Question

A sphere of mass M, radius r, and moment of inertial I = Mr2 (where is a dimensionless constant which depends on how the mass is distributed in the sphere) is placed on a track at a height h above the lowest point on the track. The sphere is released, and rolls without slipping. It reaches a horizontal surface which subsequently bends into a vertical loop-the-loop of radius R, as in Fig. 2 below.

a) When it reaches the top of the circular loop, what is its speed v, expressed sym- bolically in terms of the inputs h, R, , and g?

b) What is the magnitude and direction of the force the track exerts on the box at that point, expressed symbolically in terms of m, v, R, and g? Does the sphere’s translational acceleration have a horizontal (tangential) component at that point? A free body diagram and Newtons 2nd law written in variable form must be included for full credit.

c) What is the minimum speed vmin the sphere can have at the top of the loop, if it makes it all the way around? Express your result symbolically in terms of R, etc.

d) Derive the minimum height hmin from which the sphere must be released to make it around the loop? Express hmin symbolically in terms of R and .

e) Obtain hmin for the following three cases: (i) a uniform solid sphere ( = 2/5), (ii) a uniform spherical shell ( = 2/3), and (iii) a frictionless track? Explain the ordering of your results, from smallest to largest hmin.

Answer 1