Introduction to Classical Mechanics · Problem 3.36

Using F = ma — Morin

Physics — Mechanics · friction_inclined_plane

Problem

Comparing the times *** A block of mass mm is projected up along the surface of a plane inclined at an angle θ\theta. The initial speed is v0v_0, and the coefficients of static and kinetic friction are both equal to μ\mu. The block reaches a highest point and then slides back down to the starting point. (a) Show that for the block to in fact slide back down instead of remaining at rest at the highest point, tan⁡θ\tan \theta must be greater than μ\mu. (b) Assuming that tan⁡θ>μ\tan \theta > \mu, is the total up and down time longer or shorter than the total time it would take if the plane were frictionless? Or does the answer depend on what θ\theta and μ\mu are? (c) Assuming that tan⁡θ>μ\tan \theta > \mu, show that for a given θ\theta, the value of μ\mu that yields the minimum total time is given by μ≈(0.397)tan⁡θ\mu \approx (0.397) \tan \theta. (You will need to solve something numerically.) This minimum time turns out to be about 90% of the time it would take if the plane were frictionless.

Related theory

Browse the problem library