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HW4_prob1_Constrained2dMotion

# HW4_prob1_Constrained2dMotion - Constrained 2d Motion...

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(a) For 2-dimensional motion the kinetic energy is T = m 2 [( · x ) 2 +( · y ) 2 ] , (1) andw iththepos it ive y axis pointing down, the uniform gravitational potential is V = mgy. (2) The mass m is constrained to move on the curve y = ax 2 . (3) This constraint removes one degree of freedom and links · x and · y · y =2 ax · x, (4) enabling T to be simpli f ed to T = m 2 [1 + 4 a 2 x 2 ]( · x ) 2 . (5) Thus, in terms of x , the Lagrangian is L = m 2 [1 + 4 a 2 x 2 ]( · x ) 2 + mgax 2 . (6) To f nd the x -EOM we need the two partial derivatives ∂L · x = m [1 + 4 a 2 x 2 ] · x, (7) and ∂L ∂x =4 ma 2 x ( · x ) 2 +2 mgax. (8) With d dt ( ∂L · x )= m [1 + 4 a 2 x 2 ] ·· x +8 ma 2 x ( · x ) 2 , (9) we f nd for the EOM [1 + 4 a 2 x 2 ] ·· x +4 a 2 x ( · x ) 2 2 gax =0 . (10) To f nd the required expression for ·· s ,we f rst write · s in terms of · x using the de f nition of ds

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HW4_prob1_Constrained2dMotion - Constrained 2d Motion...

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