Question

consider a modified version of the previous table with a new column that calculates the product of the masses of objects 1 and 2:

$	f_a	$ (n)	force factor	$m_a \cdot m_b (\text{kg}^2)$
0.0834	2	$2.0 \cdot 10^{12} \text{kg}$
0.1667	4	$4.0 \cdot 10^{12} \text{kg}$
0.2501	6	$6.0 \cdot 10^{12} \text{kg}$

which statement describes the relationship between the variables in the table? choose two correct answers.

• the magnitude of the gravitational force is proportional to the product of the masses.
• the force factors are proportional to the products of the masses.
• there is no relationship between the magnitude of the gravitational force and the product of the masses.
• the magnitude of the gravitational force is inversely proportional to the product of the masses.
• the force factors are inversely proportional to the products of the masses.

Explanation:

1. Analyze the relationship between \(|F_A|\) (magnitude of gravitational force) and \(m_A \cdot m_B\) (product of masses):

• When \(m_A \cdot m_B = 1.0 \cdot 10^{12}\ \text{kg}^2\), \(|F_A| = 0.0417\ \text{N}\); when \(m_A \cdot m_B = 2.0 \cdot 10^{12}\ \text{kg}^2\), \(|F_A| = 0.0834\ \text{N}\) (doubled); when \(m_A \cdot m_B = 4.0 \cdot 10^{12}\ \text{kg}^2\), \(|F_A| = 0.1667\ \text{N}\) (quadrupled); when \(m_A \cdot m_B = 6.0 \cdot 10^{12}\ \text{kg}^2\), \(|F_A| = 0.2501\ \text{N}\) (increased by a factor of 6). So \(|F_A|\) is proportional to \(m_A \cdot m_B\).

1. Analyze the relationship between Force Factor and \(m_A \cdot m_B\):

• When \(m_A \cdot m_B = 1.0 \cdot 10^{12}\ \text{kg}^2\), Force Factor = 1; when \(m_A \cdot m_B = 2.0 \cdot 10^{12}\ \text{kg}^2\), Force Factor = 2 (doubled); when \(m_A \cdot m_B = 4.0 \cdot 10^{12}\ \text{kg}^2\), Force Factor = 4 (quadrupled); when \(m_A \cdot m_B = 6.0 \cdot 10^{12}\ \text{kg}^2\), Force Factor = 6 (increased by a factor of 6). So Force Factor is proportional to \(m_A \cdot m_B\).
• The statements "There is no relationship..." and "The magnitude of the gravitational force is inversely proportional..." are incorrect as the data shows a direct proportionality. "The force factors are inversely proportional..." is also incorrect.

Answer:

• The magnitude of the gravitational force is proportional to the product of the masses.
• The force factors are proportional to the products of the masses.