Analytical Mech Homework Solutions 44

Analytical Mech Homework Solutions 44 - projection on the...

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Unformatted text preview: projection on the xy-plane make an angle φ with the x-axis: x = s sin θ cos φ , and Fx = Fr sin θ cos φ = mx y = s si n θ si n φ , and Fy = Fr sin θ sin φ = my z = s cos θ , and Fz = − mg + Fr cos θ = mz Since Fr = −c2 s = −c2 ( x + y 2 + z 2 ) , the differential equations of motion are not 2 2 separable. mx = −c2 s 2 sin θ cos φ = −c2 sx dx dx ds dx m = m ⋅ = ms = −c2 sx dt ds dt ds dx c2 c = − ds = −γ ds , where γ = 2 m x m x ln x − ln x = ln = −γ s x θ x z g γ x g γx From eqn 4.3.16, + 2 max + 2 ln 1 − max = 0 x γ γ γ x 2 3 uu From Appendix D: ln (1 − u ) = −u − − − … for u < 1 23 3 γx γx γ 2 xmax 2 γ 3 xmax − + terms in γ 4 ln 1 − max = − max − 2 3 x x 2x 3x 2 3 z xmax gxmax gxmax gxmax gγ xmax + − − − + terms in γ 2 = 0 2 3 γx γx 2x 3x x 2 xmax + 3x 3x 2 z ≈0 xmax − 2γ gγ 1 xmax 3x 9 x 2 3x 2 z 2 ≈− ± + gγ 4γ 16γ 2 1 3 x 3 x 16γ z 2 ± xmax ≈ − 1 + 4γ 4γ 3g Since xmax > 0 , the + sign is used. From Appendix D: 1 2 16γ z 2 8γ z 1 16γ z 3 − 1 + = 1+ + terms in γ 3g 3g 8 3g 3x 3x 2 x z 8 x γ z 2 + + − + terms in γ 2 xmax = − 2 4γ 4γ g 3g 12 s y x = x e −γ s y = y e −γ s Similarly 4.15 z φ ...
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