Question

if the moon grows to twice its size but stays in the same place, what happens to gravity between the earth and moon? it doubles. it halves. it stays the same. it quadruples.

Explanation:

To determine the change in gravity between Earth and the Moon, we use Newton's law of universal gravitation: \( F = G\frac{Mm}{r^2} \), where \( F \) is the gravitational force, \( G \) is the gravitational constant, \( M \) and \( m \) are the masses of Earth and the Moon, and \( r \) is the distance between their centers. If the Moon's size (radius) doubles, its volume (and thus mass, assuming uniform density) becomes \( 2^3 = 8 \) times larger? Wait, no—wait, the problem says "grows to twice its size"—if we assume "size" refers to radius, then mass \( m \) is proportional to volume, so \( m' = 8m \)? But that might be overcomplicating. Wait, maybe the problem intends "twice its mass" when it says "twice its size"? Wait, no, the original problem: "If the moon grows to twice its size but stays in the same place"—maybe "size" here is mass? Wait, no, size (radius) doubling would make mass \( 8 \) times, but the options are double, halve, same, quadruple. Wait, maybe the problem has a simplification: maybe "grows to twice its size" means mass doubles (maybe assuming density is constant but the problem simplifies size to mass? Or maybe a mistake, but let's check the options. Wait, the law of gravitation: \( F \propto Mm / r^2 \). If the Moon's mass \( m \) doubles (and Earth's mass \( M \) and distance \( r \) stay the same), then \( F \) doubles? But wait, no—wait, "grows to twice its size"—if size is radius, then volume is \( 8 \) times, mass \( 8 \) times, so force would be \( 8 \) times, but that's not an option. So maybe the problem means "twice its mass" (so size in mass, not radius). Then, if \( m \) becomes \( 2m \), then \( F' = G\frac{M(2m)}{r^2} = 2G\frac{Mm}{r^2} = 2F \). But wait, the options include "It quadruples"—wait, maybe I misread. Wait, the options: red: It doubles; blue: It halves; orange: It stays the same; green: It quadruples. Wait, maybe the problem is about the Moon's radius doubling, but the formula for gravitational force between two objects depends on their masses, not their sizes (radii), unless size is related to mass. Wait, maybe the problem has a simplification where "twice its size" means twice its mass. Then, if mass of Moon doubles, force doubles. But wait, another approach: maybe the problem is wrong, but let's check the options. Wait, the correct answer—wait, no, maybe I made a mistake. Wait, Newton's law: \( F = G\frac{Mm}{r^2} \). If the Moon's mass \( m \) becomes \( 2m \) (because it grows to twice its size, assuming mass is proportional to size, which is a simplification), then \( F \) becomes \( 2F \), so it doubles. But wait, the green option is "It quadruples"—maybe the problem means the Moon's radius doubles, so mass is \( 8 \) times, but that's not an option. Alternatively, maybe the problem is about the distance? No, the problem says "stays in the same place", so \( r \) is constant. Wait, maybe the problem is using "size" as mass, so doubling mass, so force doubles. So the correct option is the red one: It doubles. Wait, but let's confirm. Wait, the law of gravitation: force is proportional to the product of masses. So if Moon's mass doubles, force doubles. So the answer is "It doubles."

Answer:

A. It doubles.