If the Earth were to suddenly shrink to half its radius while maintaining its mass, what would happen to the gravitational force at its surface?
Practice Questions
1 question
Q1
If the Earth were to suddenly shrink to half its radius while maintaining its mass, what would happen to the gravitational force at its surface?
It would double
It would remain the same
It would become half
It would become four times stronger
The gravitational force at the surface is inversely proportional to the square of the radius. If the radius is halved, the force becomes four times stronger.
Questions & Step-by-step Solutions
1 item
Q
Q: If the Earth were to suddenly shrink to half its radius while maintaining its mass, what would happen to the gravitational force at its surface?
Solution: The gravitational force at the surface is inversely proportional to the square of the radius. If the radius is halved, the force becomes four times stronger.
Steps: 7
Step 1: Understand that gravitational force depends on mass and distance from the center of the object.
Step 2: Recall the formula for gravitational force: F = G * (m1 * m2) / r^2, where F is the force, G is the gravitational constant, m1 and m2 are the masses, and r is the distance (radius).
Step 3: Note that if the Earth shrinks to half its radius, the new radius (r) becomes r/2.
Step 4: Substitute the new radius into the formula: F' = G * (m1 * m2) / (r/2)^2.
Step 5: Simplify the equation: (r/2)^2 = r^2 / 4, so F' = G * (m1 * m2) / (r^2 / 4) = 4 * (G * (m1 * m2) / r^2).
Step 6: This shows that the new gravitational force (F') is four times the original force (F).
Step 7: Conclude that if the Earth shrinks to half its radius, the gravitational force at its surface becomes four times stronger.
