A bead of mass m is located on a parabolic wire, with its axis vertical and vertex directed downward, as shown in the figure, and whose equation is x 2 = a y . If the coefficient of friction is μ , then the highest distance above the x -axis at which the particle will be in equilibrium is

Step-by-step solution

Given, x 2 = a y Assuming that θ is the angle of repose, hence, for the particle to not slide, the limiting condition is given by, m g sinθ ≤ f     f = μ N ⇒ m g sinθ ≤ μ m g cos   θ ⇒ tan θ ≤ μ From graph, ⇒ tan θ = d y d x ⇒ tan θ = 1 a d d x x 2 = 2 x a Since, x 2 = a y ⇒ tan θ = 2 y a ∴   2 y a ≤ μ ⇒ y ≤ a μ 2 4 .