Certainly!
Newton's Law of Universal Gravitation states that the gravitational force ([tex]\(F\)[/tex]) between two masses ([tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex]) separated by a distance ([tex]\(d\)[/tex]) is given by the equation:
[tex]\[ F = \frac{G m_1 m_2}{d^2} \][/tex]
where [tex]\(G\)[/tex] is the gravitational constant.
Let's analyze what happens if we double one of the masses, say [tex]\(m_1\)[/tex].
1. Initial Gravitational Force:
We start with the initial masses [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex] separated by a distance [tex]\(d\)[/tex]. The initial gravitational force is:
[tex]\[ F_{\text{initial}} = \frac{G m_1 m_2}{d^2} \][/tex]
2. Doubling One of the Masses:
Now, let's double the mass [tex]\(m_1\)[/tex], so the new mass becomes [tex]\(2m_1\)[/tex].
3. New Gravitational Force:
With the doubled mass, the new gravitational force becomes:
[tex]\[ F_{\text{new}} = \frac{G (2m_1) m_2}{d^2} \][/tex]
Simplifying this, we get:
[tex]\[ F_{\text{new}} = 2 \cdot \frac{G m_1 m_2}{d^2} \][/tex]
[tex]\[ F_{\text{new}} = 2 \cdot F_{\text{initial}} \][/tex]
The new gravitational force ([tex]\(F_{\text{new}}\)[/tex]) is exactly twice the initial gravitational force ([tex]\(F_{\text{initial}}\)[/tex]).
By using the given values and the gravitational constant ([tex]\(G = 6.67430 \times 10^{-11} \ \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\)[/tex]), we see:
- The initial gravitational force is [tex]\(6.6743 \times 10^{-11}\)[/tex] Newtons.
- When we double the mass [tex]\(m_1\)[/tex], the new gravitational force becomes [tex]\(2 \times 6.6743 \times 10^{-11} = 1.33486 \times 10^{-10}\)[/tex] Newtons.
Therefore, doubling one of the masses results in the gravitational force also being doubled.
