Final Answer:
We see that \( F_{\text{Mars on Earth}} \) is \( 4 \times \) the gravitational force \( F \) exerted by Earth on Mars. Therefore, the correct answer is: b) \( 4F \)
Step-by-step explanation:
The gravitational force between two objects is given by Newton's law of gravitation:
\[ F = \dfrac{G \cdot m_1 \cdot m_2}{r^2} \]
where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( r \) is the separation between the centers of the masses.
Given that the mass of Mars (\( m_{\text{Mars}} \)) is one-tenth the mass of Earth (\( m_{\text{Earth}} \)) and the radius of Mars (\( r_{\text{Mars}} \)) is one-half the radius of Earth (\( r_{\text{Earth}} \)), we can express the masses and radii of Mars in terms of Earth's masses and radii:
\[ m_{\text{Mars}} = \dfrac{1}{10} m_{\text{Earth}} \]
\[ r_{\text{Mars}} = \dfrac{1}{2} r_{\text{Earth}} \]
Now, let's substitute these into the gravitational force equation:
\[ F_{\text{Mars on Earth}} = \dfrac{G \cdot m_{\text{Earth}} \cdot m_{\text{Mars}}}{r_{\text{Earth}}^2} \]
Substitute the expressions for \( m_{\text{Mars}} \) and \( r_{\text{Mars}} \):
\[ F_{\text{Mars on Earth}} = \dfrac{G \cdot m_{\text{Earth}} \cdot \left(\dfrac{1}{10} m_{\text{Earth}}\right)}{\left(\dfrac{1}{2} r_{\text{Earth}}\right)^2} \]
Simplify the expression:
\[ F_{\text{Mars on Earth}} = \dfrac{G \cdot m_{\text{Earth}} \cdot \left(\dfrac{1}{10} m_{\text{Earth}}\right)}{\dfrac{1}{4} r_{\text{Earth}}^2} \]
Combine terms:
\[ F_{\text{Mars on Earth}} = 4 \cdot G \cdot m_{\text{Earth}} \cdot \left(\dfrac{1}{10} m_{\text{Earth}}\right) \]
Now, we see that \( F_{\text{Mars on Earth}} \) is \( 4 \times \) the gravitational force \( F \) exerted by Earth on Mars. Therefore, the correct answer is:
b) \( 4F \)