Fowles04 - CHAPTER 4 GENERAL MOTION OF A PARTICLE IN THREE...

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CHAPTER 4 GENERAL MOTION OF A PARTICLE IN THREE DIMENSIONS --------------------------------------------------------------------------------------------------------- Note to instructors … there is a typo in equation 4.3.14. The range of the projectile is … 2 0 sin 2 v R x g α = = … NOT 2 2 0 sin 2 v g ... α --------------------------------------------------------------------------------------------------------- 4.1 (a) ˆ ˆ ˆ V V F V i j k V x y z = −∇ = − K K K ( ) ˆ ˆ ˆ F c iyz jxz kxy = − + + K K (b) ˆ ˆ ˆ 2 2 2 F V i x j y k z α β γ = −∇ = − K K (c) ( ) ( ) ˆ ˆ ˆ x y z F V ce i j k α β γ α β γ + + = −∇ = + + (d) 1 1 ˆ ˆ ˆ sin r V V F V e e e r r r θ φ V θ θ φ = −∇ = − K K K 1 ˆ n r F e cnr = − 4.2 (a) ˆ ˆ ˆ 0 i j k F x y z x y z ∇× = = K K conservative (b) ( ) 2 ˆ ˆ ˆ ˆ 1 1 0 i j k F k x y z y x z ∇× = = − − K K conservative (c) ( ) 3 ˆ ˆ ˆ ˆ 1 1 0 i j k F k x y z y x z ∇× = = = K K conservative (d) 1
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2 ˆ ˆ ˆ sin 1 0 sin 0 0 r n e e r e r F r r kr θ φ θ θ θ φ ∇× = = K K conservative 4.3 (a) ( ) 2 3 ˆ ˆ ˆ 2 i j k F k x y z xy cx z ∇× = = K K cx x 2 0 cx x = 1 2 c = (b) 2 ˆ ˆ ˆ i j k F x y z z cxz x y y y = K K ∇× 2 2 2 2 1 1 ˆ ˆ ˆ x cx cz z i j k y y y y y y + + + = 2 2 0 x cx y y = c 1 = − also 2 2 0 z y y = cz + implies that 1 c = − as it must 4.4 (a) constant E = ( ) 2 1 , , 2 V x y z mv = + at the origin 2 1 0 2 = + D E m v at ( ) 1,1,1 2 2 1 1 2 2 E mv α β γ = + + + = D mv 2
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( ) 2 2 2 m v v α β γ = + + D ( ) 1 2 2 2 m α β γ v v = + + D (b) ( ) 2 2 0 m α β γ + = D v + ( ) 1 2 2 m α β γ = + + D v (c) x V mx F x = = − ±± mx α = − ±± 2 V y y my β = − = − ±± 2 3 V z z mz γ = − = − ±± 4.5 (a) ˆ ˆ F ix jy = + G on the path x y = : ˆ ˆ dr idx jdy = + K ( ) ( ) 1,1 1 1 1 1 0,0 0 0 0 0 1 x y F dr F dx F dy xdx ydy = + = + = K K K on the path along the x-axis: ˆ dr idx = K and on the line 1 x = : ˆ dr jdy = ( ) ( ) 1,1 1 1 0,0 0 0 1 x y F dr F dx F dy = + = K K K is conservative. F (b) ˆ ˆ F iy jx = G on the path x y = : ( ) ( ) 1,1 1 1 1 1 0,0 0 0 0 0 x y F dr F dx F dy ydx xdy = + = K K K and, with x y = ( ) ( ) 1,1 1 1 0,0 0 0 0 F dr xdx ydy = K = on the x-axis: ( ) ( ) 1,0 1 1 0,0 0 0 x F dr F dx ydx = = K K K and, with on the x-axis 0 y = ( ) ( ) 1,0 0,0 0 F dr = K K on the line 1 x = : ( ) ( ) 1,1 1 1 1,0 0 0 y F dr F dy xdy = = K 3
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and, with 1 x = ( ) ( ) 1,1 1 1,0 0 1 F dr dy = = K K on this path ( ) ( ) 1,1 0,0 0 1 1 F dr = + = K K is not conservative. F K 4.6 From Example 2.3.2, ( ) ( ) 2 e e r mg r z = − V z + ( ) 1 1 e e z mgr r V z = − + From Appendix D, ( ) 1 2 1 1 x x x + = + + ( ) 2 2 1 e e e z z mgr r r = − + + V z ( ) 2 e e mgz mgr mgz r = − + + V z With an additive constant, e mgr ( ) 1 e z mgz r V z ( ) ˆ F V k V z = −∇ = − K K z 1 ˆ 1 e e z kmg z r r = − + 2 ˆ 1 e z F kmg r = − K , 0 x mx F = = ±± 0 y my F = = ±± 2 1 e z mz mg r = − ±± 2
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