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Analytical Mech Homework Solutions 179

# Analytical Mech Homework Solutions 179 - 16 11.20 X...

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16 11.20 Generalized coordinates: X , s a ( ) θ = θ y x ( x,y ) X M m a and sin (1 cos ) x X a y a θ θ = + = cos sin x X a y a θ θ θ = + = ± ± ± ± ± θ ( ) 2 2 1 1 2 2 T MX m x y = + + ± ± ± 2 ( ) ( 2 2 2 1 1 cos sin 2 2 MX m X a a θ θ θ θ ) = + + + ± ± ± ± ( ) 1 cos V mgy mga θ = = For small oscillations, in terms of the generalized coordinates X and s ( ) 2 2 2 1 1 1 2 2 2 T MX m X s V mg a + + ± ± ± s ( ) 2 2 2 1 1 1 2 2 2 L T V MX m X s mgs a = + + ± ± ± Lagrange’s equations of motion yield … ( ) 0 0 g X s s M m X ms a + + = + + = ±± ±± ±± ±± Assuming … i t i t X Ae s Be ω ω = = we obtain the matrix equation … ( ) 2 2 2 2 0 A g a B M m m ω ω ω ω = + Setting the determinant of the above matrix equal to zero yields…
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