Suppose gravitational attraction varies inversely as the ⁵⁄² power of separation. For a circular satellite orbit, the slope of log(v₀) against log(r) is −a/b. Find a + b, where a and b are positive integers in lowest terms.
Physics · Mixed Chapter
JEE Mains

Question

Suppose gravitational attraction varies inversely as the ⁵⁄² power of separation. For a circular satellite orbit, the slope of log(v₀) against log(r) is −a/b. Find a + b, where a and b are positive integers in lowest terms.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — For the hypothetical attraction law, F = k/r⁵⁄². Step 2 — Circular-motion balance: mv₀²/r = k/r⁵⁄². Multiplying both sides by r gives mv₀² = k/r³⁄². Step 3 — Thus v₀² ∝ r⁻³⁄² and v₀ ∝ r⁻³⁄⁴. Step 4 — Take logarithms: log(v₀) = constant − (3/4)log(r). The required graph slope is −3/4. Step 5 — Hence a = 3, b = 4 and a + b = 7. Answer: 7.