- #1

Elvis 123456789

- 158

- 6

## Homework Statement

A rope of length L is falling off an incline. Part of the rope is still on top of the horizontal surface. There is no friction between the incline and the rope. The incline is at an angle theta above the horizontal.

a) What is the magnitude of the acceleration of the rope, when the part of the rope on the incline is x (i.e., the part left on the horizontal surface has a length of L-x)

b) What is the speed of the rope at this moment?

c) What is the value of x, when the acceleration reaches maximum?

## Homework Equations

∂L/∂x - d/dt(∂L/∂v) = 0 where v = x-dot

a = dv/dt = (dv/dx)*(dx/dt)

## The Attempt at a Solution

So I worked out this problem but I am unsure of my result, specifically my result for the potential energy in the lagrangian.

L = ½mv

^{2}- U(x) = ½λLv

^{2}+ (λx)(g)(½x sin(θ))

where λ=m/L is the linear mass density

the Euler-Lagrange equation then gives

1.) a = (x/L)g sin(θ)

assuming my acceleration for # 1 is correct, I used the identity a = v*(dv/dx) to find the velocity

a*dx = v*dv → ∫(x/L)g sin(θ)dx = ∫vdv

2.) v = √[v

^{2}

_{0}+ 1/L (x

^{2}- x

^{2}

_{0})g sin(θ) ]

then # 3 would be x = L

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