If the Earth were to suddenly shrink to half its radius while keeping its mass constant, what would happen to the gravitational acceleration at its surface?
Practice Questions
1 question
Q1
If the Earth were to suddenly shrink to half its radius while keeping its mass constant, what would happen to the gravitational acceleration at its surface?
It would double
It would remain the same
It would halve
It would quadruple
If the radius is halved, the gravitational acceleration would increase by a factor of 4 (g ∝ 1/R²).
Questions & Step-by-step Solutions
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Q
Q: If the Earth were to suddenly shrink to half its radius while keeping its mass constant, what would happen to the gravitational acceleration at its surface?
Solution: If the radius is halved, the gravitational acceleration would increase by a factor of 4 (g ∝ 1/R²).
Steps: 7
Step 1: Understand that gravitational acceleration (g) at the surface of a planet is determined by the formula g = G * M / R², where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.
Step 2: Note that in this scenario, the mass (M) of the Earth remains constant, but the radius (R) is halved.
Step 3: If the radius 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: g' = G * M / (R²/4) = 4 * (G * M / R²).
Step 6: Recognize that (G * M / R²) is the original gravitational acceleration (g), so g' = 4g.
Step 7: Conclude that if the radius is halved, the gravitational acceleration at the surface increases by a factor of 4.
