Work and Energy PHY 207 - energy - J. Hedberg - 2017 1. Introduction 2. Work 3. Kinetic Energy 4. Potential Energy 5. Conservation of Mecanical Energy 6. Ex: Te Loop te Loop 7. Conservative and Non-conservative Forces 8. Power 9. Variable and Constant Forces Introduction Tis is word tat means a lot of tings depending on te context: 1. Energy Consumption of a Houseold 2. Energy Drinks 3. Auras & Spiritual Energy 4. Renewable Energy Energy is anoter example of a word tat we use very often in regular speec. However, in pysics it as a specific meaning. Energy: basics James Prescott Joule 1818-1889 Energy is a scalar quantity. It as SI units of: kg m 2 / s 2 Te unit is called a Joule, after Mr. Joule 1 Joule = 1 Newton 1 meter Te different energies of a system can be transferred between eac oter. Work system K U E c E t environment eat, work Again, we migt ave word problems ere. For us, Work will be defined as te process of transferring energy between a well-defined system and te environment. Te energy may go from te system to te environment. Or, it may go from te environment to te system An example of work A constant force is applied tis box as it moves across te floor. Te work done is equal given by: W = Fd. Tis is for te case of a constant force wic is applied in te same direction as te box is moving. But wat if te force is not in te exact same direction. Page 1
Now we ave to consider te component of te force wic does point in te direction of te distance traveled. W = F d = F d = Fd cos θ Wat is te work being done on te bag by te person carrying it? Quick Question 1 I swing a ball around my ead at constant speed in a circle wit circumference 3 m. Wat is te work done on te ball by te 10 N tension force in te string during one revolution of te ball? Kinetic Energy a) 30 J b) 20 J c) 10 J d) 0.333 J e) 0 J We said tere we many different forms energy can take (cemical, termal, potential, etc) Kinetic Energy is te energy associated wit moving objects. KE = 1 m 2 v2 Some example kinetic energies 1. Ant walking: 1 10 8 J 2. Person walking: 1 10 5 J 3. Bullet: 5000 J 4. Car @ 100 kp ~ 5 10 5 J 5. Fast train ~ 1 10 10 J Quick Question 2 Two balls of equal size are dropped from te same eigt from te roof of a building. One ball as twice te mass of te oter. Wen te balls reac te ground, ow do te kinetic energies of te two balls compare? (assume no air resistance) a) Te ligter one as one fourt as muc kinetic energy as te oter does. b) Te ligter one as one alf as muc kinetic energy as te oter does. c) Te ligter one as te same kinetic energy as te oter does. d) Te ligter one as twice as muc kinetic energy as te oter does. e) Te ligter one as four times as muc kinetic energy as te oter does. Work and Kinetic Energy Page 2
Since te work is related to te amount of energy eiter entering or leaving a system, we can establis a relationsip between te work done on an object and te kinetic energy. PHY 207 - energy - J. Hedberg - 2017 Potential Energy 1 1 W = KE f K E 0 = m m 2 v2 f 2 v2 0 We can store energy in a gravitational system by separating te two objects. Te cat, wen lifted off te ground, as a gravitational potential energy. U g = mg Potential Energy is only dependent on te eigt! It doesn t matter te pat an object takes wile climbing iger away from te eart. Any route wic ends at te same point will impart te same amount of gravitational potential energy to te object. Work and Potential Energy Canging te potential energy of an object requires work. In te case of gravity: "Te cange in te gravitational potential energy of an object is equal to te negative of te work done by te gravitational force" Δ ΔU = W U G = W G Gravity does positive work initial position In tis case we let an object drop, and gravity does positive work. final position Gravity does negative work final position In tis case, we raise an object, and gravity does negative work. initial position Page 3
Quick Question 3 A beam is being raised by a construction crane wit an acceleration upwards (+y direction). Call te work done by te cable tension: and te work done by Gravity:. Wic of te following is true: a) W T > 0 & W G > 0 b) W T > 0 & W G < 0 c) W T < 0 & W G > 0 d) W T < 0 & W G < 0 e) = 0 & = 0 W T W G W T W G Quick Question 4 A beam is being lowered by a construction crane a constant velocity. Call te work done by te cable tension: and te work done by Gravity:. Wic of te following is true: W T a) W T > 0 & W G > 0 b) W T > 0 & W G < 0 c) W T < 0 & W G > 0 d) W T < 0 & W G < 0 e) = 0 & = 0 W T W G W G Work - Energy Let's now take a look at systems wic ave bot kinetic and potential energy. For example: tis box wic is located at a distance above te ground. Usually, we'll call te ground = 0. If tese are te only two types of energy in te system, ten we ave a very special situation. We can define te total mecanical energy energy of te system as te kinetic + potential. = KE + U E mecanical Conservation of Mecanical Energy If we don't worry about any of tose retarding forces like friction or air resistance, ten we can say tat te total mecanical energy of a system remains te same as time advances. or E mec = E i f mec 1 1 m + mg = m + mg 2 v2 i i 2 v2 f f Play/Pause conserved. Upon dropping a ball, te total mecanical energy is E mec = constant = KE + U grav Knowing tis conservation law, we can use eiter to find v, or v to find. Page 4
Quick Question 5 1 m 2 Te block starts from rest at point 1. Using te conservation of mecanical energy, determine te speed of te block at point 2. a) v = (2g) 2 b) v = g 2m c) v = 2g d) v = 0 Quick Question 6 1 2 m 2 Te block starts from rest at point 1. Using te conservation of mecanical energy, determine te speed of te block at point 2. a) v = (4g) 2 b) v = g 4m c) v = 2g d) v = 4g = 0 Quick Question 7 m 1 = 0-2 Te block starts from rest at point 1. Using te conservation of mecanical energy, determine te speed of te block at point 2. a) v = (4g) 2 b) v = g 4m c) v = 2g d) v = 4g A car on a ill. Play/Pause Quick Question 8 L 30º L If L = 4 meters, ow ig is te ball? Tat is, find te distance. Example Problem #1: Block on a ramp Page 5
Here's a 5 kg block on a ramp, were θ = 25, and = 20 meters. Find te work done by gravity and te final speed. We want to calculate te work done by gravity: So, we need to decompose te two vectors into components. Let's call te orizontal direction +x to te left, and te vertical +y up. Tus, in unit vector notion, te force of gravity will be: and te displacement will be: W = F G d = F G d = mgj tan θ i j Now, we can execute te dot product using unit vectors: W = F G d = [ mg j ] [ ] tan θ i j Since i i = 1 and i j = 0 tis dot product simplifies to: W = mg wic will be te work done by gravity. Notice ow tat is same as if we just let te box fall from a distance above te ground. Tis situation occurs because te force of gravity is a conservative force, i.e. it is pat independent. To find te speed at te bottom of te ramp, we can use te work-energy teorem wic relates te work done to te cange in kinetic energy: ΔKE = Work done Initially, te box is at rest, so tere is no kinetic energy: K E i = 0 Te kinetic energy at te bottom of te ramp will be given by: 1 2 1 K E f = m 2 v2 Tus, te ΔKE will be m Setting tis equal to te work done tat we calculated above: v 2 mg = 1 m 2 v2 allows us to solve for v: v = 2g Tis result sould look familiar. Notice: we use regular, non-rotated coordinates for tis problem. Try it wit rotated axes and see wat you get. Sould it be different? Page 6
Ex: Te Loop te Loop PHY 207 - energy - J. Hedberg - 2017 v top r We saw tat if te velocity of te object was large enoug, ten it would remain inside in contact wit te loop. v Play/Pause A successful loop-te-loop looks like tis. Te ball remains in pysical contact wit te road at all times. (Tis is anoter way of saying te normal force is not zero.) Quick Question 9 After te ball looses contact wit te loop, wic of te following dotted lines sows te most likely trajectory? Play/Pause An unsuccessful loop-te-loop looks like tis. Te ball looses pysical contact wit te road at all times. (Tis is anoter way of saying te normal force becomes zero.) In te case were te velocity is not fast enoug to get te ball all te way around te loop, te ball looses contact wit te road near te top. success fail At te top of te loop, te sum of forces pointing towards te ground will be given by: F ground = w + F N = ma Page 7
Since tis is circular motion, we ave PHY 207 - energy - J. Hedberg - 2017 ma = m v2 r Tus we can write: mg weigt v 2 + F N = m r Quick Question 10 Wat is te minimum velocity at te top of te loop needed to complete te loop-te-loop for a regular, no friction, sliding object. a) v > rg b) v > (rg) 2 c) v > r g d) v > e) v > 2rg Quick Question 11 v top r Wat is an expression for tat would lead to a speed so te ball maintains contact at all times? a) > 5r b) > 3r 2 c) > 3r d) > 5r 2 Conservative and Non-conservative Forces Most of te forces we ave seen can be classified as non-conservative. Te exception is Gravity. Here are two definitions for a conservative force: 1. A force is conservative if te work it does to move an object from point A to point B does not depend on te route cosen (aka te pat) to get from point A to point B. 2. A force is conservative if tere is zero net work done wile moving an object around a closed loop, tat is, around a pat tat as te same point for te beginning and te end. Examples of Conservative Forces Gravitational force Elastic spring force Electric force and Non-conservative forces: Static and kinetic frictional forces Air resistance Tension Normal force Page 8
Sown are two pats tat a box can take to get from point A to point B If we look at te cange in potential energy, ΔU grav, ten we'll see tat te Work done by gravity on te box in eiter case is equal to zero. And, for any oter pat you can imagine, it's also zero. W = Δ U G = mg( ) = 0 Here is an example of a conservative force interacting wit a box. If we slide te box from point A to B along pat 1, we can calculate te work done by figuring out te work done by gravity to go from te ground up to te apex, and ten back down to te ground. Adding tese two works togeter will result in zero, since on te way up, gravity is pointing against te displacement vector: [ F G d cos( 180 )] and on te way down, gravity will be in te same direction as te displacement vector: [ F G d cos( 0 ]. Tus, tese two values are equal in magnitude but opposite in sign and so wen tat are added togeter will be equal to zero. In te case of pat 2, te work done by gravity will be zero also, since tere is no cange in eigt along te pat. Tus, no matter ow we cose to go from te A to B, te net work done by gravity will be zero. Tis is te definition of a conservative force. Moving a box along 2 pats in te presence of friction. Te situation is very different if we ask about te work done against friction during tese two pats. Pat 2 will require more work tan pat 1. Now, if you want to move te box from A to B, but are asking about te work done by friction on te box during te motion, te sorter pat (pat 1) will result in less work. Since te force of friction is always opposite to te direction of motion, tere will never be te case of canceling out like we ad in te gravitational case. Pat 2 will lead to more work being done by kinetic friction, wic is terefore considered a non-conservative force. And so, forces like friction and air resistance are considered nonconservative. We can owever account for tese forces in our analysis of certain systems. If te cange in energy is non-zero, bewteen te initial and final times of a mecanical system, ten tere must be some non-conservative forces doing work during te motion. Example Problem #2: W NC = E f E 0 Now, we ave a ramp tat as some friction. How can we describe te motion of a box on tis ramp? Page 9
Example Problem #3: PHY 207 - energy - J. Hedberg - 2017 A cild, starting from rest, slides down a slide. On te way down, a kinetic frictional force (a nonconservative force) acts on er. Te cild as a mass of 50.0 kg, and te eigt of te slide is 18 m. If te kinetic frictional force does 6.50 10 3 Joules of work, ow fast is te cild going at te bottom of te slide? Example Problem #4: If te object slides down a ill wit an elevation of 10 m, were will it stop? Consider te sloped part frictionless and te roug part at te bottom aving a equal to 0.3. μ k 10 m µ = 0 µ =.3 Power Again, ere's anoter word we ave to be very careful wit. We migt be used to calling anyting tat seems strong 'more powerful'. For us owever, we'll need to restrict our use of te word power to describe ow fast a process converts energy. cange in energy Power = P = time Te SI unit of power is called te watt: 1watt = 1joule/second = 1kg m 2 / s 3 A unit of power in te US Customary system is orsepower: 1 p = 746 W Units of [power - time] can also be used to express energy (e.g. kilowattour) 1kW = (1000W)(3600s) = 3.6 10 6 J We sould be able to understand tese mysterious documents now. How many Joules of energy did I use? How many pusups would I ave to do to generate tat muc energy? Example Problem #5: Page 10
I used to work out on a rowing macine. Eac time I pulled on te rowing bar (wic simulates te oars), it moved a distance of 1.1 m in a time of 1.6 s. Tere was a little display tat sowed my power and it said 90 Watts. How large was te force tat I applied to te rowing bar? Example Problem #6: A go kart weigs 1000 kg and accelerates at 4 m/s 2 for 4 seconds. Wat is te average power? Variable and Constant Forces F In te most basic situation, te force applied to an object was constant. Wen tis is te case, te Work was given by: x W = F d Tis is also equal to te "area under te curve" Work done by a variable force F te curve" x However, it's also possible tat te work will vary as a function of distance. In tis case, te work will not be given by te standard form, because, te F is not constant. W F d However, te work is still equal to te "area under Work done by a variable force F We don't ave te tools (i.e. integration) to be able to do muc wit tis at tis level of pysics. x Work done by a variable force To deal wit tese situations, we'll ave to use some integration tecniques. F In tis case: x Here's an applied force tat varies as te square of te distance. We could figure out te area under te curve, wic does equal te work, but performing te integral W = x i x f F(x)dx x f W = x 2 1 dx = x i 3 x3 xf x i Page 11
In te case of 3d: If our forces are variable and 3 dimensional, (and peraps not pointed directly in te direction of te displacement), ten we ave to use some more developed vector calc: Or, if we integrate over te distance: r f dw = F dr = F x dx + F y dy + F z dz W = dw = F x dx + F y dy + F z dz x i y i r i x f Since work is a scalar, ten tese terms will just add up witout any complications. y f z i z f Recover te work-kinetic energy equation? W = x i F(x)dx We sould be able to sow tat tis gets back to kinetic energy someow... x f Based on te above: ΔU(x) = W = FΔx Now we can take te derivative wit respect to x: F(x) = du(x) dx Example Problem #7: Draw plots of te gravitational potential energy and te gravitational force for an object as a function of eigt above te eart. Consider only small distances above te surface of te eart. Example Problem #8: If te force from an elastic band wen stretced is given by F = kx, wat would te cange in potential energy be for an object attaced to te rubber band and pulled back by a distance d? (k is a constant tat is determined by te rubber band material ) Plot te force and as functions of distance. U rubber band Page 12
Quick Question 12 A U Here is te potential energy of an unknown as a function of distance along te x-axis. Were would tat force be te largest, and pointed in te positive direction? B D E F x C Example Problem #9: A diatomic molecule as two atoms (H 2 for example). If te potential energy of suc a system is given by: U = A B r 12 Find te equilibrium separation of suc a molecule. Tat is, find a distance r tat will lead to zero net force on eac atom.(a and B are just constants) r 6 1.0 0.8 0.6 0.4 0.2 0.0-0.2-0.4 1.0 0.5 1.0 1.5 2.0 2.5 3.0 Here are plots of te Potential function, and its derivative (i.e. force). 1 1 U = r 12 r 6 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0-0.5-1.0 Force Microscopy More info: Intro to AFM Emmy Emmy Noeter was a teoretical pysicist wo did pioneering work in te study of conservation principles. Page 13
23 Marc 1882 14 April 1935 Page 14