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Chapter 7 Conservation of Energy

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Recap – Work & Energy The total work done on a particle is equal to the change in its kinetic energy

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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.

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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

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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

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Conservative Forces The work done on a particle by a conservative force is independent of the path taken between any two points

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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

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Potential-Energy Function potential-energy function associated with gravity (taking +y to be up) The value of U 0 = U(y 0 ) can be set to any convenient value

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Potential-Energy Function of a Spring By convention, one chooses U 0 =U(0) = 0

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Force & Potential-Energy Function In 1-D, given the potential energy function associated with a force one can compute the latter using: Example:

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7-1 Conservation of Energy

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Conservation of Energy Energy can be neither created nor destroyed Closed System Open System

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Conservation of Mechanical Energy If the forces acting are conservative then the mechanical energy is conserved

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Example 7-3(1) How high does the block go? Initial mechanical energy of system Final mechanical energy of system

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Example 7-3(2) Forces are conservative, therefore, mechanical energy is conserved Height reached

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Example 7-4(1) How far does the mass drop? Final mech. energy Initial mech. energy

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Example 7-4(2) Final mech. energy = Initial mech. energy

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Example 7-4(3) Solve for d Since d 0

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Example 7-4(4) Note is equal to loss in gravitational potential energy

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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.

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Work-Energy Theorem Work done, on a system, by external forces is equal to the change in energy of the system The energy in a system can be distributed in many different ways

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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

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Example 7-11(2) Initial Energy Take potential energy of system to be zero initially Kinetic energy of system is zero initially

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Example 7-11(3) Final Energy Kinetic and potential energies of system have changed

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Example 7-11(4) Subtract initial energy from final energy But since no external forces act, W ext = 0, so E f = E i

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Example 7-11(5) And the answer is… Try to derive this.

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E = mc 2 In a brief paper in 1905 Albert Einstein wrote down the most famous equation in science E = mc 2

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Suns Power Output Power 1 Watt = 1 Joule/second 100 Watt light bulb = 100 Joules/second Suns power output x Watts x Watts

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Suns Power Output Mass to Energy x Watts Kg/s = x Watts / (3 x 10 8 m/s) 2 The Sun destroys mass at ~ 4 billion kg / s

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Problems To go…

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Ch. 7, Problem 19

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Ch. 7, Problem 29

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Ch. 7, Problem 74

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