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Unformatted text preview: Kinetic Energy, Work, and Power Shina Tan Kinetic energy of a single particle The αth particle in a system of n particles obeys Newton’s Second Law m α ¨ r α ( t ) = F α ( t ) , (1) where F α ( t ) = F ( e ) α ( t ) + ∑ β f αβ ( t ), F ( e ) α ( t ) is the force exterted by objects external to the system, and f αβ ( t ) is the force exterted by the βth particle in the system. Taking the scalor products of both sides of Eq. (1) with the instantaneous velocity v α ( t ) = ˙ r α ( t ) , and noting that d dt h 1 2 m α v 2 α ( t ) i = d dt h 1 2 m α v α ( t ) · v α ( t ) i = 1 2 m α ˙ v α ( t ) · v α ( t ) + 1 2 m α v α ( t ) · ˙ v α ( t ) = m α ˙ v α ( t ) · v α ( t ) = m α ¨ r α ( t ) · v α ( t ) , we get d dt h 1 2 m α v 2 α ( t ) i = F α ( t ) · d r α ( t ) dt . The left hand side is the rate of change of the kinetic energy, and the right hand side F α ( t ) · v α ( t ) is called the power exerted on the particle. Integrating this equation over time from t 1 to t 2 , we get 1 2 m α v 2 α ( t 2 ) 1 2 m α v 2 α ( t 1 ) = Z t 2 t 1 F α ( t ) · d r α ( t ) . (2) The left hand side is the change of the kinetice energy 1 2 m α v 2 α , and the right hand side is the work done on the particle. 1 Kinetic energy of a system of particles Definition 1– The kinetic energy of a system of particles is defined as T ( t ) = X α 1 2 m α v 2 α ( t ) . (3)...
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This note was uploaded on 01/22/2012 for the course PHYS 3202 taught by Professor Tan during the Spring '11 term at Georgia Tech.
 Spring '11
 Tan
 mechanics, Energy, Force, Kinetic Energy, Power, Work

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