91Ó°ÊÓ

According to the universal law of gravitational, every object attracts another object towards its center with a force known as gravitational force. The gravitational force exists due to the mass of the object.

The two identical spaceships of mass \(5 \times {10^5}{\rm{ kg}}\) are \(50{\rm{ m}}\) from each other. (a) How much charge should each ship carry to counteract the gravitational attraction between them? (b) Is the charge on the ship affected by the distance between the ships' centres?

The gravitational force of attraction between two identical space ships of mass \(m\) and separated by a distance \(r\) is,

\({F_g} = \frac{{G{m^2}}}{{{r^2}}}\)

Here, \(G\) is the universal gravitational constant.

The electrostatic force between two identical space ships having same magnitude of charge \(q\) and separated by a distance \(r\) is,

\({F_e} = \frac{{K{q^2}}}{{{r^2}}}\)

Here, \(K\) is the electrostatic force constant.

Since, the gravitational force of attraction is balanced by the electrostatic force. Therefore,

\(\begin{array}{c}{F_e} = {F_g}\\\frac{{K{q^2}}}{{{r^2}}} = \frac{{G{m^2}}}{{{r^2}}}\end{array}\)

Rearranging the above expression in order to get an expression for the charge,

\(q = \sqrt {\frac{{G{m^2}}}{K}} \ldots \left( {1.1} \right)\)

Substitute \(6.67 \times {10^{ - 11}}{\rm{ N}} \cdot {{\rm{m}}^2}/{\rm{k}}{{\rm{g}}^2}\) for \(G\), \(5 \times {10^5}{\rm{ kg}}\) for \(m\), and \(9 \times {10^9}{\rm{ N}} \cdot {{\rm{m}}^2}/{{\rm{C}}^2}\)for\(K\),

\(\begin{array}{c}q = \sqrt {\frac{{\left( {6.67 \times {{10}^{ - 11}}{\rm{ N}} \cdot {{\rm{m}}^2}/{\rm{k}}{{\rm{g}}^2}} \right) \times {{\left( {5 \times {{10}^5}{\rm{ kg}}} \right)}^2}}}{{\left( {9 \times {{10}^9}{\rm{ N}} \cdot {{\rm{m}}^2}/{{\rm{C}}^2}} \right)}}} \\ = 4.3 \times {10^{ - 5}}{\rm{ C}}\end{array}\)

Hence, the charge on the ship is \(4.3 \times {10^{ - 5}}{\rm{ C}}\).

From equation \(\left( {1.1} \right)\), it is clear that the charge is independent of the distance between the centers of space ship.

Hence, the charge on the ship does not depend on the distance between the centers of the ships.