Question

(9.3) Suppose you are buying some spherical glass beads. The store has two types, each with a different diameter and dril holes with different diameters, as shown in the figure. You discover that both beads have the same height, h (a) Guess which bead has more material i. Provide some justification for your intuition but no math is required (b) Check your guess: Use cylindrical shells to compute the volume of a spherical bead of radius R, with a hole of radius r<R drilled through the center. Express your answer in terms of h

second picture is problem, first is information relevant to problem

Answer 1

Answer 2

The spherical beading more radius (R) will have less material as more of the material near its center is removed To maintain same height of both beads: if R > r then 0 > d: to have same height more width should be cut More mass of the sphere is sphere is concentrated near its center so, even though the bead with radius 'R' has higher volume because of higher 'R': higher values also has been removed out of it And the volume removed is more, as it is near the center compared to the extra volume of sphere (R) has over sphere (r) near the surface \r\n So lesser radius bead will have more material b) volume of bead = volume of sphere - volume of out of past = pi h'^2/3 (3r - h') times 2 = pi r^h^2- pi h'^3/3 2h' + h = 2r h = 2 r - h/2 rake r = R