homework4sol.20100217.4b7cc95861c116.31632260

# homework4sol.20100217.4b7cc95861c116.31632260 - TAM 412...

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TAM 412, Homework 4, Selected answers Harry Dankowicz Mechanical Science and Engineering University of Illinois at Urbana-Champaign [email protected] 2. Here, T = 1 2 ˙ q 2 and U = 1 2 q 2 which implies that F q = U q = q and d dt μ T ˙ q T q = F q ¨ q = q or, alternatively, d dt μ ( T U ) ˙ q ( T U ) q = 0 ¨ q + q = 0 On the other hand, ˜ q = q 2 ˙ ˜ q = 2 q ˙ q , such that T = ˙ ˜ q 2 q and U = 1 2 ˜ q which implies that F ˜ q = U ˜ q = 1 2 and d dt μ T ˙ ˜ q T ˜ q = F ˜ q ¨ ˜ q ˜ q ˙ ˜ q 2 q 2 + ˙ ˜ q 2 q 2 = 1 2 4. See Lagrange’s equations under Figure 3.2 on page 79. Here F = F x e x + F y e y + F z e z = x e x + y e y + z e z ( x 2 + y 2 + z 2 ) 3 / 2 such that F q i = U q i , where U = 1 p x 2 + y 2 + z 2 = 1 r and thus ˜ Q 1 = F r = U r = 1 r 2 ˜ Q 2 = F θ = U ∂θ = 0 ˜ Q 3 = F ϕ = U ∂ϕ = 0 It remains to show that θ ( t ) = π / 2, ˙ θ ( t ) = ¨ θ ( t ) = 0 satis fi es the equations on page 79 for some r ( t ) and ϕ ( t ). 1

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5. Here T = 2 ³ ˙ θ 2 + ˙ ϕ 2 ´ and U = 2 cosh ϕ and d dt μ ( T U ) ˙ θ ( T U ) ∂θ = 0 d dt μ ( T U )
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