Grade 10Mechanics
(a) Fig. shows a planetary object of uniform density ρ and radius R. show that the compressive stress S (defined as force per unit cross-sectional area) near the center is given by
(Hint: Construct a narrow column of cross-sectional area A extending from the center to the surface. The weight of the material in the column is mgav where m is the mass of material in the column and gav is the value of g midway between center and surface.) (b) In our solar system, objects (for example, as-teroids small satellites, comets) with “diameters” less than 600 km can be very irregular in shape (see Fig., which shows Hyperion, a small satellite of Saturn), whereas those with larger diameters are spherical. Only if the rocks have sufficient strength to resist gravity can an object maintain a non-spherical shape. Calculate the maximum compressive stress that can be sustained by the rocks making up asteroids. As-sume a density of 4000 kg/m3. (c) What is the largest possible size of a no spherical self-gravitating satellite made of concrete? Assume that concrete has a maximum compressive stress of 4.0 × 107 N/m² and a density ρ = 3000 kg/m3.
r3
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