Remove the assumption of smooth surfaces as stated in Prob. 3 /88 and assume a coefficient of kinetic friction μ_k between the particle and rotating tube. Determine the radial position r of the particle as a function of time t if it is released with no relative velocity at r = r₀ when t = 0. Assume that static friction is overcome.

Data from Problem 88

The particle P is released at time t = 0 from the position r = r₀ inside the smooth tube with no velocity relative to the tube, which is driven at the constant angular velocity ω₀ about a vertical axis. Determine the radial velocity v_r, the radial position r, and the transverse velocity v_θ as functions of time t. Explain why the radial velocity increases with time in the absence of radial forces. Plot the absolute path of the particle during the time it is inside the tube for r₀ = 0.1 m, l = 1 m, and ω₀ = 1 rad /s.

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