physics

A small block of mass m=1 kg glides down (without friction) a circular track of radius R=2 m, starting from rest at height R. At the bottom of the track it hits a massless relaxed spring with spring constant k= 10 N/m, which starts to be compressed as the block continues to move horizontally. Note that we assume no energy loss during this “collision". There is friction between the block and the horizontal surface, and it is not uniform. As a function of distance, the friction coefficient varies like μ(x)=αx, with α= 1.1 m−1. Assume for simplicity that static and kinetic friction coefficients are the same, and use g=10 m/s2.

What is the maximum distance x1 that the block moves horizontally away from the track at x=0? (in meters)
What time t1 does it take for the block to travel between x=0 (relaxed spring) and x=x1 (block at first stop)? (in seconds)

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  1. fall distance R = 2
    Ke = (1/2) m v^2 = m g R

    v = sqrt (2 g R) = 2 sqrt g =2 sqrt 10

    normal force = m g = 2 g
    friction force = mu m g = -2 g mu
    mu = 1.1 x
    so friction force = -2.2 g x
    F = m a
    m a = -2.2 g x - k x
    d^2x/dt^2 = -2.2 g x = -22 x - 10 x = -32 x
    dx/dt = Vinitial - 16 x^2
    when is v = dx/dt = 0?
    -16 x^2 = Vinitial = 2 sqrt (10)
    that gives you x when v = 0
    integrate again for t

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