Suppose the Earth and the Moon carried positive charges of equal magnitude. How large would the charge need to be to produce an electrostatic repulsion equal to \(1.00 \%\) of the gravitational attraction between the two bodies?

We will use Newton's law of universal gravitation to find the force between the Earth and the Moon. The formula for gravitational force is: \(F_g = G\frac{m_1m_2}{r^2}\) Where: - \(F_g\) is the gravitational force - \(G\) is the gravitational constant, \(6.674 \times 10^{-11} N\cdot m^2 / kg^2\) - \(m_1\) is the mass of the Earth, \(5.972 \times 10^{24} kg\) - \(m_2\) is the mass of the Moon, \(7.342 \times 10^{22} kg\) - \(r\) is the distance between the Earth and the Moon, \(3.844 \times 10^8 m\) Now, we will substitute these values into the equation: \(F_g = 6.674 \times 10^{-11} \frac{(5.972 \times 10^{24})(7.342 \times 10^{22})}{(3.844 \times 10^8)^2}\) Calculate \(F_g\) to get the gravitational force between the Earth and the Moon.

The question asks for the charge that will produce an electrostatic force equal to 1% of the gravitational force. First, we need to find 1% of the gravitational force: \(F_e = 0.01 \times F_g\) Calculate \(F_e\) using the value of \(F_g\) obtained in Step 1.

We will use Coulomb's law to find the charge required to produce the desired electrostatic force. The formula for electrostatic force is: \(F_e = k\frac{q_1q_2}{r^2}\) Since the question states that the charges on the Earth and the Moon are equal, we can simplify the equation to: \(F_e = k\frac{q^2}{r^2}\) Where: - \(F_e\) is the electrostatic force - \(k\) is the electrostatic constant, \(8.987 \times 10^9 N\cdot m^2 / C^2\) - \(q\) is the charge on the Earth and the Moon - \(r\) is the distance between the Earth and the Moon, \(3.844 \times 10^8 m\) Now, rearrange the equation to solve for \(q\): \(q^2 = \frac{F_e \cdot r^2}{k}\) Substitute the values of \(F_e\), \(r\), and \(k\) into the equation and solve for \(q^2\). Finally, take the square root of both sides to find \(q\). This value is the charge required to produce an electrostatic force equal to 1% of the gravitational force between the Earth and the Moon.