Let's solve this problem step by step using the conceptual understanding of gravitational force.
The gravitational force between two objects is given by Newton's law of universal gravitation, which states that the force [tex]\( F \)[/tex] is proportional to the product of the masses of the two objects [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex], and inversely proportional to the square of the distance [tex]\( d \)[/tex] between them. Mathematically, it is expressed as:
[tex]\[ F = G \frac{m_1 m_2}{d^2} \][/tex]
where [tex]\( G \)[/tex] is the gravitational constant.
Given in the table:
- Initial mass of Object 1 ([tex]\( m_1 \)[/tex]): 1 kg
- Mass of Object 2 ([tex]\( m_2 \)[/tex]): 1 kg
- Distance between the objects ([tex]\( d \)[/tex]): 1 m
- Gravitational force ([tex]\( F \)[/tex]): 4 N
Now, the table asks us to find the new gravitational force when the mass of Object 1 is doubled to 2 kg (i.e., [tex]\( m_1' = 2 \)[/tex] kg) while the mass of Object 2 and the distance between the objects remain the same.
To determine the effect of doubling the mass of Object 1, we observe that the force [tex]\( F \)[/tex] is directly proportional to the mass [tex]\( m_1 \)[/tex]. Therefore, if we double [tex]\( m_1 \)[/tex], the new force [tex]\( F' \)[/tex] can be computed as follows:
[tex]\[ F' = G \frac{(2 \cdot m_1) m_2}{d^2} \][/tex]
[tex]\[ F' = 2 \left( G \frac{m_1 m_2}{d^2} \right) \][/tex]
Since [tex]\( G \frac{m_1 m_2}{d^2} = F = 4 \)[/tex] N, we can substitute and get:
[tex]\[ F' = 2 \cdot 4 \][/tex]
[tex]\[ F' = 8 \text{ N} \][/tex]
Therefore, the number that should be in the cell with the question mark is 8. When you double the mass of one of the objects, the gravitational force between the objects also doubles. Hence, the correct answer is:
The number is eight because when you double the mass of one of the objects, the force between the objects also doubles.
