To analyze the weight you would experience on each planet, we need to understand how gravitational force (which we typically refer to as weight) works. The gravitational force that a planet exerts on an object is directly proportional to the mass of the planet and the object, and inversely proportional to the square of the distance between the center of the planet and the object.
The formula for gravitational force (weight) is:
[tex]\[ F = G \frac{m1 \cdot m2}{d^2} \][/tex]
where:
- [tex]\( F \)[/tex] is the gravitational force.
- [tex]\( G \)[/tex] is the gravitational constant.
- [tex]\( m1 \)[/tex] is the mass of the planet.
- [tex]\( m2 \)[/tex] is the mass of the object (e.g., you).
- [tex]\( d \)[/tex] is the distance between the center of the planet and the object.
Let's denote:
- [tex]\( M \)[/tex] as the mass of each planet.
- [tex]\( R \)[/tex] as the radius of planet B.
- [tex]\( r \)[/tex] as the radius of planet A.
According to the problem, planet A is half the size of planet B. This implies that [tex]\( r = \frac{R}{2} \)[/tex].
Now, let's calculate the weight on each planet.
### Weight on Planet A
Using the formula for weight, and substituting [tex]\( r = \frac{R}{2} \)[/tex]:
[tex]\[ F_A = G \frac{M \cdot m}{r^2} = G \frac{M \cdot m}{\left(\frac{R}{2}\right)^2} = G \frac{M \cdot m}{\frac{R^2}{4}} = G \frac{M \cdot m \cdot 4}{R^2} \][/tex]
Therefore:
[tex]\[ F_A = 4 \frac{G M \cdot m}{R^2} \][/tex]
### Weight on Planet B
Using the formula for weight on planet B:
[tex]\[ F_B = G \frac{M \cdot m}{R^2} \][/tex]
### Comparison
From the calculations:
[tex]\[ F_A = 4 \cdot \left(G \frac{M \cdot m}{R^2}\right) = 4 \cdot F_B \][/tex]
This indicates that the weight on planet A ([tex]\(F_A\)[/tex]) is 4 times greater than the weight on planet B ([tex]\(F_B\)[/tex]).
Now, let’s review the provided options:
A. You would weigh less on planet A because the distance between you and the planet's center of gravity would be smaller.
B. You would weigh more on planet A because the distance between you and the planet's center of gravity would be smaller.
C. You would weigh the same on both planets because the masses of the planets are the same.
D. You would weigh the same on both planets because your mass would be the same on both.
Given that [tex]\( F_A \)[/tex] is 4 times [tex]\( F_B \)[/tex], the correct statement is:
B. You would weigh more on planet A because the distance between you and the planet's center of gravity would be smaller.
