Fisica Generale - Alan Giambattista, Betty McCarty Richardson Chapter 6: Conservation of Energy •Work by a Constant Force •Kinetic Energy •Potential Energy •Work by a Variable Force •Springs and Hooke’s Law •Conservation of Energy •Power Copyright © 2008 – The McGraw-Hill Companies s.r.l. 1 Fisica Generale - Alan Giambattista, Betty McCarty Richardson §6.1 The Law of Conservation of Energy The total energy of the Universe is unchanged by any physical process. The three kinds of energy are: kinetic energy, potential energy, and rest energy. Energy may be converted from one form to another or transferred between bodies. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 2 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Copyright © 2008 – The McGraw-Hill Companies s.r.l. 3 Fisica Generale - Alan Giambattista, Betty McCarty Richardson §6.2 Work by a Constant Force Work is an energy transfer by the application of a force. For work to be done there must be a nonzero displacement. The unit of work and energy is the joule (J). 1 J = 1 Nm = 1 kg m2/s2. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 4 Fisica Generale - Alan Giambattista, Betty McCarty Richardson It is only the force in the direction of the displacement that does work. An FBD for the box at left: F rx y rx N w x F The work done by the force F is: WF Fx rx F cos x Copyright © 2008 – The McGraw-Hill Companies s.r.l. 5 Fisica Generale - Alan Giambattista, Betty McCarty Richardson The work done by the force N is: WN 0 The normal force is perpendicular to the displacement. The work done by gravity (w) is: Wg 0 The force of gravity is perpendicular to the displacement. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 6 Fisica Generale - Alan Giambattista, Betty McCarty Richardson The net work done on the box is: Wnet WF WN Wg F cos x 0 0 F cos x Copyright © 2008 – The McGraw-Hill Companies s.r.l. 7 Fisica Generale - Alan Giambattista, Betty McCarty Richardson In general, the work done by a force F is defined as W Fr cos where F is the magnitude of the force, r is the magnitude of the object’s displacement, and is the angle between F and r (drawn tail-to-tail). Copyright © 2008 – The McGraw-Hill Companies s.r.l. 8 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example: A ball is tossed straight up. What is the work done by the force of gravity on the ball as it rises? y r FBD for rising ball: x w Wg wy cos 180 mgy Copyright © 2008 – The McGraw-Hill Companies s.r.l. 9 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example: A box of mass m is towed up a frictionless incline at constant speed. The applied force F is parallel to the incline. What is the net work done on the box? y F N F x w Apply Newton’s 2nd Law: F F x F w sin 0 y N w cos 0 Copyright © 2008 – The McGraw-Hill Companies s.r.l. 10 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example continued: The magnitude of F is: F mg sin If the box travels along the ramp a distance of x the work by the force F is WF Fx cos 0 mgx sin The work by gravity is Wg wx cos 90 mgx sin Copyright © 2008 – The McGraw-Hill Companies s.r.l. 11 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example continued: The work by the normal force is: WN Nx cos 90 0 The net work done on the box is: Wnet WF Wg WN mgx sin mgx sin 0 0 Copyright © 2008 – The McGraw-Hill Companies s.r.l. 12 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example: What is the net work done on the box in the previous example if the box is not pulled at constant speed? F x F w sin ma F ma w sin Proceeding as before: Wnet WF Wg WN ma mgx sin mgx sin 0 ma x Fnet x Copyright © 2008 – The McGraw-Hill Companies s.r.l. 13 Fisica Generale - Alan Giambattista, Betty McCarty Richardson §6.3 Kinetic Energy 1 2 K mv 2 is an object’s translational kinetic energy. This is the energy an object has because of its state of motion. It can be shown that, in general Wnet K . Copyright © 2008 – The McGraw-Hill Companies s.r.l. 14 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example: The extinction of the dinosaurs and the majority of species on Earth in the Cretaceous Period (65 Myr ago) is thought to have been caused by an asteroid striking the Earth near the Yucatan Peninsula. The resulting ejecta caused widespread global climate change. If the mass of the asteroid was 1016 kg (diameter in the range of 4-9 miles) and had a speed of 30.0 km/sec, what was the asteroid’s kinetic energy? 1 2 1 16 K mv 10 kg 30 103 m/s 2 2 24 4.5 10 J 2 This is equivalent to ~109 Megatons of TNT. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 15 Fisica Generale - Alan Giambattista, Betty McCarty Richardson §6.4 Gravitational Potential Energy Part 1 Objects have potential energy because of their location (or configuration). There are potential energies associated with different (but not all!) forces. Such a force is called a conservative force. In general Wcons U Copyright © 2008 – The McGraw-Hill Companies s.r.l. 16 Fisica Generale - Alan Giambattista, Betty McCarty Richardson The change in gravitational potential energy (only near the surface of the Earth) is U g mgy where y is the change in the object’s vertical position with respect to some reference point that you are free to choose. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 17 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example: What is the change in gravitational potential energy of the box if it is placed on the table? The table is 1.0 m tall and the mass of the box is 1.0 kg. First: Choose the reference level at the floor. U=0 here. U g mgy mg y f yi 1.0 kg 9.8 m/s 2 1.0 m 0 m 9.8 J Copyright © 2008 – The McGraw-Hill Companies s.r.l. 18 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example continued: Now take the reference level (U=0) to be on top of the table so that yi = -1.0 m and yf = 0.0 m. U g mgy mg y f yi 1 kg 9.8 m/s 2 0.0m 1.0 m 9.8 J The results do not depend on the location of U=0. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 19 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Mechanical energy is E K U Whenever nonconservative forces do no work, the mechanical energy of a system is conserved. That is Ei=Ef or K= -U. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 20 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example (text problem 6.27): A cart starts from position 4 with v = 15.0 m/s to the left. Find the speed of the cart at positions 1, 2, and 3. Ignore friction. E4 E3 U 4 K 4 U 3 K3 1 2 1 2 mgy4 mv4 mgy3 mv3 2 2 v3 v42 2 g y4 y3 20.5 m/s Copyright © 2008 – The McGraw-Hill Companies s.r.l. 21 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example continued: E4 E2 Or use E3=E2 U 4 K4 U 2 K2 1 1 mgy4 mv42 mgy2 mv22 2 2 v2 v42 2 g y4 y2 18.0 m/s E4 E1 U 4 K 4 U1 K1 Or use E3=E1 1 2 1 2 mgy4 mv4 mgy1 mv1 2 2 E2=E1 v1 v42 2 g y4 y1 24.8 m/s Copyright © 2008 – The McGraw-Hill Companies s.r.l. 22 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example (text problem 6.82): A roller coaster car is about to roll down a track. Ignore friction and air resistance. m=988 kg 40 m 20 m y=0 (a) At what speed does the car reach the top of the loop? Ei E f U i Ki U f K f 1 2 mgyi 0 mgy f mv f 2 v f 2 g yi y f 19.8 m/s Copyright © 2008 – The McGraw-Hill Companies s.r.l. 23 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example continued: (b) What is the force exerted on the car by the track at the top of the loop? FBD for the car: y Apply Newton’s Second Law: x N w v2 Fy N w mar m r v2 N wm r v2 N m mg 2.9 10 4 N r Copyright © 2008 – The McGraw-Hill Companies s.r.l. 24 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example continued: (c) From what minimum height above the bottom of the track can the car be released so that it does not lose contact with the track at the top of the loop? Using conservation of mechanical energy: Ei E f U i Ki U f K f 1 2 mgyi 0 mgy f mvmin 2 Solve for the starting height 2 vmin yi y f 2g Copyright © 2008 – The McGraw-Hill Companies s.r.l. 25 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example continued: What is vmin? v=vmin when N=0. This means that v2 N wm r 2 vmin w mg m r vmin gr The initial height must be 2 vmin gr yi y f 2r 2.5r 25.0 m 2g 2g Copyright © 2008 – The McGraw-Hill Companies s.r.l. 26 Fisica Generale - Alan Giambattista, Betty McCarty Richardson What do you do when there are nonconservative forces? For example, if friction is present E E f Ei Wfric The work done by friction. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 27 Fisica Generale - Alan Giambattista, Betty McCarty Richardson §6.5 Gravitational Potential Energy Part 2 The general expression for gravitational potential energy is: GM1M 2 U r r where U r 0 Copyright © 2008 – The McGraw-Hill Companies s.r.l. 28 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example: What is the gravitational potential energy of a body of mass m on the surface of the Earth? GM e m GM1M 2 U r Re r Re Copyright © 2008 – The McGraw-Hill Companies s.r.l. 29 Fisica Generale - Alan Giambattista, Betty McCarty Richardson §6.6 Work by a Variable Force Work can be calculated by finding the area underneath a plot of the applied force in the direction of the displacement versus the displacement. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 30 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example: What is the work done by the variable force shown below? Fx (N) F3 F2 F1 x1 x2 x3 x (m) The work done by F1 is W1 F1 x1 0 The work done by F2 is W2 F2 x2 x1 The work done by F3 is W3 F3 x3 x2 The net work is then W1+W2+W3. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 31 Fisica Generale - Alan Giambattista, Betty McCarty Richardson By hanging masses on a spring we find that stretch applied force. This is Hooke’s law. For an ideal spring: Fx = -kx Fx is the magnitude of the force exerted by the free end of the spring, x is the measured stretch of the spring, and k is the spring constant (a constant of proportionality; its units are N/m). A larger value of k implies a stiffer spring. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 32 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example (text problem 6.48): (a) If forces of 5.0 N applied to each end of a spring cause the spring to stretch 3.5 cm from its relaxed length, how far does a force of 7.0 N cause the same spring to stretch? For springs Fx. This allows us to write F1 x1 . F2 x2 F2 7.0 N Solving for x2: x2 x1 3.5 cm 4.9 cm. F1 5.0 N Copyright © 2008 – The McGraw-Hill Companies s.r.l. 33 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example continued: (b) What is the spring constant of this spring? F1 5.0 N k 1.43 N/cm. x1 3.5 cm Or F2 7.0 N k 1.43 N/cm. x2 4.9 cm Copyright © 2008 – The McGraw-Hill Companies s.r.l. 34 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example (text problem 6.46): An ideal spring has k = 20.0 N/m. What is the amount of work done (by an external agent) to stretch the spring 0.40 m from its relaxed length? Fx (N) kx1 x1=0.4 m x (m) W Area under curve 1 1 1 2 kx1 x1 kx12 20.0 N/m 0.4 m 1.6 J 2 2 2 Copyright © 2008 – The McGraw-Hill Companies s.r.l. 35 Fisica Generale - Alan Giambattista, Betty McCarty Richardson §6.7 Elastic potential energy The work done in stretching/compressing a spring transfers energy to the spring. 1 2 U s kx 2 Copyright © 2008 – The McGraw-Hill Companies s.r.l. 36 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example: A box of mass 0.25 kg slides along a horizontal, frictionless surface with a speed of 3.0 m/s. The box encounters a spring with k = 200 N/m. How far is the spring compressed when the box is brought to rest? Ei E f U i Ki U f K f 1 2 1 2 0 mv kx 0 2 2 m v 0.11 m x k Copyright © 2008 – The McGraw-Hill Companies s.r.l. 37 Fisica Generale - Alan Giambattista, Betty McCarty Richardson §6.8 Power Power is the rate of energy transfer. Average Power E Pav t Instantaneous Power P Fv cos The unit of power is the watt. 1 watt = 1 J/s = 1 W. Copyright © 2008 – The McGraw-Hill Companies s.r.l. 38 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Example (text problem 6.73): A race car with a mass of 500.0 kg completes a quarter-mile (402 m) race in a time of 4.2 s starting from rest. The car’s final speed is 125 m/s. What is the engine’s average power output? Neglect friction and air resistance. E U K Pav t t 1 2 mv K 2 f 9.3 105 watts t t Copyright © 2008 – The McGraw-Hill Companies s.r.l. 39 Fisica Generale - Alan Giambattista, Betty McCarty Richardson Summary •Conservation of Energy •Calculation of Work Done by a Constant or Variable Force •Kinetic Energy •Potential Energy (gravitational, elastic) •Power Copyright © 2008 – The McGraw-Hill Companies s.r.l. 40

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# Chapter 6: Energy