A marble of mass m is going around the funnel slowly making its way down to the

Question

A marble of mass m is going around the funnel slowly making its way down to the bottom hole (which is negligibly small). As this marble goes around it experiences kinetic friction mu_k only in the up/down direction (NOT the tangential direction). You need to find the time tau it will take for the marble to make its way down, and the full magnitude of the final velocity on exit (NOT only the vertical component, but the horizontal one as well). Your final expressions should only have g, R, omega, theta, and mu_k as variables.

For the solution, assume that angle theta is large, so that the radial acceleration is approximately v^2/R.

Kinetic friction down/up along the side. No tangential friction! Find vf = f(R, Omega, theta, muK) T = f(R, Omega, theta, muK)

Explanation / Answer

force down the incline is given by

mg sin theta - f_r = ma

where f_r is frictional force

f_r = m u_k

a = g sin theta - u_k

0.5 a t^2 = L

where is the length of the funnels

t = [2L/(g sin theta - u_k)]^(0.5)

final velocity v = at

==> v = [2L (g sin theta - u_k)]^(0.5)

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