A bathroom towel is placed on a horizontal rack as shown from the side. The horizontal edges of the towel are parallel with the rack and there's no friction between the rack and the towel, so the towel starts sliding under the influence of gravity. Assume that the radius of the rack is negligible compared to the length of the towel, the length of the towel is constant, and that the towel is ideally bendeable (that is no energy needed to bend it). (a) Using the length of the hanging portion of the towel as a general- ized coordinate, construct the Lagrangian of the system. (b) Derive the equation of motion. (c) Determine the position of the lower edge of the towel as the function of time. (It is enough to give a general solution, no specific boundary conditions are needed.) (b) The general solution is √2 √gt √2 √gt x(t) = C₁ e √ī + C₂ e + ~2 The boundary conditions determine C1 and C2 x(0) == ½-½ + C₁ + C₂ = 2/1 1/125 C1 C2 3
A bathroom towel is placed on a horizontal rack as shown from the side. The horizontal edges of the towel are parallel with the rack and there's no friction between the rack and the towel, so the towel starts sliding under the influence of gravity. Assume that the radius of the rack is negligible compared to the length of the towel, the length of the towel is constant, and that the towel is ideally bendeable (that is no energy needed to bend it). (a) Using the length of the hanging portion of the towel as a general- ized coordinate, construct the Lagrangian of the system. (b) Derive the equation of motion. (c) Determine the position of the lower edge of the towel as the function of time. (It is enough to give a general solution, no specific boundary conditions are needed.) (b) The general solution is √2 √gt √2 √gt x(t) = C₁ e √ī + C₂ e + ~2 The boundary conditions determine C1 and C2 x(0) == ½-½ + C₁ + C₂ = 2/1 1/125 C1 C2 3
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Please look at the picture with red circled text first. Where did that highlighted term come from?
Context is in second picture.
Transcribed Image Text:A bathroom towel is placed on a horizontal rack as shown from the
side. The horizontal edges of the towel are parallel with the rack and
there's no friction between the rack and the towel, so the towel starts
sliding under the influence of gravity. Assume that the radius of the
rack is negligible compared to the length of the towel, the length of the
towel is constant, and that the towel is ideally bendeable (that is no
energy needed to bend it).
(a) Using the length of the hanging portion of the towel as a general-
ized coordinate, construct the Lagrangian of the system.
(b) Derive the equation of motion.
(c) Determine the position of the lower edge of the towel as the function of time. (It is enough to give
a general solution, no specific boundary conditions are needed.)
(b) The general solution is
√2 √gt
√2 √gt
x(t) = C₁ e
√ī
+ C₂ e
+
~2
Transcribed Image Text:The boundary conditions determine C1
and
C2
x(0) == ½-½ + C₁ + C₂ = 2/1
1/125
C1 C2 3
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