This figure shows the force on a 0.1-kilogram mass at various distances from Earth. Notice that the force decreases as the square of the distance. Now, consider satellites A and B, both circling Earth. If A has ten times the mass of B but B is 1/10 of the distance to the center of Earth as is A, what is the gravitational force on A relative to that of B due to Earth?
A) 1/100 times as much
B) 1/10 the amount
C) the two are equal
D) 10 times as much
E) 100 times as much
If A has 10 times the mass, Fa/Fb = 10
If A is 10 times as far, Fa/Fb = 1/100
Now combine the two.
According to the given figure, the gravitational force on a mass decreases as the square of the distance from Earth increases.
Let's calculate the ratio of the gravitational force on A relative to B due to Earth.
Given:
Mass of A = 10 times the mass of B (m_A = 10m_B)
Distance of B from the center of Earth = 1/10 of the distance of A from the center of Earth (d_B = 1/10 * d_A)
The force of gravity can be calculated using the formula:
F = G * (m1 * m2) / r^2
Where:
F is the force of gravity,
G is the gravitational constant,
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
Let's calculate the force on A (F_A):
F_A = G * (m_A * m_Earth) / (d_A)^2
And the force on B (F_B):
F_B = G * (m_B * m_Earth) / (d_B)^2
Dividing F_A by F_B to find the ratio:
F_A / F_B = [G * (m_A * m_Earth) / (d_A)^2] / [G * (m_B * m_Earth) / (d_B)^2]
Canceling out the G and m_Earth terms:
F_A / F_B = (m_A / d_A^2) / (m_B / d_B^2)
Substituting the given values:
F_A / F_B = [(10m_B) / (d_A)^2] / [(m_B) / (d_B)^2]
Simplifying:
F_A / F_B = (10m_B / d_A^2) * (d_B^2 / m_B)
F_A / F_B = 10 * (d_B / d_A)^2
Since d_B = (1/10) * d_A:
F_A / F_B = 10 * [(1/10) * d_A / d_A]^2
F_A / F_B = 10 * (1/10)^2
F_A / F_B = 10 * 1/100
F_A / F_B = 1/10
Therefore, the gravitational force on A relative to that of B due to Earth is 1/10th the amount.
So, the answer is B) 1/10 the amount.
To find the gravitational force on A relative to that of B due to Earth, we need to compare the two forces.
According to the information given, the force between two objects due to gravity decreases as the square of the distance between them increases. So, if B is 1/10 the distance to the center of Earth as A, the force on B will be (1/10)^2 = 1/100 times the force on A.
Additionally, A has ten times the mass of B. From Newton's law of universal gravitation, we know that the force is directly proportional to the masses of the objects. Therefore, A will experience 10 times the force compared to B.
Combining these two factors, the gravitational force on A relative to that of B due to Earth is (10 times) x (1/100 times) = 1/10 the amount.
So, the correct answer is B) 1/10 the amount.