If Earth's radius were reduced by 1% while its mass remained unchanged, what would be the effect on the gravitational acceleratio…

Question

If Earth's radius were reduced by 1% while its mass remained unchanged, what would be the effect on the gravitational acceleration at its surface?

1. A. Increase by approximately 1%
2. B. Decrease by approximately 2%
3. C. Increase by approximately 2% (Correct answer)
4. D. Decrease by approximately 1%

Correct Answer

Option C — Increase by approximately 2%

Detailed Solution & Explanation

The correct answer is Increase by approximately 2%.

Key Points

• Surface gravity is g = GM/R² — inversely proportional to the square of the radius, with mass unchanged.
• Reducing R by 1% multiplies R² by (0.99)² ≈ 0.9801, so g is multiplied by 1/0.9801 ≈ 1.0203.
• That is an increase of about 2%, not 1 — because the dependence is on R², a 1% change in R gives roughly a 2% change in g.

Additional Information

• The general shortcut: for y ∝ xⁿ, a small percentage change in x produces roughly n times that percentage change in y — with a sign flip when n is negative.
• The same reasoning explains why g is slightly greater at the poles: Earth's polar radius is about 21 km smaller than its equatorial radius.
• g also falls with altitude as g' = g(1 − 2h/R) near the surface, and with depth as g' = g(1 − d/R).

प्रश्न (हिन्दी में)

यदि पृथ्वी की त्रिज्या 1% कम कर दी जाए जबकि द्रव्यमान अपरिवर्तित रहे, तो सतह पर गुरुत्वाकर्षण त्वरण पर क्या प्रभाव पड़ेगा?

1. A. लगभग 1% की वृद्धि
2. B. लगभग 2% की कमी
3. C. लगभग 2% की वृद्धि
4. D. लगभग 1% की कमी