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class14 - example A chain of uniform mass density length b...

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example: A chain of uniform mass density ρ , length b , and mass M (where ρ = M/b ) hangs from both ends. At time t = 0, the ends are adjacent, but one is released. Find the tension in the chain at the fixed point, after the other has fallen a distance x . Assume free-fall. solution : Assume free-fall: that is, the only forces acting on the system at time t are the tension (vertically upward at the fixed end) and the gravitational force Mg pulling the chain down. The center of mass momentum reacts to these forces such that ˙ P = Mg - T (20) The free side of the chain, with mass ρ ( b - x ) / 2, moves at the speed ˙ x , and the other side is not moving. The total momentum of the system is therefore P = ρ b - x 2 ˙ x (21) so ˙ P = ρ 2 - ˙ x 2 + ¨ x ( b - x ) (22) Now, the kinematic equations for free-fall are x = gt 2 / 2, so ˙ x = gt = 2 gx, ¨ x = g (23) so ˙ P = ρ 2 ( gb - 3 gx ) = Mg - T (24) finally T = Mg 2 3 x b + 1 (25) 1.2.1 Rocket Motion An important application of linear momentum conservation is rocket propulsion. Consider the rocket as a rigid body of mass M
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