Answer:
1) 2.72 × 10⁻¹² N
2) 3.82 × 10⁻¹⁷ C
Explanation:
Question 1
To calculate the gravitational force between the two objects, we can use Newton's law of universal gravitation:
[tex]\sf F= G\cdot \dfrac{m_1 \cdot m_2}{r^2}[/tex]
where:
- F is the gravitational force in newtons (N).
- G is the gravitational constant (6.674 × 10⁻¹¹ N · m²/kg²).
- m₁ and m₂ are the masses of the two objects in kilograms (kg).
- r is distance between the centers of the two objects in meters (m).
Given values:
- m₁ = 1.67 × 10⁻⁷ kg
- m₂ = 8.46 × 10⁻⁶ kg
- r = 5.89 × 10⁻⁶ m
Substitute these values into the formula:
[tex]\sf F = (6.674 \times 10^{-11}) \dfrac{(1.67 \times 10^{-7}) \cdot (8.46 \times 10^{-6})}{(5.89 \times 10^{-6})^2}[/tex]
Solve for F:
[tex]\sf F = (6.674 \times 10^{-11}) \dfrac{14.1282 \times 10^{-13}}{34.6921 \times 10^{-12}}\\\\\\\\F = (6.674 \times 10^{-11}) \dfrac{14.1282}{34.6921} \times 10^{-1}\\\\\\\\F =\dfrac{94.2916068}{34.6921} \times 10^{-12}\\\\\\\\F=2.72\times 10^{-12}[/tex]
So, the gravitational force between the two objects is approximately 2.72 × 10⁻¹² N.
[tex]\dotfill[/tex]
Question 2
The electrostatic force F between two point charges q₁ and q₂ separated by a distance r can be calculated using Coulomb's law:
[tex]\sf F = k \cdot \dfrac{|q_1 \cdot q_2|}{r^2}[/tex]
where:
- F is the electrostatic force between charges in Newtons (N).
- k is Coulomb's constant (8.98755 × 10⁹ N m² / C²)
- q₁ and q₂ are the two charges in Coulombs (C)
- r is the shortest distance between the charges in meters (m).
Given values:
- q₁ = 2.75 × 10⁻¹⁶ C
- r = 5.89 × 10⁻⁶ m
Substitute these values into the formula:
[tex]\sf F = 8.98755\times 10^9\cdot \dfrac{|2.75 \times 10^{-16} \cdot q_2|}{(5.89 \times 10^{-6})^2}[/tex]
Solve for F:
[tex]\sf F = 8.98755\times 10^9\cdot \dfrac{2.75 \times 10^{-16} \cdot q_2}{34.6921 \times 10^{-12}}\\\\\\\\F= 8.98755\times 10^9\cdot\dfrac{2.75q_2}{34.6921}\times 10^{-4}\\\\\\\\F= \dfrac{24.7157625\times 10^{5}}{34.6921}\cdot q_2[/tex]
To find the charge on the second object such that the gravitational and electrostatic forces are equal in magnitude, we set the gravitational force calculated in question 1 equal to the electrostatic force calculated above, and solve for the unknown charge (q₂):
[tex]\sf |F_{\text{grav}}| = |F_{\text{elec}}|[/tex]
[tex]\sf \dfrac{94.2916068\times 10^{-12}}{34.6921} =\dfrac{24.7157625\times 10^{5}}{34.6921}\cdot q_2\\\\\\\\q_2=\dfrac{\dfrac{94.2916068\times 10^{-12}}{34.6921} }{\dfrac{24.7157625\times 10^{5}}{34.6921}}\\\\\\\\q_2=\dfrac{94.2916068\times 10^{-12}}{24.7157625\times 10^{5}}\\\\\\\\q_2=\dfrac{94.2916068}{24.7157625}\times 10^{-17}\\\\\\\\q_2=3.82\times 10^{-17}\; C[/tex]
So, the charge on the second object would be approximately 3.82 × 10⁻¹⁷ C in order for the gravitational and electrostatic forces to be equal in magnitude.
