Answer: C) \( 1.22 \times 10^5 \, \text{kg} \)
Solution:
The gravitational force between two masses is given by:
\[
F = G \frac{m_1 m_2}{r^2}
\]
Here:
- \( F = 1.0 \, \text{N} \)
- \( G = 6.67 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \)
- \( r = 1.0 \, \text{m} \)
- \( m_1 = m_2 = m \) (since the spheres are identical)
Substituting these values:
\[
1.0 = 6.67 \times 10^{-11} \cdot \frac{m \cdot m}{(1.0)^2}
\]
Simplifying:
\[
1.0 = 6.67 \times 10^{-11} \cdot m^2
\]
\[
m^2 = \frac{1.0}{6.67 \times 10^{-11}}
\]
\[
m^2 = 1.5 \times 10^{10}
\]
Taking the square root:
\[
m = \sqrt{1.5 \times 10^{10}}
\]
\[
m = 1.22 \times 10^5 \, \text{kg}
\]
Final Answer: The mass of each sphere is \( 1.22 \times 10^5 \, \text{kg} \).
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