Mathematic Methods HW Solutions 47

# Mathematic Methods HW Solutions 47 - Chapter 9 47 5.5 L = 1...

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Unformatted text preview: Chapter 9 47 5.5 L = 1 2 m ˙ x 2- 1 2 kx 2 m ¨ x + kx = 0 5.6 L = 1 2 m ( r 2 ˙ θ 2 + r 2 sin 2 θ ˙ φ 2 )- mgr cos θ braceleftBigg a ¨ θ- a sin θ cos θ ˙ φ 2- g sin θ = 0 ( d/dt )(sin 2 θ ˙ φ ) = 0 5.8 L = 1 2 m (2˙ r 2 + r 2 ˙ θ 2 )- mgr 2¨ r- r ˙ θ 2 + g = 0 ( d/dt )( r 2 ˙ θ ) = 0 5.9 L = 1 2 m (2˙ r 2 + r 2 ˙ θ 2 )- mgr 2¨ r- r ˙ θ 2 + g = 0 r 2 ˙ θ = const. 5.10 L = 1 2 m 1 (4˙ r 2 + r 2 ˙ θ 2 ) + 2 m 2 ˙ r 2- m 1 gr √ 3 + m 2 g ( l- 2 r ) 4( m 1 + m 2 )¨ r- m 1 r ˙ θ 2 + m 1 g √ 3 + 2 m 2 g = 0 r 2 ˙ θ = const. 5.11 L = 1 2 ( m + Ia- 2 ) ˙ z 2- mgz (If z is taken as positive down, ( ma 2 + I )¨ z + mga 2 = 0 change the signs of z and ¨ z .) 5.12 L = 1 2 m (˙ r 2 + r 2 ˙ θ 2 )- bracketleftbig 1 2 k ( r- r ) 2- mgr cos θ bracketrightbig ¨ r- r ˙ θ 2 + k m ( r- r )- g cos θ = 0, d dt ( r 2 ˙ θ ) + gr sin θ = 0 5.13 L = 1 2 m [˙ r 2 (1 + 4 r 2 ) + r 2 ˙ θ 2 ]- mgr 2 ¨ r (1 + 4 r 2 ) + 4 r ˙ r 2- r ˙ θ 2 + 2 gr = 0, r...
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