A spherical shell has an outside radius of 2.45 cm and an inside radius of a. Th

Question

A spherical shell has an outside radius of 2.45 cm and an inside radius of a. The shell wall has uniform thickness and is made of a material with density 4.74 g/cm3. The space inside the shell is filled with a liquid having a density of 1.19 g/cm3. (a) Find the mass m of the sphere, including its contents, as a function of a. (b) For what value of a does m have its maximum possible value? (c) What is this maximum mass? (d) Explain whether the value from part (c) agrees with the result of a direct calculation of the mass of a solid sphere of uniform density made of the same material as the shell.

Explanation / Answer

47. (a) m 346 g (14.5 g/cm3)a3 (b) a 0 (c) 346 g (d) yes (e) a 2.60 cm (f) 90.6 g (g) yes (h) 218 g (i) No; 218 g is not equal to 314 g. (j) Parts (b), (c), and (d) describe a uniform solid sphere with r 4.70 g/cm3 as a approaches zero. Parts (e), (f), and (g) describe a uni- form liquid drop with r 1.23 g/cm3 as a approaches 2.60 cm. The function m(a) is not a linear function, so a halfway between 0 and 2.60 cm does not give a value for m halfway between the minimum and maximum values. The graph of m versus a starts at a 0 with a horizontal tan- gent. Then it curves down more and more steeply as a increases. The liquid drop of radius 1.30 cm has only one eighth the volume of the whole sphere, so its presence brings down the mass by only a small amount, from 346 g to 314 g. (k) The answer would not change as long as the wall of the shell is unbroken.

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