a rope of length l and uniform linear density is hanging over the edge of a table such that the

a) The net force acting on the rope is the weight of the hanging part of the rope, which is give

Question

A rope of length 𝐿 and uniform linear density 𝜌 is hanging over the edge of a table such that the vertically hanging piece has a length 𝑥0 at time 𝑡 = 0. The position of the bottom end of the rope, i.e. point 𝑃, with respect to the edge of the table is given at time 𝑡 by 𝑥(𝑡). Initially (at 𝑡 = 0), the rope is given an upward velocity 𝑣(0) = −𝑣0, where 𝑣0 is a positive constant. Assume that point 𝑃 does not reach the ground at any time in the problem. a) Determine the net force (𝐹𝑛𝑒𝑡) acting on the rope in terms of 𝜌, 𝑥, 𝑔 (ignore friction and the resistance of the rope to bending). b) Find the speed of the rope 𝑣(𝑥) as a function of 𝑥 in terms of 𝐿, 𝑔, 𝑥0. c) Find 𝑥(𝑡) in terms of 𝑥0, 𝑔, 𝐿 (hint: replace 𝐹𝑛𝑒𝑡 in part (a) with 𝑚𝑥̈ ). Use the initial conditions provided to find the unknown constants. d) Find the time when 𝑥(𝑡) = 0. e) Determine the minimum initial speed 𝑣0 that allows the entire chain to climb on top of the table. f) Show that if 𝑣0 is 𝛼 times larger than the speed found in (e), then the time required for the entire chain to climb on to the table is 𝑡 = 1/2 √𝐿/𝑔 ln (1+𝛼/𝛼−1)

          A rope of length 𝐿 and uniform linear density 𝜌 is hanging over the edge of a table such that the vertically
hanging piece has a length 𝑥0 at time 𝑡 = 0. The position of the bottom end of the rope, i.e. point 𝑃, with
respect to the edge of the table is given at time 𝑡 by 𝑥(𝑡). Initially (at 𝑡 = 0), the rope is given an upward velocity 𝑣(0) = −𝑣0, where 𝑣0 is a positive constant. Assume that point 𝑃 does not reach the ground at any time in the problem.
a) Determine the net force (𝐹𝑛𝑒𝑡) acting on the rope in terms of 𝜌, 𝑥, 𝑔 (ignore friction and the resistance of the rope to bending).
b) Find the speed of the rope 𝑣(𝑥) as a function of 𝑥 in terms of 𝐿, 𝑔, 𝑥0.
c) Find 𝑥(𝑡) in terms of 𝑥0, 𝑔, 𝐿 (hint: replace 𝐹𝑛𝑒𝑡 in part (a) with 𝑚𝑥̈ ). Use the initial conditions provided to find the unknown constants.
d) Find the time when 𝑥(𝑡) = 0.
e) Determine the minimum initial speed 𝑣0 that allows the entire chain to climb on top of the table.
f) Show that if 𝑣0 is 𝛼 times larger than the speed found in (e), then the time required for the entire
chain to climb on to the table is 𝑡 = 1/2 √𝐿/𝑔 ln (1+𝛼/𝛼−1)

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