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Chapter 7 Conservation of Energy Recap – Work & Energy The total work done on a particle is equal to the change in its kinetic energy W mv mv 1 2 2 f 1 2 2 i Potential Energy The total work done on an object equals the change in its kinetic energy But the total work done on a system of objects may or may not change its total kinetic energy. The energy may be stored as potential energy. Potential Energy – A Spring Both forces do work on the spring. But the kinetic energy of the spring is unchanged. The energy is stored as potential energy Conservative Forces If the ski lift takes you up a displacement h, the work done on you, by gravity, is –mgh. But when you ski downhill the work done by gravity is +mgh, independent of the path you take Conservative Forces The work done on a particle by a conservative force is independent of the path taken between any two points Potential-Energy Function If a force is conservative, then we can define a potential-energy function as the negative of the work done on the particle s2 U U 2 U1 F ds s1 Potential-Energy Function potential-energy function associated with gravity (taking +y to be up) The value s0 of s (mg ˆj ) (dx iˆ dy ˆj dz kˆ) U0 = U(y0) s0 can be set y to any mg dy y0 convenient mg ( y y0 ) U U 0 mg ( y y0 ) value s U U U 0 F ds Potential-Energy Function of a Spring U U 0 kx 1 2 2 By convention, one chooses U0 =U(0) = 0 Force & Potential-Energy Function In 1-D, given the potential energy function associated with a force one can compute the latter using: dU F dx Example: U kx 1 2 2 dU F kx dx 7-1 Conservation of Energy Conservation of Energy Energy can be neither created nor destroyed Esys 0 Closed System Open System Ein Esys 0 Eout Conservation of Mechanical Energy If the forces acting are conservative then the mechanical energy is conserved Emech K U constant Example 7-3 (1) How high does the block go? Initial mechanical energy of system Ei kx 1 2 2 Final mechanical energy of system E f mgh Example 7-3 (2) Forces are conservative, therefore, mechanical energy is conserved 1 2 kx mgh 2 Height reached 2 kx h 2mg Example 7-4 (1) How far does the mass drop? Initial mech. energy Ei mgyi ky mv 2 i 1 2 2 i 1 2 Final mech. energy E f mgy f ky mv 1 2 2 f 1 2 2 f Example 7-4 (2) Final mech. energy = Initial mech. energy mgy f ky mv 2 f 1 2 2 f 1 2 mgyi ky mv 2 i 1 2 2 i 1 2 mg (d ) (d ) m(0) 2 1 2 mg (0) (0) m(0) 1 2 2 1 2 2 1 2 2 Example 7-4 (3) Solve for d ( kd mg )d 0 1 2 Since d ≠ 0 2mg d k 2mg Example 7-4 Note Espring kd 1 2 (4) 2 is equal to loss in gravitational potential energy Egrav mgd 2mg Conservation of Energy & Kinetic Friction Non-conservative forces, such as kinetic friction, cause mechanical energy to be transformed into other forms of energy, such as thermal energy. Work-Energy Theorem Work done, on a system, by external forces is equal to the change in energy of the system Wext Esys The energy in a system can be distributed in many different ways Example 7-11 (1) Find speed of blocks after spring is released. Consider spring & blocks as system. Write down initial energy. Write down final energy. Subtract initial from final Wext Esys Wext Esys Example 7-11 Initial Energy Ei kx 1 2 2 i Kinetic energy of system is zero initially (2) Take potential energy of system to be zero initially Wext Esys Example 7-11 E f Es Em1 (3) Final Energy Em2 Eother 0 m1v m2 v m2 g s k m1 g s 1 2 2 1 2 Kinetic and potential energies of system have changed 2 Wext Esys Example 7-11 (4) Subtract initial energy from final energy Wext E f Ei But since no external forces act, Wext = 0, so Ef = Ei Wext Esys Example 7-11 (5) And the answer is… kx 2m2 g s 2k m1 g s v m1 m2 2 i Try to derive this. Wext Esys E = mc2 In a brief paper in 1905 Albert Einstein wrote down the most famous equation in science E= 2 mc Sun’s Power Output Power 1 Watt = 1 Joule/second 100 Watt light bulb = 100 Joules/second Sun’s power output 3.826 x 1026 Watts Sun’s Power Output Mass to Energy Kg/s = 3.826 x 1026 Watts / (3 x 108 m/s)2 The Sun destroys mass at ~ 4 billion kg / s Problems To go… Ch. 7, Problem 19 Ch. 7, Problem 29 Ch. 7, Problem 74