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solutions_HW4

# solutions_HW4 - ˙ θ ˙ y 1 = a(cos ϕ ˙ ϕ l 2(cos θ ˙...

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Homework 4 Classical Mechanics (Phys701) Due: October 28, 2011 Problem 1-14 Two points of mass m are joined by a rigid weightless rod of length l , the center of which is constrained to move on a circle of radius a . Express the kinetic energy in generalized coordinates. Write down the Lagrange equations. Solution: The position of the masses are determined by two angles. Angle ϕ ﬁxes the postion of the center of the rod on the circle, and angle θ gives the orientation of the rod. The coordinates of the ﬁrst mass are x 1 = a cos ϕ + l 2 cos θ y 1 = a sin ϕ + l 2 sin θ The coordinates of the second mass are x 1 = a cos ϕ - l 2 cos θ y 1 = a sin ϕ - l 2 sin θ Diﬀerentiating with respect to time one gets ˙ x 1 = - a (sin ϕ ) ˙ ϕ - l 2 (sin θ )
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Unformatted text preview: ˙ θ ˙ y 1 = a (cos ϕ ) ˙ ϕ + l 2 (cos θ ) ˙ θ ˙ x 2 =-a (sin ϕ ) ˙ ϕ + l 2 (sin θ ) ˙ θ ˙ y 2 = a (cos ϕ ) ˙ ϕ-l 2 (cos θ ) ˙ θ The kinetic energy T = m ( ˙ x 2 1 + ˙ y 2 1 + ˙ x 2 2 + ˙ y 2 2 ) / 2 becomes T = m ( a 2 ˙ ϕ 2 + l 2 4 ˙ θ 2 ) while the potential energy is equal to zero. Lagrange equations read ¨ ϕ = 0 , ¨ θ = 0 or ˙ ϕ = ω 1 = const ˙ θ = ω 2 = const Note that this independence of ϕ and θ motions happens only because we considered the particles of identical masses. Otherwise the energy T would have had a term proportional to ˙ ϕ · ˙ θ . 1...
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