Question
A small bead of mass m is constrained toslide without friction insde a circular vertical hoop ofradius r which rotates about a vertical axis(figure)at a frequency of f. (a) Determine the angle where the bead will be inequilibrium-that is, where it will have no tendency to move up ordown along the hoop. (b) If f = 2.00 rev/s and r = 22.0 cm, whatis ? (c) Can the bead ride as high asthe center of the circle(=90o)? Explain. A small bead of mass m is constrained toslide without friction insde a circular vertical hoop ofradius r which rotates about a vertical axis(figure)at a frequency of f. (a) Determine the angle theta where the bead will be inequilibrium-that is, where it will have no tendency to move up ordown along the hoop. (b) If f = 2.00 rev/s and r = 22.0 cm, whatis theta? (c) Can the bead ride as high asthe center of the circle(theta=90 degree)? Explain. Figure.
Explanation / Answer
(a)The forces acting on the bead in the x and y directionsare: Fx = w2 = w * sin or m * ax = mg * sin or ax = g * sin ----------(1) and Fy = n + (-w1) = 0 or n = w1 = w * cos -----------(2) From equation (1) we get ax = g * sin or (v2/r) = g * sin where v = (2r/T) or ((2r/T)2/r) = g * sin or ((2r * f)2/r) = g * sin or 42 * r * f2 = g *sin or sin = (42 * r *f2/g) or = sin-1(42 * r *f2/g) where r is the radius of the circle,f is the frequency and gis the acceleration due to gravity and has a value 9.8m/s2. (b)The centripetal force acting on the bead is equal to the xcomponent of weight of the bead therefore we get (m * v2/r) = mg * sin or g * sin = (v2/r) or sin = (v2/g * r) or = sin-1(v2/g * r) v = r * w or = sin-1((r * w)2/g * r) =sin-1(r2 * w2/g * r) =sin-1(r * w2/g) r = 22.0 cm = 22.0 * 10-2 m and w = 2.00 rev/s =2.00 * 2 rad/s = 4 rad/s = 4 * 3.14 rad/s = 12.56rad/s (c)When = 90o then we have sin(90o) = (r * w2/g) or 1 = (r * w2/g) or g = r * w2 But the above equation is not true.This is because the lefthand side values and the right hand side values are not equal.Thebead cannot ride as high asthe center of the circle,this is becausethe acceleration of the bead is very small when compared tothe acceleration due to gravity.