A uniform solid sphere of mass and radius is released from rest at the top of an…
PHYSICS · Multi-part · CBSE · Physics · System of Particles and Rotational Motion · System of Particles and Rotational Motion
Problem
A uniform solid sphere of mass m and radius R is released from rest at the top of an incline of length d that makes an angle θ with the horizontal. The sphere rolls without slipping down the incline and then continues to roll across a horizontal surface that is long enough to be considered effectively frictionless for rolling motion.
(a) Derive an expression for the magnitude of the sphere’s linear acceleration while it is on the incline in terms of m, g, R, and θ.
(b) Using your result from part (a), obtain an algebraic expression for the time required for the sphere to travel the distance d down the incline.
(c) Derive an expression for the sphere’s linear speed at the bottom of the incline in terms of the vertical height h through which it descends.
(d) Suppose a second rigid object of the same mass and radius but unknown internal mass distribution is released from rest at the same height and takes time t2 to reach the bottom. Explain how measurements of t2 and the given quantities could be used to determine the object’s moment of inertia about its center of mass as a multiple of mR2. Clearly outline the reasoning steps without performing numerical calculations.