If the distance between two masses is increased by a factor of 5, by what factor would the mass of one of them have to be altered to maintain the same gravitational force? Would this be an increase or decrease in the mass?
Gravitational force is given by:
F = \(G \times \frac{m \times M}{d^2}\)
Distance between two masses is increased s.t new distance is D = 5d
New gravitational force F1 = F
Let on of the mass is changed to m1 so as to maintain the same gravitational force.
F1 = \(G \times \frac{m_1 \times M}{D^2}\)
D = 5d
F = F1
= \(G \times \frac{m \times M}{d^2}\) = \(G \times \frac{m_1 \times M}{D^2}\)
\(G \times \frac{m \times M}{d^2}\) = \(G \times \frac{m_1 \times M}{25d^2}\)
\( \frac{m_1}{m}\) = 25
m1 = 25 cm
Hence one of the masses should be increased by 25 times in order to have the same gravitational force.