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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 Rx g α == … NOT 22 0 sin 2 v g ... --------------------------------------------------------------------------------------------------------- 4.1 (a) ˆ ˆˆ VV FV i jk V x yz ∂∂∂ =−∇ =− KK K () ˆ F c iyz jxz kxy + + (b) ˆ 22 2 i x j y k z βγ (c) ( ) ˆ xyz c e i j k αβ γ −++ = + + (d) 11 ˆ sin r e e e rr r θφ V θ ∂∂ K 1 ˆ n r Fe c n r 4.2 (a) ˆ 0 ijk F ∇× = = conservative (b) 2 ˆ ˆ 11 0 Fk yx z = = − − conservative (c) 3 ˆ ˆ yxz = = = conservative (d) 1
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2 ˆˆˆ sin 1 0 sin 00 r n ee r e r F rr kr θφ θ θθ φ ∂∂ ∂ ∇× = = KK conservative 4.3 (a) () 23 ˆ ˆˆ 2 ijk Fk xyz xy cx z ∂∂∂ = = c x x 20 cx x −= 1 2 c = (b) 2 ˆ F x yz zc x zx yy y = 22 11 ˆ x cx cz z ij k yy  + − + +   =− 0 xc x = −− c 1 = − a l s o 0 z = cz + implies that 1 c = − as it must 4.4 (a) constant E = 2 1 ,, 2 Vxyz m v =+ at the origin 2 1 0 2 D Em v a t 1,1,1 v αβ γ =+++ = D m v 2
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() 22 2 m vv α β γ = −+ + D 1 2 2 2 m αβ =− + + D (b) 2 2 0 m + = D v 1 2 2 m  =+ +   D v (c) x V mx F x == ±± mx 2 V y y my 2 3 V z z mz 4.5 (a) ˆˆ Fi xj y G on the path x y = : dr idx jdy K ( ) 1,1 1 1 1 1 0,0 0 0 0 0 1 xy 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 0 0 1 Fd r x y + = K K K is conservative. F (b) yj x G on the path x y = : ( ) 1 1 1 1 0 0 0 0 F dr F dx F dy ydx xdy + = K K K and, with x y = ( ) 1 1 0 0 0 r x d x y d y K = on the x-axis: ( ) 1,0 1 1 0 0 x r x y d x = K K K and, with on the x-axis 0 y = ( ) 1,0 0 r = K K on the line 1 x = : ( ) 1 1 1,0 0 0 y r y x d y = K 3
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and, with 1 x = () ( ) 1,1 1 1,0 0 1 Fd r d y == ∫∫ K K on this path ( ) 0,0 011 r =+= K K is not conservative. F K 4.6 From Example 2.3.2, 2 e e r m g rz =− Vz + 1 1 e e z m g r r  +   From Appendix D, 1 2 11 xx x + =−+ + 2 2 1 e ee zz m g r rr + + 2 e e mgz m g r m g z r + + With an additive constant, e mgr 1 e z m g z r ≈− ˆ FV k V z =−∇ KK z 1 ˆ 1 z kmg z  +   2 ˆ 1 e z Fk m g r K , 0 x mx F ±± 0 y my F = = 2 1 e z mz mg r 2 1 e dz z mz mg dz r ± ± 0 0 2 1 z h v e z zdz g dz r D ± 2 2 1 2 z e h vg h r −= −− D 2 2 0 2 ez e rv h g −+ = D hr 4
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2 2 2 1 22 ee e rr hr g =− D z v ( ) , e hz r ± 2 2 1 z e v gr D h From Appendix D, () 2 1 2 11 28 xx x + =+ − + 24 222 4 zz e vv h gg r =−+ + + DD 1 e h r  ≈+   From Example 2.3.2, 2 2 g D v h = 1 2 1 2 e v gr D And with 1 x x −≈ + , 1 e h r 4.7 For a point on the rim measured from the center of the wheel: ˆˆ cos sin ri b j b θ K vt t b θω == D , so sin cos v j v K ² Relative to the ground, ( ) 1s i n c o s v j v vi =− − K For a particle of mud leaving the rim: sin yb D and cos y = − So cos yy g t v g t =−= and 2 1 sin cos 2 v t g t D At maximum height, 0 v y = : cos v g t D 2 cos 1 cos sin cos 2 v g hb       D 2 2 cos sin 2 v g + D Maximum h
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Fowles04 - CHAPTER 4 GENERAL MOTION OF A PARTICLE IN THREE...

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