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A spherical body of radius R is made of a fluid of uniform density held in equilibrium by its own gravity. Let P(r) be the pressure at a radial distance r (with r < R) measured from the centre. Which of the following statement(s) is/are correct?

1. P(r = 0) = 0
2. P(r = 3R/4) / P(r = 2R/3) = 63/80
3. P(r = 3R/5) / P(r = 2R/5) = 16/21
4. P(r = R/2) / P(r = R/3) = 20/27

Correct answer: P(r = 3R/5) / P(r = 2R/5) = 16/21

Solution

Hydrostatic equilibrium of a uniform self-gravitating sphere gives P(r) = (2/3) pi G rho² (R² - r²), so P(r) is proportional to (R² - r²) = (1 - (r/R)²). Thus P(r1)/P(r2) = (1 - (r1/R)²)/(1 - (r2/R)²). Check option C: r1 = 3R/5 -> 1 - 9/25 = 16/25; r2 = 2R/5 -> 1 - 4/25 = 21/25; ratio = 16/21. Correct. (Option A is wrong since P(0) is maximum, not zero. The other ratios do not match: B gives (1-9/16)/(1-4/9) = (7/16)/(5/9) = 63/80 — actually B also evaluates to 63/80, so B is also correct. D: (1-1/4)/(1-1/9) = (3/4)/(8/9) = 27/32, not 20/27, so D wrong.) Both B and C are correct; the primary intended answer is the ratio 16/21 (option C).

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