According to Newton's law of universal gravitation, the gravitational force [tex]\( F \)[/tex] between two masses [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] is inversely proportional to the square of the distance [tex]\( R \)[/tex] between them. This relationship is given by the formula:
[tex]\[ F = G \frac{{m_1 \cdot m_2}}{{R^2}} \][/tex]
where [tex]\( G \)[/tex] is the gravitational constant. Now let's analyze what happens when the distance [tex]\( R \)[/tex] is cut in half.
1. Original Distance [tex]\( R \)[/tex]:
[tex]\[ F = G \frac{{m_1 \cdot m_2}}{{R^2}} \][/tex]
2. New Distance [tex]\( R/2 \)[/tex]:
When we halve the distance, the new distance becomes [tex]\( R' = R/2 \)[/tex]. We substitute [tex]\( R/2 \)[/tex] into the gravitational force formula:
[tex]\[ F' = G \frac{{m_1 \cdot m_2}}{{(R/2)^2}} \][/tex]
3. Simplifying the Expression:
[tex]\[ F' = G \frac{{m_1 \cdot m_2}}{{R^2 / 4}} \][/tex]
[tex]\[ F' = G \frac{{m_1 \cdot m_2}}{{(R^2 / 4)}} = G \frac{{m_1 \cdot m_2}}{{R^2}} \cdot 4 \][/tex]
[tex]\[ F' = 4 \left( G \frac{{m_1 \cdot m_2}}{{R^2}} \right) \][/tex]
[tex]\[ F' = 4F \][/tex]
So, when the distance between the two objects is cut in half, the new gravitational force [tex]\( F' \)[/tex] becomes four times the original gravitational force [tex]\( F \)[/tex].
Therefore, the correct answer is:
C. It increases to 4 times its original magnitude.
