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Chapter 6 Conservation of Energy 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 MFMcGraw Ch06 - Energy - Revised: 2/20/10 2 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. MFMcGraw Ch06 - Energy - Revised: 2/20/10 3 MFMcGraw Ch06 - Energy - Revised: 2/20/10 4 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. MFMcGraw Ch06 - Energy - Revised: 2/20/10 5 Work - Example Only the force in the direction of the displacement that does work. An FBD for the box at left: F Drx y q Drx N q x w F The work done by the force F is: MFMcGraw Ch06 - Energy - Revised: 2/20/10 6 Work - Example The work done by the normal force N is: The normal force is perpendicular to the displacement. The work done by gravity (w) is: The force of gravity is perpendicular to the displacement. MFMcGraw Ch06 - Energy - Revised: 2/20/10 7 Work - Example The net work done on the box is: MFMcGraw Ch06 - Energy - Revised: 2/20/10 8 Work Done In general, the work done by a force F is defined as where F is the magnitude of the force, Dr is the magnitude of the object’s displacement, and q is the angle between F and Dr (drawn tail-to-tail). MFMcGraw Ch06 - Energy - Revised: 2/20/10 9 Work - Example Example: A ball is tossed straight up. What is the work done by the force of gravity on the ball as it rises? y Dr FBD for rising ball: x w MFMcGraw Ch06 - Energy - Revised: 2/20/10 10 Inclined Plane-V Constant A box of mass m is towed up a frictionless incline at constant speed. The applied force F is parallel to the incline. y Question: What is the net work done on the box? F F N x An FBD for the box: q q w Apply Newton’s 2nd Law: MFMcGraw Ch06 - Energy - Revised: 2/20/10 11 Inclined Plane-V Constant Example continued: The magnitude of F is: If the box travels along the ramp a distance of Dx the work by the force F is The work by gravity is MFMcGraw Ch06 - Energy - Revised: 2/20/10 12 Inclined Plane-V Constant Example continued: The work by the normal force is: The net work done on the box is: MFMcGraw Ch06 - Energy - Revised: 2/20/10 13 Inclined Plane-Acceleration Example: What is the net work done on the box in the previous example if the box is not pulled at constant speed? Proceeding as before: New Term MFMcGraw Ch06 - Energy - Revised: 2/20/10 14 Kinetic Energy 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 Net Work = Change in K MFMcGraw Ch06 - Energy - Revised: 2/20/10 15 Kinetic Energy 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? This is equivalent to ~109 Megatons of TNT. MFMcGraw Ch06 - Energy - Revised: 2/20/10 16 Gravitational Potential Energy Part 1- Close to Earth’s Surface Potential energy is an energy of position. There are potential energies associated with different forces. Forces that have a potential energy associated with them are called conservative forces. In general Not all forces are conservative, i.e. Friction. MFMcGraw Ch06 - Energy - Revised: 2/20/10 17 Gravitational Potential Energy The change in gravitational potential energy (only near the surface of the Earth) is where Dy is the change in the object’s vertical position with respect to some reference point. You are free to choose to location of this where ever it is convenient. MFMcGraw Ch06 - Energy - Revised: 2/20/10 18 GPE Problem The table is 1.0 m tall and the mass of the box is 1.0 kg. Ques: What is the change in gravitational potential energy of the box if it is placed on the table? U=0 First: Choose the reference level at the floor. U = 0 here. MFMcGraw Ch06 - Energy - Revised: 2/20/10 19 GPE Problem 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. The results do not depend on the location of U = 0. MFMcGraw Ch06 - Energy - Revised: 2/20/10 20 Total Mechanical Energy Mechanical energy is The total mechanical energy of a system is conserved whenever nonconservative forces do no work. That is E i = Ef or DK = -DU. Then if DK increases DU decreases and vice versa MFMcGraw Ch06 - Energy - Revised: 2/20/10 21 Mechanical Energy Problem 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. MFMcGraw Ch06 - Energy - Revised: 2/20/10 22 Mechanical Energy Problem Or use E3=E2 Or use E3=E1 E2=E1 MFMcGraw Ch06 - Energy - Revised: 2/20/10 23 Roller Coaster Problem 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? MFMcGraw Ch06 - Energy - Revised: 2/20/10 24 Roller Coaster Problem Example continued: (b) What is the force exerted on the car by the track at the top of the loop? Apply Newton’s Second Law: FBD for the car: y x N w MFMcGraw Ch06 - Energy - Revised: 2/20/10 25 Roller Coaster Problem 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: Solve for the starting height MFMcGraw Ch06 - Energy - Revised: 2/20/10 26 Roller Coaster Problem Example continued: What is vmin? v = vmin when N = 0. This means that The initial height must be MFMcGraw Ch06 - Energy - Revised: 2/20/10 27 Nonconservative Forces What do you do when there are nonconservative forces? For example, if friction is present The work done by friction. MFMcGraw Ch06 - Energy - Revised: 2/20/10 28 Gravitational Potential Energy Part 2 - Away from Earth’s Surface The general expression for gravitational potential energy is: MFMcGraw Ch06 - Energy - Revised: 2/20/10 29 Gravitational Potential Energy Example: What is the gravitational potential energy of a body of mass m on the surface of the Earth? MFMcGraw Ch06 - Energy - Revised: 2/20/10 30 Planetary Motion A planet of mass m has an elliptical orbit around the Sun. The elliptical nature of the orbit means that the distance between the planet and Sun varies as the planet follows its orbital path. Take the planet to move counterclockwise from its initial location. QUES: How does the speed of a planet vary as it orbits the Sun once? The mechanical energy of the planet-sun system is: B r A MFMcGraw Ch06 - Energy - Revised: 2/20/10 31 Planetary Motion B r A At point “B” the planet is the farthest from the Sun. At point “A” the planet is at its closest approach to the sun. Starting from point “B” (where the planet moves the slowest), as the planet moves in its orbit r begins to decrease. As it decreases the planet moves faster. At point “A” the planet reaches its fastest speed. As the planet moves past point A in its orbit, r begins to increase and the planet moves slower. MFMcGraw Ch06 - Energy - Revised: 2/20/10 32 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. MFMcGraw Ch06 - Energy - Revised: 2/20/10 33 Example: What is the work done by the variable force shown below? Fx (N) F3 F2 The net work is then W1+W2+W3. F1 x1 x2 x3 x (m) The work done by F1 is The work done by F2 is The work done by F3 is MFMcGraw Ch06 - Energy - Revised: 2/20/10 34 Spring Force By hanging masses on a spring we find that stretch µ 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. MFMcGraw Ch06 - Energy - Revised: 2/20/10 35 Spring Force (a) A force of 5.0 N applied to the end of a spring cause the spring to stretch 3.5 cm from its relaxed length. Ques: How far does a force of 7.0 N cause the same spring to stretch? For springs Fµx. This allows us to write Solving for x2: MFMcGraw Ch06 - Energy - Revised: 2/20/10 36 Spring Force Example continued: (b) What is the spring constant of this spring? Or MFMcGraw Ch06 - Energy - Revised: 2/20/10 37 Spring Force 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? MFMcGraw Ch06 - Energy - Revised: 2/20/10 38 Elastic Potential Energy The work done in stretching or compressing a spring transfers energy to the spring. Below is the equation of the spring potential energy. The spring is considered the system MFMcGraw Ch06 - Energy - Revised: 2/20/10 39 Elastic Potential Energy 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. Ques: How far is the spring compressed when the box is brought to rest? MFMcGraw Ch06 - Energy - Revised: 2/20/10 40 Power Power is the rate of energy transfer. Average Power Instantaneous Power The unit of power is the watt. 1 watt = 1 J/s = 1 W. MFMcGraw Ch06 - Energy - Revised: 2/20/10 41 Power - Car Example 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. (Neglect friction and air resistance.) Ques: What is the engine’s average power output? MFMcGraw Ch06 - Energy - Revised: 2/20/10 42 Summary • Conservation of Energy • Calculation of Work Done by a Constant or Variable Force • Kinetic Energy • Potential Energy (gravitational, elastic) • Power MFMcGraw Ch06 - Energy - Revised: 2/20/10 43