Chapter 10: Work & Energy

Transcription

1 Chapter 0: Work & Energy WORK My family owned at one point a Paletria in Tucson, AZ. As many already know, it is very hot in Tucson (usually have 00+ days over 00 o F or 40 o C) and therefore, the paleta business was very good. I always remember my parents putting me to work by moving boxes of fruit here, there, everywhere it seemed like the work was endless. I was always volunteered to move these boxes of fruit. Picturewise, I imagine me working moving the boxes of fruit as So work depends on two things: (i) how much force I used to move the box and (ii) how far I had to move the box. Definition The work done by a constant force is Work Force displacement F x Units of Work W F x Nm Joule J Force along the displacement displacement Force along the force displacement Work only makes sense in the language of energy. In fact, energy is defined by the Work-energy theorem. Most people expect that if you do work, you should get something as a result. Physics is no different when a net force performs work on an object there is always a result for the effort. The result is a change in energy. Work-Energy Theorem When a F ext does work on a system (or object), the energy of the system changes from E to E : Wtotal Wext E E E To interpret this statement, when work is done on an object, the work does not disappear but reappears (i.e. transferred) as something, which we call energy. So, what is energy? Energy is the measure of a system s capacity to do work. Stated differently, energy is stored work. Interpretation. Work depends on both displacement and force, and there are two ways to do work: Examples of this is (i) U-Haul truck ramps and using (ii) car jacks for raising car to change tires.

2 . There are only two ways to do work on system. 3 ways to do work Positive work When the displacement and force point in the same direction, the positive work is done on the system and the system s energy changes by an overall increase. That is, there is more stored work as energy in the system or, one can think of it as there is a flow of energy into the system. Negative work When the displacement and force point in opposite directions, negative work is done on the system and the system s energy changes by an overall decrease. That is, there is less stored work as energy in the system Zero work When the displacement and force are perpendicular to each other, no work is done on the system and the system s energy does not change. That is, perpendicular forces do not change the energy of the system. Wext 0 E E E However, if there is no change in the energy of the system (W ext 0), it can mean two things: (i) forces are perpendicular to the displacement (which we just discussed) or (ii) one external force is working to increases the energy of the system while a second external force is decreasing at the same rate of increase, so the total external work is zero; then the energy of the system does not change. Wext Wincreases Wdecreases 0 E E Stated differently, the amount of energy flowing is equal to the amount of energy flowing out of the system, leaving the total energy of the system unchanged. 3. The Meaning of a Joule of Work Work and energy units are commonly measured on two different scales, the Joule (metric system) or calorie (imperial system): 3 calorie 4.86 J and 000cal Cal J food calorie Physically, the work done by a Joule is the work done in lifting a quarterpounder a distance of m. But let try to put this into an everyday type of situation. Let s say that you go out and have a McDonald s thick chocolate milkshake (a mere 60 Calories). After enjoying it you start feeling guilty and feel the need to work off the shake by bench pressing the calories off. Let s say you use 60 lbs 70 N as your bench-pressing weight. How many times

3 must you lift it to compensate for the calories in the milkshake? I raise the bar 0.65 m above my chest, and lower it the same distance. How much work is done ON the bar? Lifting up phase: Lowering phase: W 0 W F x 460 J F x F antiparallel to x up W 0 W 460 J down 70 N 0.65 m The net work done on the bar is W W W J net lifting lowering Does this make sense? Absolutely since the displacement is zero there is no net work done on the bar. However, your body does work since your heart applies a force and moves your blood some distance. The work that your body does is twice the work of lifting the bar: W W J 000 J body lifting Converting Joules into more familiar units of food calories, Cal W 000 J 0.38 Cal body J So the number of reps that are required to "burn-off" a 60 Cal milkshake from McDonald s is number reps to 60 Cal 4,900 reps burn off milk shake 0.38 Cal However, your body immediately takes 5% of these calories, reducing this to 3650 reps Does it pay to count calories when trying to be healthy - absolutely! 4. Forces that do or do not do work Suppose a stone block is dragged to the right. From the work-energy theorem, if work is done, we know that the energy of the system either increases or decreases. So, the key here is to identify the system (this is the block) and determine which forces will change or not change the energy of the system. I start by circling the block with a purple circle and then identifying which forces are acting on it (we do this by drawing a FBD). Which forces are working to change the energy of the block? As stated above, forces that are perpendicular to the displacement do not change the energy of the system. From the FBD, we see that N, mg and F y are perpendicular to the displacement (N, mg, F y d) and therefore, these forces workings do not work on the block: W W W 0 N mg F y There are two forces, F x and f k, that are parallel/antiparallel to the displacement and do work and will change the energy of the system. The parallel force F x will increase the energy of the block, while the antiparallel force f k will decrease the energy of the system such that the total external workings acting on the block is W W W ext Fy f 3

4 There are three possibilities: () W ext > 0, () W ext < 0, (3) W ext = 0.. If W ext > 0, then this means that W Fx > W f, and the energy of the system increases, causing the block to accelerate. That is, there is more energy flowing into the system then there is flowing out, increasing the energy of the system.. If W ext < 0, then this means that W f > W F, and the energy of the system decreases and the block will decelerate. In other words, there is more energy flowing out of the system then there is flowing in, decreasing the energy of the system. 3. If W ext = 0, then this means that W Fx = W f, and the energy of the system does not change and the block remains in equilibrium. That is, the amount of energy flowing due to W Fx is also flowing out due to the negative work W f. Example 0. A girl files a kite with the string at a 30 0 angle to the horizontal. The tension in the string is 4.5 N. How much work does the string do on the girl if the girl a. Stands still? b. Runs a horizontal distance of 0 m away from the kite? c. Walks a horizontal distance of 0 m toward from the kite? Solution a. The girl is standing still and there is zero displacement, so there is zero work done on the girl by the string: Wa Txx T x(0 m) 0 J Wa b. As the girl walks in the opposite direction of the tension, the tension is doing negative work and reduces the energy of of the system or girl. The work of the string is W T x 4.5Ncos30 m 43 J W b x c. If instead, the girl walks towards the kite, the tension and the displacement are in the same direction, so positive work is being done by the tension and increases the energy of the system. The work of the string is W T x 4.5 N cos30 0 m 78 J W c x c The Work-Energy Theorem and the definition of Energy As we have already defined the work-energy theorem, anytime work is done on a system, this work does not disappear but gets transferred into energy. There are three classes of energy that we will study, kinetic energy KE, potential energy PE and thermal or frictional energy, which is defined by the work-energy theorem: b 4

5 K kinetic energy Wext E E E U potential energy Eth thermal energy. KINETIC ENERGY KE When external work is done to change the motion of a system, this work is transferred into the change in kinetic energy and the system s energy changes. Suppose an automobile (our system) that needs a push start to get started and a sucker has been recruited to do this. I start by isolating the car (the system) and identifying which forces do work. Since the normal and the weight are perpendicular to the displacement, neither of these workings will not change the energy of the automobile. However, the force F is parallel to the displacement, and therefore, will increase the energy of the automobile in the form of a change in motion where in this case, the automobile is accelerating. To calculate the external work (W ext = W F = F x), we do the following: (i) replace the force F using NL and (ii) replace x using kinematics. When this is done, the work WF is determined to be NL: F ma Kinematics : v ext WF F x ma mv ext mv v ax a We see here that the work-energy theorem starts with the work expression, force times displacement, and rewrites this to define a new expression called kinetic energy: v v W mv mv K where K kinetic energy mv Units: [K] = Joules = J Remarks. Only changes in KE are meaningful. That is, it is meaningless to speak about KE at a single point. Remember, it is the external work on the system that changes the KE and this change is the only thing that is meaningful. In some sense, KE has to be measure relative to a reference KE point. W K K ext K W K ext. What the work-energy theorem also tells us is that there is a new way to describe acceleration. If an automobile is speeding up, the external work done increases the overall energy (KE) of the system. On the other hand, if an automobile is slowing down, then the external work decreases the overall energy (KE) of the automobile. 3. KE depends on mass & velocity, however, the meaning of KE is the amount of work required to change the KE of an object. Velocity dependence A slower automobile requires less external work to bring to a stop compared to a faster one. Therefore, the small automobile has a smaller change in KE while the faster automobile has a larger change in KE. 5

6 Mathematically, we write 0 mv K W ext v K 0 m Of course, this is no surprise, one expects to do more work to stop a faster moving car. Mass dependence A smaller automobile requires less external work to bring to a stop compared to a larger truck traveling at the same speed. Therefore, the small automobile has a smaller change in KE while the larger truck has a larger change in KE. W ext Mathematically, we write 0 mv K W m 0 v ext K W ext Of course, this is no surprise, one expects to do more work to stop a larger truck than a smaller moving automobile. 4. Car s driving at different speeds require more work to stop: speed v: 0 mv 0 K W ext speed v: 0 m(v ) 0 4K 4W ext speed 3v: 0 m(3v ) 0 9K 9W ext In other words, an automobile (assuming tires lock, which they do not) moving twice as fast leaves a skid mark 4 times longer whereas a car traveling three times faster leaves a s kid mark 9 times longer. Question: which requires more work to catch a bowling ball or a baseball with the same KE? Neither! In catching either ball, the final KE is zero and depends only on the initial KE values. According to the work-nrg theorem, they require the same amount of work and therefore, equally safe: K v m 0 m 0 v W ext. Potential Energy PE According to the work-energy theorem, anytime an external force does work on a system, there is a change in energy of the system. This change in energy is dependent 6

7 on a change in position/configuration of the system. A potential energy function can only be defined when a conservative force does the work. The work by a conservative force can only occur if the work from point A to point B is path independent (and consequently, the work only depends on the end points). Unfortunately, to properly define this more mathematics is required (knowledge of line integrals is required), however, we can physically interpret this it. Before doing this, the key signature of the work done by a conservative force and its relation to the potential energy function is Wext U U U There are two conservative forces that we will study in this course, the spring and gravitational forces. Spring Force and Spring Potential Energy The Spring force is given by Hooke s law: F F k x [k] = [F]/[x] = N/m spring The spring constant k tells us the strength of the spring. If a spring is strong (harder to stretch), the value of k is large or a small k-value states it s easy to stretch. The important part of Hooke s law thought is the x-term. This term says that when an object is moving away from the equilibrium point, the spring force always points back towards the equilibrium. In fact, the spring force is known as a restoring force for this reason. DEMO Use a dynamics cart with two springs Suppose that our system is the block that is attached to a spring and experiences the force of Carlos hand to compress it. What is not drawn is the gravitational force, which is perpendicular to the displacement and will not change the energy of the block. What will change the energy of the block is Carlos hand F hand and the spring s force F spring. The work-energy claim states that whenever work is done, there is a change in energy. Question: The work W hand increases the energy of the block and this is transferred into the spring and the spring increases in energy. The key point here is that the increase in energy of the spring is associated with the negative work of the spring such that Wspring E Uspring US The definition of the work is that it is the area of the force vs. position curve ELASTIC (Spring) POTENTIAL ENERGY U S As stated earlier, the spring force is a conservative force and therefore, a potential energy function can be defined. In order to derive this potential energy function, one has to determine the work where W area of F vs. x curve In the case of the spring force (F S = kx) the area is half of a rectangular as the block is compressing the spring from its equilibrium point to some point x. This negative work of the spring is equal to the change in potential energy of the spring: W kx kx U U U or U kx S S S S S The reference point where the PE is measure is where the PE is at its zero point or equilibrium since S equilibrium S U (x ) 0 at x 0 7

8 Force of Gravity and the Gravitational Potential Energy U g The force of gravity is a conservative force and therefore, one can define a gravitational PE function GPE. The key similarity between the spring and gravitational PE is that they are equal to the negative work done by the conservative force. Physically what does this gravitational PE function mean? Suppose the system is defined as the brick sitting on top of the ground and an external force is applied to lift the brick off the ground. There are two forces acting on the brick, F ext and F g. The work that the external force F ext does changes the energy of the brick by changing its height. However, associated with this increase in energy is the negative work of the gravitational force. This is exactly the same situation we encountered with the work of the spring force and its associated spring PE. Physically, what happens is that as the brick is lift, the work done is stored in the earth s gravitational. This increases in gravitational PE is exactly the same as the negative work of the gravitational force acting on the brick. Analogy As we saw with the case of the spring PE, the work done is stored in the springs themselves, which is easy to see. On the other hand, the work done to increase the gravitational PE is said to be transferred to earth s gravitational field. Since the gravitational force is a noncontact force, this is much harder to visualized but there a similarly of the spring PE that we can use. Suppose we have a spring attached to a brick where there is an external work that changes the position of the brick. Where did all the work go in stretching the spring, into the springs themselves. In essence, that is what happens when a brick is lifted off of the ground and stores PE into earth s gravitational field. Calculating the work done by gravity from the area under the F g vs. y curve and the work-energy theorem, we write Wg mgy mgy Ug Ug U g or Ug mgy As stated earlier, only changes in PE are physically meaningful. However, to simplify a calculation or discussion, we sometimes would like to say that a certain GPE value is associated with a particular system configuration or at a certain height between the object and earth. We do this by defining a reference point (U g (y ref ) = 0) from which the PE is measured and then we determine the height change from there. If we are interested in measuring height changes from the ground (because the ground is where the object will fall to), then choose the ground to be the reference point (y = 0). On the other hand, if we are dropping something relative to a table top, then choose the table top to be the reference point (y = 0). U (y ) 0 at y 0 U U (y) 0 mgy g ref ref g g arbitrary since only changes are meaningful Example 0. a. A 0-kg car accelerates from 0.0 m/s to 8.6 m/s. (i) determine which forces change the energy and (ii) draw a work-energy bar diagram of the situation. (iii) Calculate the work done by the car that changes it energy. Solution 8

9 The system is the automobile. Which forces change the energy of the automobile? The normal and the weight are perpendicular to the displacement, and will not change the energy of the car. However, we see that the frictional force (since its parallel to the displacement) will change the energy of the automobile. Using the work-energy theorem, W K mv mv f 0 kg (8.6) (0.0) m /s 398,000 J W f b. There is well known ultramarathon race called the Badwater 35. The race starts at the lowest US continental point in Death Valley (85.0 m (-8 ft) below sea level) and ends at the base of nearby Mt. Whitney (Whitney Portal) at an elevation of 440 m (+8374 ft). (i) Determine which forces change the energy and (ii) draw a work-energy bar diagram of the situation. (iii) Calculate the change in gravitation potential energy of a 65.0 kg runner who makes it from the floor of Death Valley to Mt. Whitney Portal? Solution Here is the elevation profile of the race. Our system is the runner and we are looking at what forces change the energy of the system. If we look at the elevation profile, note that the majority of the work of the runner is traveling the horizontal distance (35 miles) compared to the vertical distance (.8 miles). If we look at the change in gravitational PE, then we are only concerned with the negative vertical work done (against gravity) and this elevation profile changes to the image on the right. The negative work of gravity produces an increase in gravitational PE given by W U mg y 65 kg 9.8m/s 4505 m g g J Ug Question, does the runner consume a large number of food carloie running up this vertical distance? To answer this question, we need to convert this gravitational PE into food calories ( Cal = 400 J): 6 Cal Ug.870 J 683 Cal 400 J 9

10 This is a surprisingly a low value in calories for a race of this magnitude. Following How many calories are consummed running the 35 miles (this point was clairified to me by Nate thanks)? Since the body has to overcome friction, and much of the Badwater 35 race is run on the side of the road, let s assume that the coefficient of friction is 0.5 (a Google search of Jones and Childers report yielded this value). Then the runner has to a least do the work of friction to run the race, and this work is W N x 0.5 (65kg 9.8 m/s )(7,000 m) f m65kg x35mi7km 7 Cal 6.90 J 6,500 Cal 400 J Although one would expect that running 35 miles almost continuously would consume enmorous amounts of calories, things are not so straightforward. The human body can only process around 350 Cal and 3 oz per hour. If the average runner takes about 6-30 hours, the maximum number of calories is Maximum number of calories 350 Cal/hr 30 hr 0,500 Cal Although this estimate of the number of calories is off, it is in the ball park. c. How far must you stretch a spring with k = 000 N/m to store 00 J of energy? Solution The elastic potential energy of a spring is U S = kx /, where x is the magnitude of the stretching or compression relative to the unstretched or uncompressed length. U S = 0 when the spring is at its equilibrium length and x = 0. Solving for x: U kx x U /k (00 J)/(000 N/m) 0.63 m U S s S Conservation of Mechanical Energy DEMO Bowling ball pendulum Suppose that the only external forces that act on the system are conservative forces. In our case, this implies the gravitational and spring forces. If all the forces acting between the objects of a system are conservative, then we can define something called the total mechanical energy E. Using both version of the work-energy theorem, W = K (KEversion) and W = U (PE-version), we equating the work, we have reorganizing W K U K U K U constant EMech Remarks. Energy is fundamental and is NOT describable by displacements, velocities or whatever! Energy is something about the system that involves no time that is the same value regardless of what energy state one looks at. For example, we can use a simple method of keeping track of energy using ENERGY DIAGRAMS. PE KE PE KE decreases increases increases decreases constant energy state- energy state- constant. How is conservation of energy used? The total energy E Mech is typically not important. What is important is understanding how energy behaves and most useful. We can better understand the processes and the changes that occur in nature if we analyze them in terms of the transformations of 0

11 energy from one form into another. In other words, by keeping track of energy changes we employ an extremely useful energy bookkeeping system: large KE (= ½mv ) values implies fast speeds High GPE (= mgy) values implies high heights Large SPE (= ½kx ) value implies a large spring compression 3. Why is conservation of energy so important? Since conservative forces are not path dependent and only dependent on the end points, mechanical energy also depends on the ends too. To explain this further, suppose the system moves from energy state- to energy state-. What occurs inbetween state- and state- is typically unmeasurable and therefore, one cannot state anything about it. The power of conservation of energy is it allows one not to worry about those in-between states and focus only what can be measured: the state- and state-. Problem Solving Strategies Step : For each state, identify the associated KE s and PE s (U g, U S ). Step : When employing conservation of energy, one has to define two energy states: a known energy state E = (K, U ) where parameters (v, y, x ) are all known and an unknown energy state E = (K, U ) where one of the parameters these are unknown (v, y, x ). Step 3: Apply conservation of energy and solve for the appropriate unknown parameters. Conservation of energy sets up an equation that allows one to solve for the unknown parameter. HWR Conservation of energy DEMO Dropping a SuperBall mv E E K U K U g g mgh mv mgh The question of the elasticity of the ball can be glossed over by saying that to a good approximation the ball rebounds with unchanged speed. In real life, that does not happen because sound and heat energy are lost with the impact with the floor. When speaking about E Mech, energy loss implies that there is less available energy to the system (that is, overall energy reduction) as shown in the energy diagram on the right. DEMO Air track cart bouncing off end of track Being an air track, there is very little friction and I would expect the speed just before the cart hits the spring is equal to the speed just after the spring pushes it away (expect v = v 3 ). Why? A perfect spring is a conservative force whose work of the spring does not change the total mechanical energy of the system. Question: what is the maximal compression of the spring? Applying conservation of energy leads to

12 mv K U U K U U g S g S kx DEMO pendulum bob mv k mv kx mv kx x Example 0.3 A 500 kg car is approaching the hill at 0 m/s when it suddenly runs out of gas. a. Can the car make it to the top of the hill by coasting? b. If your answer in part (a) is yes, what is the car's speed after coasting down the other side? Solution Assume there is zero rolling friction since friction is not mentioned. Set U g = 0 on the other side of the hill. Drawing out an energy diagram, a. Setting U (y = 0) = 0, conservation of energy shows that E E K Ug K Ug we can solve for the KE at the top K : K U g, it makes it to the top K K Ug K U g, it does not make it to the top Solving for the individual energies K and U g, we find that K mv (500 kg)(0 m/s) 75,000 J Ug mgy (500 kg)(9.80 m/s )(5.0 m) 74,000 J K K U ( ) 000 J 0 g Since K > U g, the car has enough energy to make it to the top and over the top. b. Setting U 3 (y 3 = 0) = 0 (careful here with the change in reference PE), conservation of energy shows that E E E 3 K Ug K3 Ug3 we solve for v 3 : K mv K U for K solving g 4 K U g ( ) 0 J 3 3 m (500 kg) v 4 m/s v A higher speed on the other side of the hill is reasonable because the car has increased its kinetic energy and lowered its potential energy compared to its starting values. Note that the shape of the hill is irrelevant because gravitational potential energy depends only on height. Conceptual Question

13 Consider two possible paths by which the person can reach the water. On path, the person lets go of the rope on the downward part of the swing. On path, the person swings upward before letting go. Ignoring air resistance, for which path is the speed upon entering the water greater? What causes the KE to change in this system - the tension in the rope, gravity,...? Is there work being done by the rope? WORK and ENERGY From the work-energy theorem, only external nonconservative works can either increase or decrease the energy of the system. Typically, it goes like this: An external push F push (W ext ) will increase the available energy An external frictional force (W f ) will decrease the available energy to the system Work Done by an External Force W ext Let's look at the bowling ball pendulum again. Suppose I now give the pendulum bob a shove so that this W ext increase the energy in the system. That is, this external work is transformed into an initial KE and according to the work-energy theorem reads as W U K U U external An energy bar diagram tells us that the pendulum bob will come back to a higher height. Work done by an external frictional force Friction reduces the overall energy that is available in the system. That is, the system will have less available energy to operate with. There are two ways to look at this and it is easiest to see this with energy diagrams. Method-: Reduction in the initial energy of the system Method-: Final energy of the system and WNC share the initial available energy. Equationwise, this is how they compare: W E E vs. E E W NC NC Method- Method- As you can see, the two methods are identical, however, I actually prefer Method- more than Method- simply by the fact that I do not deal with the minus sign. Although technically we should be calling this the work-energy theorem since that is exactly what it is, you will hear we call it conservation of total energy since the W ext and W NC are just forms of energy (energy and work are the same thing). Since most of the time the NC force is friction then W NC = f k d = Nd. DEMO 3 Inclined air track cart bouncing off end of track 3

14 Example 0.4 A 0-kg child slides down a 3.0-m-high playground slide. She starts from rest, and her speed at the bottom is.0 m/s. a. What energy transfers and transformations occur during the slide? b. What is the total change in the thermal energy of the slide and the seat of her pants? Solution a. The thermal energy of the slide and the child s pants changes during the slide. If we consider the system to be the child and slide, total energy is conserved during the slide. b. Applying conservation of energy: E E K U K U Solving for E th, we write g g mv mgh mv mgh Eth E mgh mv th Solving for the individual energies, U mgy (0 kg)(9.80 m/s )(3.0 m) 590 J g K mv (0 kg)(.0 m/s) 40 J and substituting them in gives Eth 590 J 40 J 550 J At the top of the slide, the child has gravitational potential energy of 590 J. This energy is transformed partly into the kinetic energy of the child at the bottom of the slide. Note that the final KE of the child is only 40 J, much less than the initial GPE of 590 J. The remainder is the change in thermal energy of the child s pants and the slide. That is, most of the gravitational potential energy is converted to thermal energy, and only a small amount is available to be converted to KE. 4