2WorkWork tells us how much a force or combination of forces changes the energy of a system.Work is the product of the force or its component in the direction of motion and its displacement.W = Fdcos F: force (N)d : displacement (m): angle between force and displacement
3Units of Work SI System: Joule (N m) British System: foot-pound 1 Joule of work is done when 1 N acts on a body moving it a distance of 1 meterBritish System: foot-pound(not used in Physics B)cgs System: erg (dyne-cm)Atomic Level: electron-Volt (eV)
4Force and direction of motion both matter in defining work! There is no work done by a force if it causes no displacement.Forces can do positive, negative, or zero work. When an box is pushed on a flat floor, for example…The normal force and gravity do no work, since they are perpendicular to the direction of motion.The person pushing the box does positive work, since she is pushing in the direction of motion.Friction does negative work, since it points opposite the direction of motion.
5QuestionIf a man holds a 50 kg box at arms length for 2 hours as he stands still, how much work does he do on the box?None, 0 J, No Movement.
6QuestionIf a man holds a 50 kg box at arms length for 2 hours as he walks 1 km forward, how much work does he do on the box?None, force and movement are perpendicular to each other (cos 90 = 0)
7QuestionIf a man lifts a 50 kg box 2.0 meters, how much work does he do on the box?W =fdcosQW = (mg)dcosQW = (50)(9.8)(2) cos 0W = 980 J
8Work and Energy Work changes mechanical energy! If an applied force does positive work on a system, it tries to increase mechanical energy.If an applied force does negative work, it tries to decrease mechanical energy.The two forms of mechanical energy are called potential and kinetic energy.
9Sample problemJane uses a vine wrapped around a pulley to lift a 70-kg Tarzan to a tree house 9.0 meters above the ground.How much work does Jane do when she lifts Tarzan?m = 70 kg, d = 9 m, g = 9.8 m/s2, Q = 0W = ?W = FdcosQ = (mg)dcosQW = (70)(9.8)(9)cos 0How much work does gravity do when Jane lifts Tarzan?Same, but in the opposite direction.
10Sample problemJoe pushes a 10-kg box and slides it across the floor at constant velocity of 3.0 m/s. The coefficient of kinetic friction between the box and floor is 0.50.How much work does Joe do if he pushes the box for 15 meters?m = 10 kg, m = 0.5, d = 15 m, Q = 0, v = 15 m/s (constant) W = ?W = FappdcosQ , since constant velocity Fapp = FkW = (mmg)dcosQW = (.5)(10)(9.8)(15)cos 0b) How much work does friction do as Joe pushes the box?Same but opposite.
11Sample problem F = 16 N, d = 10 m, Q = 60 W = ? W = FdcosQ A father pulls his child in a little red wagon with constant speed. If the father pulls with a force of 16 N for 10.0 m, and the handle of the wagon is inclined at an angle of 60o above the horizontal, how much work does the father do on the wagon?F = 16 N, d = 10 m, Q = 60W = ?W = FdcosQW = (16)(10)cos 60W =
12Kinetic Energy Energy due to motion K = ½ m v2 K: Kinetic Energy m: mass in kgv: speed in m/sUnit: Joules
13Sample problem A 10.0 g bullet has a speed of 1.2 km/s. What is the kinetic energy of the bullet?Do on your own, remember to convert units to proper units.What is the bullet’s kinetic energy if the speed is halved?What is the bullet’s kinetic energy if the speed is doubled?
14The Work-Energy Theorem The net work due to all forces equals the change in the kinetic energy of a system.Wnet = DKWnet: work due to all forces acting on an objectDK: change in kinetic energy (Kf – Ki)
15Sample problemAn 8.0-g acorn falls from a tree and lands on the ground 10.0 m below with a speed of 11.0 m/s.What would the speed of the acorn have been if there had been no air resistance?Did air resistance do positive, negative or zero work on the acorn? Why?
16Sample problemAn 8.0-g acorn falls from a tree and lands on the ground 10.0 m below with a speed of 11.0 m/s.How much work was done by air resistance?What was the average force of air resistance?
17Constant force and work F(x)The force shown is a constant force.W = FDx can be used to calculate the work done by this force when it moves an object from xa to xb.The area under the curve from xa to xb can also be used to calculate the work done by the force when it moves an object from xa to xbxxaxb
18Variable force and work F(x)The force shown is a variable force.W = FDx CANNOT be used to calculate the work done by this force!The area under the curve from xa to xb can STILL be used to calculate the work done by the force when it moves an object from xa to xbxaxbx
19SpringsWhen a spring is stretched or compressed from its equilibrium position, it does negative work, since the spring pulls opposite the direction of motion.Ws = - ½ k x2Ws: work done by spring (J)k: force constant of spring (N/m)x: displacement from equilibrium (m)The force doing the stretching does positive work equal to the magnitude of the work done by the spring.Wapp = - Ws = ½ k x2
20Springs: stretching m m Ws = negative area = - ½ kx2 100 -100 -200 200 100-100-200200F(N)12345x (m)mFsmxWs = negative area= - ½ kx2FsFs = -kx (Hooke’s Law)
21Sample problemA spring with force constant 250 N/m is initially at its equilibrium length.How much work must you do to stretch the spring m?Do on your own.How much work must you do to compress it m?
22Sample problemIt takes 1000 J of work to compress a certain spring 0.10 m.What is the force constant of the spring?Do on your ownTo compress the spring an additional 0.10 m, does it take 1000 J, more than 1000 J, or less than 1000 J? Verify your answer with a calculation.
23Sample ProblemHow much work is done by the force shown when it acts on an object and pushes it from x = 0.25 m to x = 0.75 m?Figure from “Physics”, James S. Walker, Prentice-Hall 2002
24Sample ProblemHow much work is done by the force shown when it acts on an object and pushes it from x = 2.0 m to x = 4.0 m?Figure from “Physics”, James S. Walker, Prentice-Hall 2002
25Power Power is the rate of which work is done. P = W/t W: work in Joulest: elapsed time in secondsWhen we run upstairs, t is small so P is big.When we walk upstairs, t is large so P is small.
26Unit of Power SI unit for Power is the Watt. 1 Watt = 1 Joule/s Named after the Scottish engineer James Watt ( ) who perfected the steam engine.British systemhorsepower1 hp = 746 W
27How We Buy Energy…The kilowatt-hour is a commonly used unit by the electrical power company.Power companies charge you by the kilowatt-hour (kWh), but this not power, it is really energy consumed.1 kW = 1000 W1 h = 3600 s1 kWh = 1000 J/s • 3600s = 3.6 x 106J
28Sample problemA man runs up the 1600 steps of the Empire State Building in 20 minutes. If the height gain of each step was 0.20 m, and the man’s mass was 80.0 kg, what was his average power output during the climb? Give your answer in both watts and horsepower.
29Sample problemCalculate the power output of a 0.10 g fly as it walks straight up a window pane at 2.0 cm/s.
30Force typesForces acting on a system can be divided into two types according to how they affect potential energy.Conservative forces can be related to potential energy changes.Non-conservative forces cannot be related to potential energy changes.So, how exactly do we distinguish between these two types of forces?
31Conservative forces Work is path independent. Work can be calculated from the starting and ending points only.The actual path is ignored in calculations.Work along a closed path is zero.If the starting and ending points are the same, no work is done by the force.Work changes potential energy.Examples:GravitySpring forceConservation of mechanical energy holds!
32Non-conservative forces Work is path dependent.Knowing the starting and ending points is not sufficient to calculate the work.Work along a closed path is NOT zero.Work changes mechanical energy.Examples:FrictionDrag (air resistance)Conservation of mechanical energy does not hold!
33Potential Energy Energy due to position or configuration “Stored” energyFor gravity: Ug = mgym: massg: acceleration due to gravityh: height above the “zero” pointFor springs: Us = ½ k x2k: spring force constantx: displacement from equilibrium position
34Conservative forces and Potential energy Wc = -UIf a conservative force does positive work on a system, potential energy is lost.If a conservative force does negative work, potential energy is gained.For gravityWg = -Ug = -(mgy – mgyo)For springsWs = -Us = -(½ k x2 – ½ k xo2)
35More on paths and conservative forces. Figure from “Physics”, James S. Walker, Prentice-Hall 2002Q: Assume a conservative force moves an object along the various paths. Which two works are equal?A: W2 = W3(path independence)Q: Which two works, when added together, give a sum of zero?A: W1 + W2 = 0orW1 + W3 = 0(work along a closed path is zero)
36Sample problem Figure from “Physics”, James S Sample problem Figure from “Physics”, James S. Walker, Prentice-Hall 2002A box is moved in the closed path shown.How much work is done by gravity when the box is moved along the path A->B->C?How much work is done by gravity when the box is moved along the path A->B->C->D->A?
38Sample problem Figure from “Physics”, James S Sample problem Figure from “Physics”, James S. Walker, Prentice-Hall 2002A box is moved in the closed path shown.How much work would be done by friction if the box were moved along the path A->B->C?How much work is done by friction when the box is moved along the path A->B->C->D->A?
40Sample problemA diver drops to the water from a height of 20.0 m, his gravitational potential energy decreases by 12,500 J. How much does the diver weigh?
41Sample problemIf 30.0 J of work are required to stretch a spring from a 2.00 cm elongation to a 4.00 cm elongation, how much work is needed to stretch it from a 4.00 cm elongation to a 6.00 cm elongation?
42Law of Conservation of Energy In any isolated system, the total energy remains constant.Energy can neither be created nor destroyed, but can only be transformed from one type of energy to another.
43Law of Conservation of Mechanical Energy E = K + U = ConstantK: Kinetic Energy (1/2 mv2)U: Potential Energy (gravity or spring)E = U + K = 0K: Change in kinetic energyU: Change in gravitational/spring potential energyU – Uo + K – Ko = 0U + K = Uo + Ko**** mgy + ½ mv2 = mgyo + ½ mvo2 ****
44Roller Coaster Simulation Roller Coaster Physics Simulation(demonstrates gravitational potential energy and kinetic energy)
45Pendulums and Energy Conservation Energy goes back and forth between K and U.At highest point, all energy is U.As it drops, U goes to K.At the bottom , energy is all K.
46Pendulum Energy ½mvmax2 = mgy K1 + U1 = K2 + U2 For minimum and maximum points of swinghK1 + U1 = K2 + U2For any points 1 and 2.
47Springs and Energy Conservation Transforms energy back and forth between K and U.When fully stretched or extended, all energy is U.When passing through equilibrium, all its energy is K.At other points in its cycle, the energy is a mixture of U and K.
48Spring Energy K1 + U1 = K2 + U2 = E m -x m ½kxmax2 = ½mvmax2 m x All U All UAll KK1 + U1 = K2 + U2 = EFor any two points 1 and 2m-xm½kxmax2 = ½mvmax2For maximum and minimum displacements from equilibriummx
52Sample problem Problem copyright “Physics”, James S Sample problem Problem copyright “Physics”, James S. Walker, Prentice-Hall 2002A 0.21 kg apple falls from a tree to the ground, 4.0 m below. Ignoring air resistance, determine the apple’s gravitational potential energy, U, kinetic energy, K, and total mechanical energy, E, when its height above the ground is each of the following: 4.0 m, 2.0 m, and 0.0 m. Take ground level to be the point of zero potential energy.
53Sample problem Problem copyright “Physics”, James S Sample problem Problem copyright “Physics”, James S. Walker, Prentice-Hall 2002A 1.60 kg block slides with a speed of m/s on a frictionless, horizontal surface until it encounters a spring with a force constant of 902 N/m. The block comes to rest after compressing the spring 4.00 cm. Find the spring potential energy, U, the kinetic energy of the block, K, and the total mechanical energy of the system, E, for the following compressions: 0 cm, 2.00 cm, 4.00 cm.
54Law of Conservation of Energy E = U + K + Eint= ConstantEint is thermal energy.U + K + Eint = 0Mechanical energy may be converted to and from heat.
55Work done by non-conservative forces Wnet = Wc + WncNet work is done by conservative and non-conservative forcesWc = -UPotential energy is related to conservative forces only!Wnet = KKinetic energy is related to net force (work-energy theorem)K = -U + WncFrom substitutionWnc = U + K = ENonconservative forces change mechanical energy. If nonconservative work is negative, as it often is, the mechanical energy of the system will drop.
56Sample problem Problem copyright “Physics”, James S Sample problem Problem copyright “Physics”, James S. Walker, Prentice-Hall 2002Catching a wave, a 72-kg surfer starts with a speed of 1.3 m/s, drops through a height of 1.75 m, and ends with a speed of 8.2 m/s. How much non-conservative work was done on the surfer?
57Sample problem Problem copyright “Physics”, James S Sample problem Problem copyright “Physics”, James S. Walker, Prentice-Hall 2002A 1.75-kg rock is released from rest at the surface of a pond 1.00 m deep. As the rock falls, a constant upward force of 4.10 N is exerted on it by water resistance. Calculate the nonconservative work, Wnc, done by the water resistance on the rock, the gravitational potential energy of the system, U, the kinetic energy of the rock, K, and the total mechanical energy of the system, E, for the following depths below the water’s surface: d = 0.00 m, d = m, d = 1.00 m. Let potential energy be zero at the bottom of the pond.
58Pendulum labFigure out how to demonstrate conservation of energy with a pendulum using the equipment provided.The photo-gates must be set up in “gate” mode this time.The width of the pendulum bob is an important number. To get it accurately, use the caliper.Turn in just your data, calculations, and result. Clearly show the speed you predict for the pendulum bob from conservation of energy, the speed you measure using the caliper and photo-gate data, and a % difference for the two.