Analytical Mech Homework Solutions 149

Analytical Mech Homework Solutions 149 - 1 2 1 22 mx ma mgx...

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t θ = ± x T 22 2 11 25 V m x m a m g x =−= + + ± ± LT The equation of constraint is … (, ) 0 fx x a = −= The 2 Lagrange equations with multipliers are … 0 LdL f xd tx x λ ∂∂ += ∂∂∂ ± 0 f dt θθ ± L mx x = ± ± dL mx dt x = ±± ± L mg x = f x = 0 mg mx and from the θ -equation … m g 2 2 0 5 ma a θλ Differentiating the equation of constraint … x a = and substituting into the above … 5 7 x g = and 2 7 mg =− (b) The generalized force that is equivalent to the tension T is … 2 7 x f Qm x λλ == = g 10.16 For the velocity of a differential mass element, dm , of the spring at a distance v G x below the support … x vv x ′ = GG , m dm dx x = () 2 2 00 11 11 22 22 mx xm Tm v v d mm x x d xx x  ′′ =+   ∫∫ 3 m x
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