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Fowles02

# Fowles02 - CHAPTER 2 NEWTONIAN MECHANICS RECTILINEAR MOTION...

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CHAPTER 2 NEWTONIAN MECHANICS: RECTILINEAR MOTION OF A PARTICLE 2.1 (a) ( ) 1 x F ct m = + D ±± ( ) 2 0 1 2 t F c x F ct dt t t m m = + = + D D ± m 2 2 0 2 6 t F F c c 3 x t t dt t m m m m = + = + D D t (b) sin F x ct m D = ±± ( ) 0 0 sin cos 1 cos t t F F F x ct dt ct ct m cm cm = = − = D D D ± ( ) 0 1 1 cos 1 sin t F F x ct dt ct cm cm c = = D D (c) ct F x e = ±± m D ( ) 0 1 t ct ct F F x e e cm cm = = D D ± ( ) 2 1 1 1 ct ct F F x e t e cm c c c m = = D D ct 2.2 (a) dx dx dx dx x x dt dx dt dx = ± ± ± = = ±± ± ( ) 1 dx x F cx dx m = + D ± ± ( ) 1 xdx F cx dx m = + D ± ± 2 2 1 1 2 2 cx x F x m = + D ± ( ) 1 2 2 x x F cx m = + D ± (b) 1 cx dx x x F dx m D ± ± e = = ±± 1

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1 cx xdx F e dx m = D ± ± ( ) ( ) 2 1 1 1 2 cx cx F F x e cm cm = − = D D ± e ( ) 1 2 2 1 cx F x e cm = D ± (c) ( ) 1 cos dx x x F dx m D ± ± cx = = ±± cos F xdx cx dx m = D ± ± 2 1 sin 2 F x cx cm = D ± 1 2 2 sin F x cx cm = D ± 2.3 (a) ( ) ( ) 2 2 x x cx F cx dx F x C = − + = − + D D D V x (b) ( ) x cx cx x F F e dx e C c = − = + D D D V x (c) ( ) cos sin x x F F cx dx cx C c = − = − + D D D V x 2.4 (a) ( ) ( ) dV x kx dx = − = − F x ( ) 2 0 1 2 x kx dx kx = = V x (b) T T ( ) ( ) x V x = + D ( ) ( ) ( ) 2 1 2 T V x k A x = = D T x (c) 2 1 2 E T kA = = D (d) turning points @ T x ( ) 1 0 1 x A = ± 2.5 (a) ( ) 3 2 kx F x kx A + so ( ) 3 4 2 2 2 0 1 1 2 4 x kx kx kx dx kx A A = = V x (b) ( ) ( ) 4 2 2 1 1 2 4 kx T V x T kx A = = + D D T x (c) E T = D 2
(d) V has maximum at ( ) x ( ) 0 m F x 3 2 0 m m kx A = kx m x A = ± ( ) 4 2 2 2 1 1 1 2 4 4 m kA x kA kA A = = V If turning points exist. ( m E V x < ) Turning points @ T x let u ( ) 1 0 2 1 x = 2 2 1 1 0 2 4 ku E ku A + = solving for u , we obtain 1 2 2 2 4 1 1 E kA = ± u A or 1 2 1 2 4 1 1 E x A kA = ± 2.6 ( ) x v x x α = = ± 2 2 3 x x x x α α = − = − ±± ± ( ) 2 3 m F x mx x α = = − ±± 2.7 sin F Mg θ 2.8 dx F mx mx dx = = ± ±± ± 3 x bx = ± 4 3 dx

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Fowles02 - CHAPTER 2 NEWTONIAN MECHANICS RECTILINEAR MOTION...

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