Irodov Problem 1.222

Problem Statement

Two point masses m₁ and m₂ are separated by distance d. Find the point on the line joining them where the gravitational field is zero.

Given

m₁, m₂, separation d. Field zero at distance x from m₁.

Concepts & Formulas

Fields from both masses cancel: Gm₁/x² = Gm₂/(d−x)².

Step-by-Step Solution

Step 1: At distance x from m₁: g₁ = Gm₁/x² (toward m₁) and g₂ = Gm₂/(d−x)² (toward m₂).
Step 2: Set g₁ = g₂: m₁/x² = m₂/(d−x)².
Step 3: √(m₁)/x = √(m₂)/(d−x) → x·√m₂ + x·√m₁ = d·√m₂… wait: (d−x)/x = √(m₂/m₁) → d/x = 1 + √(m₂/m₁) → x = d/(1+√(m₂/m₁)) = d√m₁/(√m₁+√m₂).

Worked Calculation

x = d√m₁/(√m₁ + √m₂) from m₁.

Boxed Answer

x = d√m₁/(√m₁ + √m₂)

Physical Interpretation

The null point lies closer to the lighter mass — it takes more distance to weaken the stronger field of the heavier body to the same level.