Question 1 (40 pt) A rope of length 𝑎 slides under constant gravity 𝑔 from a tilted frictionless tableto

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Question 1 (40 pt)
A rope of length 𝑎 slides under constant
gravity 𝑔 from a tilted frictionless tabletop which makes
an angle 𝜃 to horizontal as shown in the figure below.
The rope has a total mass 𝑚. The tabletop is completely fixed
throughout the motion.
𝜃
horizontal line
𝑥
(a) Write down the Lagrangian of the rope in term
of 𝑥. [5pt]
(b) Find the generalized momentum. [5pt]
(c) Write down the Hamiltonian of the rope. [5pt]
(d) Find the Hamilton equations of motion of the rope.
[5pt]
(e) If 𝜃 = 𝜋 and the rope is initially
released from rest with the lower end at the edge of 2
the tabletop. Find the time, 𝑇 , at which the upper
end of the rope reaches the edge 1
of the table. [5pt]
(f) For 𝜃 = 0, consider the rope is initially (𝑡 = 0)
moving towards the edge of the table
with speed 𝑣(0) = 𝑣0 and the right end of the rope is
at the edge of the table,
𝑥(0) = 0. Solve the Hamilton equation of motion to
find 𝑥(𝑡). [10pt]
(g) Continue on (f). Take 𝑎 = 1 and 𝑣 = 1.
Find the time, 𝑇 , at which the left end of
the rope reaches the edge of the table. [5pt]