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(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/m 3 . (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/m 3 .

(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.

Grade:10

1 Answers

Jitender Pal
askIITians Faculty 365 Points
6 years ago
Consider a small horizontal slice of the column of thickness dr.
The corresponding figure is shown below:

235-257_im1.JPG
The mass contained in the sphere of radius r is M (r) is
M (r) = \frac{4}{3} \pir3
Here, density of the rock is\rho .
The weight of the material above the slice exerts a force F (r) on the top of the slice. There is a force of gravity of the slice which is given by


235-1594_im2.JPG

235-1918_im3.JPG
(b)
It is given that the diameter of the asteroids is 600 km.
The radius of the asteroids is equal to half the diameter. The radius of the asteroid is
235-1577_im4.JPG
235-1398_im5.JPG

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