2 Introduction Energy – ability to perform work. Unit of energy (and of work) – Joules (J)1 J = 1 N- m = 1 kg- m2 / s2Forms of energy:Light, chemical, nuclear, mechanical, electrical, sound, heat, kinetic, elastic, magnetic etc.Each type of energy is calculated differently.Maybe do example of conservation (like cookies) and counter example. Force is NOT conserved.
3 Law of Conservation of Energy In any natural process, total energy is always “conserved”, i.e. energy can not be created nor destroyed.Can be transformed from one form to another.Can be transferred from one system toanother.In science, any law of conservation is a very powerful tool in understanding the physical universe.Maybe do example of conservation (like cookies) and counter example. Force is NOT conserved.
4 Work by Constant Force Work – transfer of energy as a result of force. Constant force – its magnitude and direction unchanged.The force acts on the object throughout the process.Only component of force parallel to direction of motion performs work!
5 Work by Constant ForceWork done by force F in moving the object at constant speed through displacement Dx :W = (F cos q) DxF cos q = component of F parallel to Dx.Work is a scalar quantity.Unit = Joule (J) J = 1 N-mFxfNW
6 Work by Constant Force W = (F cos q) Dx 1. If Dx = 0, then W = 0 If you held a 50 kg bag on your head without moving for 3 hours, would you have done work?
7 W = (F cos q) Dx 2. If q = 0, cos 0 = 1 ie F and Dx are parallel and W = F DxxF
8 W = (F cos q) Dx3. If q = 90o, cos 90o = 0, ie F and Dx are perpendicular to each otherand W = 0FxfNWWork done by the normal force N is zero
9 W = (F cos q) Dx4. If q = 180o, cos 180o = -1, ie F and Dx are antiperpendicular to each otherand W = negative.FxfNWWork done by the frictional force f is - F Dx
10 Example:A 30 N chest was pulled 5 m across the floor at a constant speed by applying a force of 50 N at an angle of 30o. How much work was done by tension T?W = F Dx cos q= (50 N) (5 m) cos (30)= 217 JoulesTmgNf30o50 N
11 Example:A 30 N chest was pulled 5 m across the floor at a constant speed by applying a force of 50 N at an angle of 30o. How much work was done by gravity?TmgNf30o50 Nmgfmg90oDxW = mg Dx cos q= 30 x 5 cos(90) = 0
12 Example:A 30 N chest was pulled 5 m across the floor at a constant speed by applying a force of 50 N at an angle of 30o. How much work was done by friction f?50 NmgfTo find the magnitude of f:Consider x-component of forces acting.Since constant velocity: Fnet = 0So Tcos30o – f = 0Thus f = 43.3 N.W = 43.3 x (cos180o) x 5 = -217 JTmgNffDx180o
13 Work by Variable Force W = Fx Dx Work = area under F vs x plot F = kxSpring F = k xArea = ½ k (x)2 = Wspring
14 ExampleA dart gun with a spring constant k = N/m is compressed 8.0 cm. How much energy will it transfer to a dart when the spring is released?W = ½ k(x)2
15 ExampleAs you ride in an elevator going upward at constant speed, work done on you by the normal force from the floor of the elevatoris negative.is zero.is positive.is zero, negative or positive depending on the speed of the elevator.
16 Kinetic EnergySuppose a constant force F is applied horizontally on a small rigid object of mass m.Its velocity will change, say from vo to v as it moves through a distance x (without rotation). Work done on the objectW = Fx x [F parallel to x]= m ax x [Recall: v2 = vo2 + 2a(x-xo)]= ½ m (v2 – v02) = ½ mv2 - ½ mv02The quantity ½ . mass . (velocity)2 is called kinetic energy of the object.
17 An object of mass m, moving with velocity v, has kinetic energy K = ½ mv2K m: If m is doubled, K will double.K v2 If v is doubled, K will quadruple.Can K ever be zero?Can K ever be negative?W = K Can W ever be negative?
18 Kinetic EnergyFvovDxIf more than one force acts on the object, the change in kinetic energy is the total work done by all the forces acting on the object.Total work Wtotal = DKThis is called the Work- Energy Principle
19 Total Work Total work Wtotal = DK = change in K.E. Wtotal = First find work done by each force ,then add them.= first add all the forces and then then calculate work done by the net forceQ: What is the total work done on an object moving with constant velocity?(A) Positive (B) Negative (C) Zero
20 WT = F Dx cos q = (50 N)(5 m) cos (30) = 217 J Example 1:A 30 N chest was pulled 5 m across the floor at a constant speed by applying a force of 50 N at an angle of 30o. How much work was done by all the forces acting on it?WT = F Dx cos q = (50 N)(5 m) cos (30) = 217 JWN = 0, Wmg = 0 Wf = -217JWtotal = (-217) = 0TmgNf30oT = 50 N
21 Example 2:A 30 N chest was pulled 5 m across the floor by applying a force of 50 N at an angle of 30o. How much work was done by all the forces acting on it if its speed changed from 2 m/s to 5 m/s in the process?
22 ExampleWFNVYou are towing a car up a hill with constant velocity. The work done on the car by the normal force is:1. positive 2. negative 3. zeroTcorrectSince the direction of the force is positive the value of work will be positive.it's negative because it's trying to slow down the car. The normal force does no work because it acts in a direction perpendicular to the displacement of the car.Work done by gravity?Work done by tension?
23 Example: Block w/ friction A block is sliding on a surface with an initial speed of 5 m/s. If the coefficient of kinetic friction between the block and table is 0.4, how far does the block travel before stopping?xymgNfY direction: F=maN-mg = 0N = mgWorkWN = 0Wmg = 0Wf = f Dx cos(180)= -mmg DxW = D K-mmg Dx = ½ m (vf2 – v02)-mg Dx = ½ (0 – v02)mg Dx = ½ v02Dx = ½ v02 / mg= 3.1 meters5 m/s
24 POTENTIAL ENERGY Potential Energy (U) = stored energy. It is the energy an object possess due to its position or its configuration.There are different types of potential energy:Gravitational potential energy.Elastic potential energy.Electrical potential energy.
25 Gravitational Potential Energy To move mass m from initial point to final point at constant velocity, an external force FEXT, equal but opposite to the force of gravity must be applied.WEXT = F cos y = mgh= (mg)(cos0)h= mghWg = mgcos(180o)(h)= -mghmgYo = 0Yf = hFEXT
26 Gravitational Potential Energy The quantity mgh is called gravitational potential energy near the earth’s surface:U = mghwhere U = 0 wherever we have chosen h = 0More general way of writing gravitational potential energy isU = mg hFEXTU = 0mg
27 The gravitational potential energy of an object of mass m at a height h is Ug = mghUg = mgh and Wg = -mgh ie Ug = - Wg.Ug mUg hA reference level where h = 0 can be chosen to be at any convenient location.Only change in Ug is important.Ug = mgh = mg(hf – hi)
28 Gravitational Potential Energy Example 3: A block slid down a planeIf a block is slid down a plane.Distance moved down the plane = sVertical height moved = hHow much work will the force of gravity do?hmgSW = F cos xWg = mg . cos . SBut S = y /cosWg = (mg)(cos)(h/cos) = mgh
29 Elastic Potential Energy Elastic (Spring) force F = -kxWork done by spring force W = ½ kx2Energy stored in the spring when compressed or stretched by distance xElastic Potential Energy U = ½ kx2
30 ExampleA dart gun with a spring constant k = N/m is compressed 8.0 cm.(a) How much energy will it transfer to a dart when the spring is released?(b) With what speed will the dart (mass 200 g) leave the gun?
31 ExampleEmil throws an orange straight up and catches it at the same point it was thrown.How much work is done during the orange’s free fall?If the orange was thrown upward from a 1.2 m height above the ground and falls to the ground, how much work is done by gravity? [Mass of orange = .25 kg]
32 Conservative and Non Conservative Forces hmgDroppedSliding down a planeThrownWork done by force of gravity, Wg = - mghIndependent of path
33 Gravitational Potential Energy To move mass m from initial point to final point at constant velocity, an external force FEXT, equal but opposite to the force of gravity must be applied.WEXT = F cos y = mgh= (mg)(cos0)h= mghWg = mgcos(180o)(h)= -mghmgYo = 0Yf = hFEXT
34 Negative, zero or positive depending on the speed of the elevator 1. As you ride an elevator going upward at constant speed, the work done on you by the normal force from the floor of the elevator isnegativezeropositiveNegative, zero or positive depending on the speed of the elevator1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465
35 2. What is the total work done on an object moving with constant velocity? PositiveNegativeZero1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465
36 3. A passenger whose mass is 95 kg is seated in a jet airliner flying 1,200 m above the ground at a speed of 253 m/s. What is the kinetic energy of the passenger?931 J6.08 x 106 J3.04 x 106 J1.12 x 106 J1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465
37 4. A passenger whose mass is 95 kg is seated in a jet airliner flying 1,200 m above the ground at a speed of 253 m/s. What is the potential energy of the passenger?931 J6.08 x 106 J3.04 x 106 J1.12 x 106 J1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465
38 5. A spring has an elastic constant of 920 N/m 5. A spring has an elastic constant of 920 N/m. By how much should it be stretched in order to store 120 J of energy?0.36 m0.51 m0.26 m7.7 m1.96 m1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465
39 Conservative Forces Forces which perform same amount of work independent of the path taken are calledconservative forces. Eg: gravity, elastic force, electric force.Work done by a conservative force depends only on the initial and final position and not on the path taken to get there.Potential energy only goes with conservative forces. There is potential energy associated with gravity, elastic force, electric force.WC = - U [ Ug = mgh and Uspring = ½ kx2]
40 Non Conservative Forces Amount of work done by them depend on the path taken. Eg: friction, tension, push/pull.There is no potential energy associated with non conservative forces.
41 Work - Energy w/ Conservative Forces Total work = Work done by all types of forces.= Work by conservative forces + workby non-conservative forcesWTotal = WC + WNCBut WTotal = K and WC = - USo, K = - U + WNC ie, WNC = K + UWNC = K + U
42 Work - Energy w/ Conservative Forces WNC = K + UWhen only conservative forces are acting; WNC = 00 = K + U OR = (Kf – Ki) + (Uf – Ui)OR Kf + Uf = Ki + UiThe quantity K + U is called mechanical energy EThus Ef = Ei (Conservation of mechanical energy)
43 Skiing Example (no friction) A skier goes down a 78 meter high hill with a variety of slopes. What is the maximum speed she can obtain if she starts from rest at the top? [Assume friction is negligible]Conservation of energy:Ki + Ui = Kf + Uf½ m vi2 + m g yi = ½ m vf2 + m g yf0 + g yi = ½ vf2 + g yfvf2 = 2 g (yf-yi)vf = sqrt( 2 g (yf-yi))vf = sqrt( 2 x 9.8 x 78) = 39 m/s
44 Two similar objects are released from rest at the same time to slide down two frictionless slopes A and B of different inclines as shown in the figure below. Which of these statements is true about the motion of the two balls? They will reach the bottom with the sameAcceleration (B) Speed (C) time duration(D) Same acceleration, speed, time durationABhh30o
45 ExampleSuppose the initial kinetic and potential energies of a system are 75J and 250J respectively, and that the final kinetic and potential energies of the same system are 300J and -25J respectively. How much work was done on the system by non-conservative forces?1. 0J J J J JInitial = 75 J J = 325 J.Final = 300 J - 25 J = 275 J.Final -Initial = = -50 J. No energy goes into or comes out of the system overall, so some non-conservative force has to be accounted for.
46 ExampleA dart gun with a spring constant k = N/m is compressed 8.0 cm.How much energy will it transfer to a dart of mass 200 g when the spring is released?What would be the escape speed of the dart?If the dart was aimed upward, how high would it travel?Uspring = ½ k(x)2 K = ½ mv2, Ug = mgh
47 Power Average Power P = rate at which work is being done. = rate at which energy is being transformed.P = DW / Dt Units: J/s = WattDW/ Dt = [F Dx cos(q)]/ Dt= F (v Dt) cos(q)i.e., P = F v cos(q)
48 Example:How much electrical energy will a 75-watt light bulb use if left lit for 5 hours? [1 hour = 3,600 s]
49 Example:A hot plate used 225,000 J of electrical energy in 5 minutes. What is the power (wattage) of the hot plate?