Approximating Earth as a uniform solid sphere of mass 5.98×10²⁴ kg and radius 6.4×10⁶ m, the central pressure from self-gravity is q×8.5×10¹⁰ N m⁻². Find q.
Physics · Mixed Chapter
JEE Mains

Question

Approximating Earth as a uniform solid sphere of mass 5.98×10²⁴ kg and radius 6.4×10⁶ m, the central pressure from self-gravity is q×8.5×10¹⁰ N m⁻². Find q.

G1 Physics · Accepted Answer

Step 1 — Inside a uniform sphere, gravitational field: g(r)=GMr/R³. Density ρ=3M/(4πR³). Step 2 — Hydrostatic pressure gradient: dP/dr=−ρg(r)=−ρGMr/R³. Step 3 — Integrate centre to surface with P(R)=0: P(0)=∫(0→R)ρGMr/R³ dr = ρGM/(2R). Substitute ρ to obtain P(0)=3GM²/(8πR⁴). Step 4 — Numerical value: P(0)≈[3×6.674×10⁻¹¹×(5.98×10²⁴)²]/[8π×(6.4×10⁶)⁴]≈1.70×10¹¹ Pa. Step 5 — Compare with q×8.5×10¹⁰ Pa: q=2. Answer: 2.