PHYSICS · Multi-part · CBSE · Physics · Laws of Motion · Laws of Motion
Problem
A small bead of mass m is threaded on a smooth, rigid wire formed into a vertical circle of radius R. The bead is launched from the lowest point with an initial speed v0 and subsequently moves around the circle under the influence of gravity only. Throughout this problem, neglect any energy losses.
(a) Derive an expression for the speed v(θ) of the bead as a function of the angle θ the bead has moved through measured from the lowest point (with θ=0 at the bottom and θ=π at the top). Express your result in terms of v0, g, R, and θ.
(b) Using your expression from part (a), obtain an algebraic formula for the normal force N(θ) exerted by the wire on the bead as a function of θ.
(c) Determine the minimum launch speed vmin required for the bead to maintain contact with the wire at all points. Show all reasoning clearly.
(d) For v0=1.10vmin, compute the ratio of the normal force at the bottom of the circle to the normal force at the top. Interpret the physical meaning of this ratio.