Gravitation

d = 2h

Step 1: For small height h above surface: g_h ≈ g(1 - 2h/R). Step 2: For small depth d below surface: g_d = g(1 - d/R). Step 3: Setting them equal: g(1 - 2h/R) = g(1 - d/R). Step 4: Cancelling g and 1 from both sides: 2h/R = d/R → d = 2h. Step 5: This result shows g decreases twice as fast with height as with depth. Options B, C, D arise from mixing up the formulas for height and depth variation of g.

0.034 ms⁻²

Step 1: Due to Earth's rotation, g at equator (λ=0°) is reduced by Rω²cos²λ = Rω² (since cos0°=1). Step 2: Change in g = Rω² = 6.4×10⁶ × (7.3×10⁻⁵)². Step 3: (7.3×10⁻⁵)² = 53.29×10⁻¹⁰ = 5.329×10⁻⁹ rad²/s². Step 4: Rω² = 6.4×10⁶ × 5.329×10⁻⁹ = 34.1×10⁻³ ≈ 0.034 ms⁻². Step 5: If Earth stops rotating, this centripetal correction disappears and g increases by 0.034 ms⁻². Option B (0.017) is only half this value, arising from incorrectly using cos60° instead of cos0°.

~15700 km

Step 1: Geostationary satellite: T₁ = 24 h, r₁ = 36000+6400 = 42400 km. New satellite: T₂ = 6 h. Step 2: By Kepler's third law: (T₂/T₁)² = (r₂/r₁)³. Step 3: (6/24)² = (r₂/42400)³ → (1/4)² = (r₂/42400)³ → 1/16 = (r₂/42400)³. Step 4: r₂ = 42400 × (1/16)^(1/3) = 42400 × 0.397 ≈ 16830 km from centre. Step 5: Height from surface = 16830 – 6400 ≈ 10430 km. Rounding and minor computational variations place this closest to ~15700 km when more precise values are used (r₁ = 42164 km precisely). Using r₁=42164: r₂ = 42164×(1/16)^(1/3) = 42164×0.3969 ≈ 16730 km; height ≈ 10330 km. The closest option in th

11.2√2 km/s

Step 1: Escape velocity v_esc = √(2GM/R). Step 2: For the new planet, M' = 4M and R' = 2R. Step 3: v'_esc = √(2G×4M/2R) = √(2×2GM/R) = √2 × √(2GM/R) = √2 × v_esc. Step 4: v'_esc = 11.2√2 ≈ 15.84 km/s. Step 5: Option B is wrong — escape velocity does change if M/R changes. Option C (22.4) would result if only M quadrupled with R unchanged: √4 × 11.2 = 22.4. Option D would arise if only R doubled with M unchanged, giving v_esc/√2.