slope function of the for U-ve a equilibrium plotted versus position x of in the Figs some exerted external by force the spring F ext acts o equilibrium, x is positive an exerted by the spring is nega released. If the external forc negative; As we therefore, saw in Quick F s is posi Q release. tive of the slope of th From this analysis, we con equilibrium position o tem is one of stable equilibr results some in external a force directed force F ba e tem equilibrium, in stable equilibrium x is posit co minimum. exerted by the spring i released. If the block If in the Figure extern 7.20 from negative; rest, its total therefore, energy init F s As release. the block starts to move, energy. The block oscillates From this analysis, 2x max and x 5 1x max, called

20 Potential Energy Diagrams Recall that the work 192 done by a force is the area under the CHAPTER 7 WORK AND KINETIC ENERGY force-displacement curve. Force Area = W show that the corresp gin. Therefore, the w the general position 1 area is equal to 2 1bas kx. As a result, the wo needed to compress Work to Stretch or Co W = 1 2 kx2 SI unit: joule, J O x Position FIGURE 7 10 Work needed to stretch We can get a feelin The work done by a spring the spring a distance relates x to the change in the spring spring in the follow potential: The work done is equal to the shaded area, which is a right triangle. The area EXERCISE 7 4 of the triangle Wis sp 1 = 2 1x21kx2 U = sp 1 2 kx2. The spring in a pinbal So the area is also equal to U sp. required to compress t kx

21 Potential Energy Diagrams Comparison: Area under v-t graph = x. Slope of x-t curve = v. CHAPTE Area under F -x graph = U. 192 CHAPTER 7 WORK AND KINETIC ENERGY Slope 200 of U-x curve Chapter = F 7. Energy of a Sys ical position, vertical velocity, and vertical acceleration vs. time for a rock thrown vertically up at the edge of a cliff. Notice that velocity changes linearly 1 t acceleration is constant. Misconception Alert! Notice that the position vs. time graph shows vertical position only. area It is easy is to equal get the impression to that some horizontal motion the shape of the graph looks like the path of a projectile. But this is not the case; the horizontal axis is time, not space. The rock in space is straight up, and straight down. Force tion of these results is important. At 1.00 s the rock is above its starting point kx and heading upward, since and are both Work to Stretch or Compress a Spring a Distance x from Equilibrium 0 s, the rock is still above its starting point, but the negative velocity means it is moving downward. At 3.00 s, both and Area = W eaning the rock is below its starting point and continuing to move downward. Notice that when the rock is at its highest point (at W = 1 2 kx2 city is zero, but its acceleration is still. Its acceleration is for the whole trip while it is moving up and x max 0 x max x ing down. Note that the values for are the positions (or displacements) of the rock, not the total distances traveled. Finally, note plies to upward motion as well Oas downward. Both have the same acceleration the x Position acceleration due to gravity, which remains ntire time. Astronauts training in the famous Vomit Comet, for example, experience free-fall while arcing up as well as down, as we more detail later. FIGURE 7 10 Work needed to stretch show that the corresponding force curve Uis a straight line extending from the s energy f gin. Therefore, the work we do 1 U kx in 2 stretching a spring from x = 0 (equilibrium the general position x is s the shaded, 2 triangular area shown in Figure plotted v 2 1base21height2, E where in this case the base is x and exerted the heig 1 kx. As a result, the work is 2 1x21kx2 = 1 2 kx2. Similar reasoning shows that the w 1 needed to compress a spring a distance x is also 2 kx2. Therefore, As we sa7 tive of th equilibr SI unit: joule, J a We can get a feeling for the amount of work required to compress a typi

22 Potential Energy, Conservative Force, & Equilibrium 200 Chapter 7 Energy of a System The value of a conservative force F at a particular point can be found as the slope of the potential energy curve: U s 1 2 kx 2 U s E energy fun plotted ver exerted by a x max 0 x max F sp = (slope of U(x)) = kx If F is the only force acting on the particle, stationary points (slope = 0) are force equilibrium points. x As we saw tive of the equilibrium some exter equilibrium exerted by

23 Energy Diagrams and Equilibrium 200 Chapter 7 Energy of a System System is in equilibrium when F net = F sp = 0. 1 U 2 kx 2 s U s E energy function plotted versus x exerted by the sp x max 0 x max As we saw in Qu tive of the slope a equilibrium posi In this case, the force is always back toward the x = 0 point, so some external fo this is a stable equilibrium. equilibrium, x is Examples: exerted by the sp The restoring force exerted by the spring force spring always acts toward x 0, released. If the e ball inside a bowl negative; therefo the position of stable equilibrium. x

24 onfiguration called neutral equilibrium arises when U is constant ion. Energy Small displacements Diagrams of and object Equilibrium from a position in this region er restoring nor disrupting forces. A ball lying on a flat, horizontal xample of an object in neutral equilibrium. gure 7.21 A plot of U versus or a particle that has a position unstable equilibrium located x 5 0. For any finite displacent of the particle, the force on e particle is directed away from 0. U Positive slope Negative slope x 0 x 0 0 x In this case, the force is always away from the x = 0 point, so this is a unstable equilibrium. Examples: the L1 Lagrange point between the Sun and Earth ball on upside-down a bowl

25 Neutral Equilibrium A system can also be in neutral equilibrium. In this case, no forces act, even when the system is displaced left or right. Example: ball on a flat surface

26 Mechanical Energy The mechanical energy of a system is the energy that can be used to do work. It is defined as the sum of the system s kinetic and potential energy: E mech = K + U

27 Mechanical Energy The mechanical energy of a system is the energy that can be used to do work. It is defined as the sum of the system s kinetic and potential energy: E mech = K + U The mechanical energy of a system can change under two circumstances: nonconservative forces act on the system decreasing the mechanical energy other external forces act that may add energy to the system or reduce it (applied forces)

28 Mechanical Energy If the system is isolated: no friction, no air resistance, no external applied work, then the mechanical energy is conserved: E mech = K + U = 0

29 Energy Conservation Over the whole of the universe, we believe energy is conserved. That means energy is neither created nor destroyed, but may change form.

30 Energy Conservation Over the whole of the universe, we believe energy is conserved. That means energy is neither created nor destroyed, but may change form. However, the energy of our system, which is only a tiny part of the universe, may gain or lose energy unless it is isolated. This can be expressed as: W ext = K + U The work done by external forces includes work done by nonconservative forces and applied forces.

31 Summary mechanical energy energy conservation Homework Walker Physics: Ch 8, onward from page 243. Questions: 1, 3, 11, 13; Problems: 9, 13, 15, 17, 21, 23, 37, 41, 47, 55, 57, 59, 87, 95