1. Falling Chain. Mathematical model for the velocity v of a chain slipping off the edge of a high horizontal platform is a Bernoulli equation xv(dv/dx) + v^2 = 32x
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The given equation is: dv T0 + 0.2 = 3.21 d Let's rewrite it as: dv = (3.21 d - 0.2) / T0 Now, we can recognize this as a Bernoulli equation of the form: dv/dt = (3.21 d - 0.2) / T0 To solve this equation, we can use separation of variables. We can rewrite Show more…
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A uniform chain has a mass $m$ and length $L .$ It is placed on a frictionless table with length $l_{0}$ hanging over the edge. The chain begins to slide down. The speed $v$ with which the chain slides away from the edge is given by (A) $v=\sqrt{\frac{g l_{0}}{L}\left(L+l_{0}\right)}$ (B) $v=\sqrt{\frac{g l_{0}}{L}\left(L-l_{0}\right)}$ (C) $v=\sqrt{\frac{g}{L}\left(L^{2}-l_{0}^{2}\right)}$ (D) $v=\sqrt{2 g\left(L-l_{0}\right)}$
Assuming drag is proportional to the square of velocity, we can model the velocity of a falling object like a parachutist with the following differential equation: dv/dt = g - (cd/m)v^2 where v is velocity in m/s, t is time in seconds, g is the acceleration due to gravity, cd is a second order drag coefficient in kg/m, and m is mass in kg. Suppose we have a 90-kg object with a drag coefficient 0.225 kg/m. If the initial height is 1 km, use the Runge Kutta method to determine when the object hits the ground.
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