Homework 1 Solutions

# N l m2 l cos 0 sin 0 mg sin 0 1 5 6 7 8 9 10 38 a

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Unformatted text preview: d @L @L - Lagrange multiplier. ˙ dt @✓ = @✓ , where @✓ 2¨ 22 ml ✓ = m⌦ l cos ✓ sin ✓ + mgl sin ✓ + , ¨ ✓ = 0, ✓ = ✓0 . N = l = m⌦2 l cos ✓0 sin ✓0 mg sin ✓0 . 1 5 6 7 8 9 10 3.8 A point mass m slides without friciton inside a surface of revolution z = α sin(r/R) whose symmetry axis lied along the direction of a uniform gravitational ﬁeld g. Consider 0 &lt; r/R &lt; 1 π . 2 (a) Construct the lagrangian L and compute the equations of motion for the generalized coordinates r and φ. The lagrangian is 1 α ˙ L = m r 2 + r 2 φ2 + ˙ 2 R 2 r R r2 cos2 ˙ − mgα sin r . R The equation of motion are d ˙ mr2 φ = 0, dt α2 α r ˙ mr 1 + ¨ = mrφ2 + m cos2 R R R 2 r2 sin ˙ r α r r cos − mg cos . R R r R ˙ Note that angular momentu...
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