Physic Labs

Problem 1

A mass mm starts from rest and slides down a plane inclined by angle α\alpha to the horizontal. The kinetic-friction coefficient is constant, μk\mu_k, and the plane is long enough; assume tan⁡α>μk\tan\alpha>\mu_k so the object accelerates downhill. Take g=9.8 m/s2g=9.8\,\mathrm{m/s^2}. (a) Find the normal force and friction magnitude. (b) Find the acceleration along the plane. (c) Find speed after distance ss along the plane and the time taken to travel that distance.
problems.proof.stepOf
problems.proof.prevStepproblems.proof.nextStep
t=va=2sg(sin⁡α−μkcos⁡α),N=mgcos⁡α,  fk=μkmgcos⁡α,  a=g(sin⁡α−μkcos⁡α)t=\frac va=\sqrt{\frac{2s}{g(\sin\alpha-\mu_k\cos\alpha)}},\quad \boxed{N=mg\cos\alpha,\;f_k=\mu_kmg\cos\alpha,\;a=g(\sin\alpha-\mu_k\cos\alpha)}
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Using s=at2/2s=at^2/2 gives the positive time shown. The force, acceleration, and speed results are consistent with Newton's law and kinematics.

problems.proof.pitfall. Friction opposes sliding motion, not necessarily acceleration; choose the axis and signs before projecting forces.