To solve this problem, let's use the formula for gravitational force between two masses, which is given by:
[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]
where [tex]\( G \)[/tex] is the gravitational constant, [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] are the masses of the two objects, and [tex]\( r \)[/tex] is the distance between them.
### For the first pair of gloves:
The masses are [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex], and the distance between them is [tex]\( r \)[/tex]. The gravitational force [tex]\( F_1 \)[/tex] can be expressed as:
[tex]\[ F_1 = G \frac{m_1 m_2}{r^2} \][/tex]
### For the second pair of gloves:
The masses are doubled, so they are [tex]\( 2m_1 \)[/tex] and [tex]\( 2m_2 \)[/tex], and the distance between them is also doubled to [tex]\( 2r \)[/tex]. The gravitational force [tex]\( F_2 \)[/tex] can be expressed as:
[tex]\[ F_2 = G \frac{(2m_1)(2m_2)}{(2r)^2} \][/tex]
Simplify the expression for [tex]\( F_2 \)[/tex]:
[tex]\[ F_2 = G \frac{4m_1 m_2}{4r^2} \][/tex]
Notice that the factor of 4 in the numerator and the denominator cancel each other out:
[tex]\[ F_2 = G \frac{m_1 m_2}{r^2} \][/tex]
Hence, we see that:
[tex]\[ F_2 = F_1 \][/tex]
This means:
[tex]\[ F_2 = 1 \cdot F_1 \][/tex]
Therefore, the value of [tex]\( n \)[/tex] is:
[tex]\[ n = 1 \][/tex]
