Problem:
$F = \frac{(G m_1 m_2)}{r^2}$
An object of unknown mass is placed directly between the Earth and the Moon such that the forces of gravity acting on the object are equal from both the Earth and the Moon. The distance from the Moon to the object is exactly one-tenth of the distance from the Moon to Earth. Using the above formula, determine the distance from the Moon to the Earth.
Given is:
The factor $G≈6.67∙10−11\mathbf{N(\frac{m^2}{kg^2})}$ is a universal constant.
Mass of Earth $≈5.972∙10^{24}\mathbf{kg}$
Mass of the Moon as $≈7.348∙10^{22}\mathbf{kg}$
Mass of unknown object = $m$
My solution:
$F = \frac{(6.67 *10^-11) m (7.348*10^{24})}{R^2}$
$F = \frac{(6.67 *10^-11) (5.92* 10^22) m}{10R^2}$
As gravitational force acting on the object is equal
$\frac{(5.92\cdot 10^22)}{100r^2} = \frac{(7.348*10^{24})}{r^2}$
$R = 1.24 \cdot 10^4 KM$
Is this a viable solution to the problem? Please provide feedback. Thank you!