If the radius of a planet is halved while keeping its mass constant, how does the gravitational acceleration at its surface change?
Practice Questions
1 question
Q1
If the radius of a planet is halved while keeping its mass constant, how does the gravitational acceleration at its surface change?
It becomes four times stronger
It becomes twice stronger
It remains the same
It becomes half as strong
If the radius is halved, the gravitational acceleration becomes four times stronger, as g is inversely proportional to the square of the radius.
Questions & Step-by-step Solutions
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Q
Q: If the radius of a planet is halved while keeping its mass constant, how does the gravitational acceleration at its surface change?
Solution: If the radius is halved, the gravitational acceleration becomes four times stronger, as g is inversely proportional to the square of the radius.
Steps: 5
Step 1: Understand that gravitational acceleration (g) at the surface of a planet is calculated using 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 of the planet.
Step 2: Note that in this scenario, the mass (M) of the planet remains constant, but the radius (R) is halved. This means R becomes R/2.
Step 3: Substitute the new radius into the formula: g = G * M / (R/2)^2.
Step 4: Simplify the equation: (R/2)^2 = R^2 / 4, so g = G * M / (R^2 / 4) = G * M * 4 / R^2.
Step 5: This shows that the new gravitational acceleration is g' = 4 * (G * M / R^2), which means the gravitational acceleration is now four times stronger than before.
