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If the radius of the Earth were to shrink to half its size while keeping its mas

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Question: If the radius of the Earth were to shrink to half its size while keeping its mass constant, what would happen to the gravitational acceleration at the surface?

Options:

  1. It doubles
  2. It halves
  3. It remains the same
  4. It quadruples

Correct Answer: It quadruples

Solution: 

Gravitational acceleration is inversely proportional to the square of the radius. If the radius is halved, g becomes 4 times greater.

If the radius of the Earth were to shrink to half its size while keeping its mas

Practice Questions

Q1
If the radius of the Earth were to shrink to half its size while keeping its mass constant, what would happen to the gravitational acceleration at the surface?
  1. It doubles
  2. It halves
  3. It remains the same
  4. It quadruples

Questions & Step-by-Step Solutions

If the radius of the Earth were to shrink to half its size while keeping its mass constant, what would happen to the gravitational acceleration at the surface?
  • Step 1: Understand that gravitational acceleration (g) depends on the mass of the object and the distance from its center (radius).
  • Step 2: Recall the formula for gravitational acceleration: g = G * (M / r^2), where G is the gravitational constant, M is the mass, and r is the radius.
  • Step 3: Note that if the radius (r) is halved, we can express the new radius as r/2.
  • Step 4: Substitute the new radius into the formula: g' = G * (M / (r/2)^2).
  • Step 5: Simplify the equation: (r/2)^2 = r^2 / 4, so g' = G * (M / (r^2 / 4)) = G * (4M / r^2).
  • Step 6: This shows that g' = 4 * (G * (M / r^2)), which means the new gravitational acceleration (g') is 4 times the original gravitational acceleration (g).
  • Step 7: Conclude that if the radius of the Earth shrinks to half its size while keeping its mass constant, the gravitational acceleration at the surface becomes 4 times greater.
  • Gravitational Acceleration – Gravitational acceleration at the surface of a planet is determined by the formula g = G * M / r^2, where G is the gravitational constant, M is the mass of the planet, and r is the radius.
  • Inverse Square Law – The gravitational force (and thus acceleration) is inversely proportional to the square of the radius, meaning that if the radius decreases, the gravitational acceleration increases significantly.
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