Two spheres (m_(1) and m_(2)) are separated a distance d, apart. The gravitational force between the two spheres is 41 N. If the spheres are separated to a distance of 3 times the original distance what is the new gravitational force between the spheres? m1 m2

The gravitational force between two masses is given by Newton's law of universal gravitation:<br /><br />\[ F = \frac{G \cdot m_1 \cdot m_2}{d^2} \]<br /><br />where:<br />- \( F \) is the gravitational force,<br />- \( G \) is the gravitational constant (\(6.674 \times 10^{-11} \, \text{N}\cdot(\text{m/kg})^2\)),<br />- \( m_1 \) and \( m_2 \) are the masses of the two spheres,<br />- \( d \) is the distance between the centers of the two masses.<br /><br />Initially, the gravitational force is 41 N when the distance is \( d \). If the distance is increased to 3 times the original distance, the new distance becomes \( 3d \).<br /><br />The new gravitational force \( F' \) can be calculated using the same formula, but with the new distance:<br /><br />\[ F' = \frac{G \cdot m_1 \cdot m_2}{(3d)^2} \]<br /><br />This simplifies to:<br /><br />\[ F' = \frac{G \cdot m_1 \cdot m_2}{9d^2} \]<br /><br />Since the initial force \( F = \frac{G \cdot m_1 \cdot m_2}{d^2} = 41 \, \text{N} \), we can express \( F' \) in terms of \( F \):<br /><br />\[ F' = \frac{F}{9} \]<br /><br />Substituting the known value of \( F \):<br /><br />\[ F' = \frac{41}{9} \approx 4.56 \, \text{N} \]<br /><br />Therefore, the new gravitational force between the spheres when they are separated by a distance of 3 times the original distance is approximately 4.56 N.