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Chapter 5: Energy Energy Energy is present in a variety of forms: mechanical, chemical, electromagnetic, nuclear, mass, etc. Energy The If can be transformed from one from to another. total amount of energy in the Universe never changes. a collection of objects can exchange energy with each other but not with the rest of the Universe (an isolated system), the total energy of the system is constant. If one form of energy in an isolated system decreases, another form of energy must increase. In this chapter, we focus on mechanical energy: kinetic energy and potential energy. Work The work W done on an object by a constant force F when the object is displaced by Dx by the force: W FDx SI unit: joule (J) = newton-meter (N m) = kg m2/s2 • Work is a scalar quantity. • If the force exerted on an object is not in the same direction as the displacement: W ( F cos )Dx F Dx component of dot product the force along or the direction of inner product the displacement Work If an object is displaced vertical to the direction of a force exerted, then no work is done. W ( F cos )Dx 0 ( 90) If an object is displaced in opposite direction to that of an exerted force, the work done by the force is negative (if F<Fg). W ( F cos )Dx FDx ( 180) Work Work and dissipative forces • The friction force between two objects in contact and in relative to each other always dissipate energy in complex ways. • Friction is a complex process caused by numerous microscopic interactions over the entire area of the surfaces in contact. • The dissipated energy above is converted to heat and other forms of energy. • Frictional work is extremely important: without it Eskimos can’t pull sled, cars can’t move, etc. Work Examples • Example 5.1: Sledding through the Yukon (a) How much work is done if =0? m=50.0 kg F= 1.20x102 N W FDx Dx=5.00 m 2 (1.20 10 J )(5.00 m) 6.00 10 2 J (b) How much work is done if =30o? W ( F cos )Dx (1.20 10 2 J ) cos 30(5.00 m) 5.20 10 2 J Work Examples • Example 5.2: Sledding through the Yukon (with friction) (a) How much work is done if =0? m=50.0 kg 2N F= 1.20x10 F n mg 0 n mg y Dx=5.00 m W f Dx nDx fric k k k mgDx 4.90 10 2 J Wnet Wapp W fric Wn Wg fk=0.200 6.00 10 2 J (4.90 10 2 J ) 0 0 1.10 10 2 J (b) How much work is done if =30o? F y n mg 0 n mg W fric f k Dx k nDx k (mg Fapp sin )Dx 4.30 10 2 J Wnet Wapp W fric Wn Wg 5.20 10 2 J (4.30 10 2 J) 0 0 90.0 J Kinetic Energy Kinetic energy (energy associated with motion) • Consider an object of mass m moving to the right under action of a constant net force Fnet directed to the right. (constant acceleration) Wnet Fnet Dx (ma)Dx v 2 v02 2aDx v 2 v02 aDx 2 Wnet Wnet v 2 v02 m 2 1 1 mv2 mv02 2 2 KE f KEi DKE work-energy theorem Define the kinetic energy KE as: 1 2 KE mv 2 SI unit: J Kinetic Energy An example • Example 5.3: Collision analysis m=1.00x103 kg vi = 35.0 m/s -> 0 =8.00x103 N (a) The minimum necessary stopping distance? 1 2 1 2 1 2 Wnet mv f mvi f k Dx 0 mvi 2 2 2 Dx 76.6 m (b) If Dx=30.0 m what is the speed at impact? Wnet W fric f k Dx v 2f vi2 1 2 1 2 mv f mvi 2 2 2 f k Dx 745 m 2 / s 2 v f 27.3 m/s m Kinetic Energy Conservative and non-conservative forces • Two kinds of forces: conservative and non-conservative forces • Conservative forces : gravity, electric force, spring force, etc. A force is conservative if the work it does moving an object between two points is the same no matter what path is taken. It can be derived from “potential energy”. • Non-conservative forces : friction, air drag, propulsive force, etc. In general dissipative – it tends to randomly disperse the energy of bodies on which it acts. The dispersal of energy often takes the form of heat or sound. The work done by a non-conservative force depends on what path of an object that it acts on is taken. It cannot be derived from “potential energy”. • Work-energy theorem in terms of works by conservative and nonconservative force Wnc Wc DKE Gravitational Potential Energy Gravitational work and potential energy • Gravity is a conservative force and can be derived from a potential energy. Work done by gravity on the book: Wg FDy cos mg ( yi y f ) cos 0 mg ( y f yi ) mgDy v 2 v02 2 gDy v 2 2 gDy KEi 0 KE f 1 2 mv mgDy DKE 2 Wnet Wnc Wg DKE Wnc DKE mg ( y f yi ) Gravitational Potential Energy Gravitational work and potential energy • Gravity is a conservative force and can be derived from a potential energy. Wnc DKE mg ( y f yi ) • Let’s define the gravitational potential energy of a system consisting of an object of mass m located near the surface of Earth and Earth as: PE mgy y : the vertical position of the mass to a reference point ( often at y=0 ) g : the acceleration of gravity SI unit: J Wnc DKE DPE where DPE PE f PEi mg( y f yi ) Gravitational Potential Energy Reference levels for gravitational potential energy • As far as the gravitational potential is concerned, the important quantity is not y (vertical coordinate) but the difference Dy between two positions. • You are free to choose a reference point at any level (but usually at y=0). yi yf Gravitational Potential Energy Gravity and the conservation of mechanical energy • When a physical quantity is conserved the numeric value of the quantity remains the same throughout the physical process. • When there is no non-conservative force involved, Wnc DKE DPE 0 KEi PEi KE f PE f • Define the total mechanical energy as: E KE PE • The total mechanical energy is conserved. Ei E f 1 2 1 mvi mgyi mv2f mgy f 2 2 • In general, in any isolated system of objects interacting only conservative forces, the total mechanical energy of the system remains the same at all times. Gravitational Potential Energy Examples • Example 5.5: Platform diver (a) Find the diver’s speed at y=5.00 m. KEi PEi KE f PE f 1 2 1 2 mvi mgyi mv f mgy f 2 2 0 gyi 1 2 v f gy f 2 v f 2 g ( yi y f ) 9.90 m/s (b) Find the diver’s speed at y=0.0 m. 1 2 0 mgyi mv f 0 2 v f 2 gyi 14.0 m/s Gravitational Potential Energy Examples • Example 5.8: Hit the ski slopes (a) Find the skier’s speed at the bottom (B). KEi PEi KE f PE f 1 2 1 2 mvi mgyi mv f mgy f 2 2 0 gyi fk 0 1 2 v f gy f 2 v f 2 g ( yi y f ) 19.8 m/s (b) Find the distance traveled on the horizontal rough surface. Wnet f k d DKE d vB2 2 k g 95.2 m 1 2 1 2 1 mvC mvB k mgd mvB2 2 2 2 f k 0.210 Spring Potential Energy Spring and Hooke’s law • Force exerted by a spring Fs Fs kx Hooke’s law If x > 0, Fs <0 Fs to the left If x < 0, Fs >0 F to the right s k : a constant of proportionality called spring constant. SI unit : N/m • The spring always exerts its force in a direction opposite the displacement of its end and tries to restore the attached object to its original position. Restoring force x>0 Fs Spring Potential Energy Potential due to a spring • The spring Fs is associated with elastic potential energy. Between xi -1/2Dx and xi+1/2Dx the work exerted by the spring is approximately: Wi Fs ,i Dx Fs ,i ( x / N ) Between x=0 and x, the total work exerted -Fs by the spring is approximately: Ws lim N i 1Ws ,i N lim N i 1 Fs ,i Dx N lim N i 1 ( areai ) N width = Dx xi-1/2Dx xi+1/2Dx 1 2 Fs , N x kx 2 In general when the spring is stretched -Fi from xi to xf, the work done by the spring is: 1 1 -Ws,i= areai xi xi+1 Ws kx2f kxi2 2 2 x Spring Potential Energy Potential due to a spring (cont’d) • The energy-work theorem including a spring and gravity 1 2 1 2 Wnc kx f lxi DKE DPEg 2 2 1 elastic potential energy PEs kx2 2 Wnc ( KE f KEi ) ( PEgf PEgi ) ( PEsf PEsi ) ( KE PEg PEs )i ( KE PEg PEs ) f Extended form of conservation of mechanical energy Spring Potential Energy Examples • Example 5.9: A horizontal spring (a) Find the speed at x=0 without friction. m=5.00 kg k=4.00x102 N/m 1 2 1 2 1 2 1 2 mvi kxi mv f kx f xi=0.0500 m 2 2 2 2 vi 0, x f 0 k 2 k xi v 2f v f xi m m 0.447 m/s (b) Find the speed at x=xi/2. 2 i kx2f kx v 2f m m k 2 vf ( xi x 2f ) 0.387 m/s m k=0 Spring Potential Energy Examples • Example 5.9: A horizontal spring (cont’d) (c) Find the speed at x=0 with friction 1 2 1 2 1 2 1 2 W fric mv f mvi kx f kxi 2 2 2 2 1 2 1 2 k nxi mv f kxi 2 2 1 2 1 2 mv f kxi k nxi 2 2 k 2 vf xi 2 k gxi m 0.230 m/s m=5.00 kg k=4.00x102 N/m xi=0.0500 m k= 0.150 Spring Potential Energy Examples • Example 5.10 : Circus acrobat What is the max. compression of the spring d? ( KE PEg PEs )i ( KE PEg PEs ) f 1 0 mg (h d ) 0 0 0 kd 2 2 d 2 (0.123 m) d - 0.245 m 2 0 d 0.560 m m=50.0 kg h =2.00 m k = 8.00 x 103 N/m Spring Potential Energy Examples • Example 5.11 : A block projected up a frictionless incline (a) Find the max. distance d the block travels up the incline. 1 2 1 2 mvi mgyi kxi 2 2 1 2 1 2 mv f mgy f kx f 2 2 m=0.500 kg xi=10.0 cm k=625 N/m =30.0o kxi2 / 2 1 2 kxi mgh mgd sin d 1.28 m 2 mg sin (b) Find the velocity at hafl height h/2. 1 2 1 2 k 2 1 kxi mv f mg h xi v 2f gh 2 2 m 2 k 2 vf xi gh 2.50 m/s m Spring Potential Energy Systems and energy conservation • Work-energy theorem Wnc Wc DKE • Consider changes in potential Wnc DKE DPE ( KE f KEi ) ( PE f PEi ) ( KE f PE f ) ( KEi PEi ) E f Ei The work done on a system by all non-conservative forces is equal to the change in mechanical energy of the system. If the mechanical energy is changing, it has to be going somewhere. The energy either leaves the system and goes into the surrounding environment, or stays in the system and is converted into nonmechanical form(s). Systems and Energy Conservation Forms of energy • Forms of energy stored kinetic, potential, internal energy • Forms of energy transfer between a non-isolated system and its environment Mechanical work : transfers energy to a system by displacing it with a force. Heat : transfers energy through microscopic collisions between atoms or molecules. Mechanical waves : transfers energy by creating a disturbance that propagates through a medium (air etc.). Electrical transmission : transfers energy through electric currents. Electromagnetic radiation : transfers energy in the form of electromagnetic waves such as light, microwaves, and radio waves. Systems and Energy Conservation Energy conservation • Principle of energy conservation: Energy cannot be created or destroyed, only transferred from one form to another. • The principle of conservation of energy is not only true in physics but also in other fields such as biology, chemistry, etc. Power Power • The rate at which energy is transferred is important in the design and use of practical devices such as electrical appliances and engines. • If an external force is applied to an object and if the work done by this force is W in time interval Dt, then the average power delivered to the object during this interval is the work done divided by the time interval: W P Dt FDx Fv Dt P Fv SI unit : watt (W) = J/s = 1 kg m2/s3 W=FDt More general definition U.S. customary unit : 1 hp = 550 ft lb/s = 746 W 1 kWh = (103 W)(3600 s) = 3.60 x 105 J Power Examples • Example 5.12 : Power delivered by an elevator What is the min. power to lift the elevator with the max. load? F ma T f Mg 0 T f Mg 0 T f Mg 2.16 104 N P Fv 6.48 104 W 64.8 kW 86.9 hp M=1.00x103 kg m=8.00x102 kg f =4.00x103 N v = 3.00 m/s Power Examples • Example 5.14 : Speedboat power How much power would a 1.00x103 kg speed boat need to go from rest to 20.0 m/s in 5.00 s, assuming the water exerts a constant drag force of magnitude fd=5.00x102 N and the acceleration is constant? 1 1 Wnet DKE mv2f mvi2 2 2 1 Wengine Wdrag Wengine f d Dx mv 2f 2 v f at vi v f at a 4.00 m/s 2 v 2f vi2 2aDx 50.0 m W fric f d Dx 2.50 10 4 J Wengine P 1 2 mv f f d Dx 2.25 105 J 2 Wengine Dt 4.50 10 4 W 60.3 hp Power Energy and power in a vertical jump • Center of mass (CM) The point in an object at which all the may be considered to be concentrated. h=0.40 m depth of crouch • Stationary jump Dt=0.25 s time for extension Two phases: m=68 kg (1) Extension, (2) free flight PEi KEi PE f KE f 2 vCM 1 2 mvCM mgH H 2 2g vCM 2v 2h / Dt 3.2 m/s H 0.52 m 1 2 KE mvCM 3.5 10 2 J 2 KE P 1.4 103 W Dt