# l12anno - Work = Force times distance t2 W12 = P(t dt t1 t2...

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1 2 2 2 22 1 1 1 1 ) () 1 ( 2 ()· (( ) ) · · OC OC C OC OC t t t t t t t t Pt W t dt dt tt t dt d = = = = = = rr F r r F v r F r ± ( (() ) ) OC OC OC t td t d = = == F FF r ± Work = “Force times distance” Change of Variable
1 2 1 22 1 2 1 2 1 2 1 () 12 21 · · · ·) ( OC OC OC OC OC OC t t t t t t d d d W = = = = = = = = = =− rr Fr r r Case 1: Constant force, e.g. gravity

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() 11 1 1 22 2 2 ( 1 ) ) 2 ( ()· () ( ( )) ( ) · ( ) ( ) . .. ( (() )( ) ) rr t t t t t t P f fr tt t rt t t tr t Pt d t W t frd r dt = = = =− = + Fv ee ± ± ± Case 2: Central force r f r = − Fe r e OP r r = re O
2 2 1 2 1 2 1 1 2 1 0 0 12 0 0 ( ( 2 2 ) ) 1 ( ) ( ) is spring stiffness is () () ' ' () 1 '( ) ( natural (relaxed) spring length and Change of variable, 2 r r r r r r r r fr kr L k kr L kr L Wf r d r dr d rr r L r k r dr dr k r dd δδ δ = = =− = == = = −= 2 )( ) ii r = Spring 0 r = −− Fe r e OP r r = re O

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1 2 2 1 2 2 2 2 2 21 1 2 2 2 ( ) Newton's law of Universal gravitation Universal gravitation c earth's earth's mass onstant ( ) At earth's surface radius, 11 ( () ) e e eee e e r r r e r e mm fr G r G fr mg Gm gr r m mgr Wf
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## This note was uploaded on 01/26/2012 for the course TAM 212 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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l12anno - Work = Force times distance t2 W12 = P(t dt t1 t2...

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