To determine the effect on gravitational force when both the masses double and the distance doubles, we can use the formula for gravitational force:
\[ F = G \frac{m_1 m_2}{d^2} \]
Where:
- \( F \) is the gravitational force,
- \( G \) is the gravitational constant,
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( d \) is the distance between the centers of the two objects.
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Doubling the masses: If both masses double, the new gravitational force due to the mass change becomes: \[ F' = G \frac{(2m_1)(2m_2)}{d^2} = G \frac{4m_1 m_2}{d^2} \] This indicates the force increases by a factor of 4 due to the mass increase.
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Doubling the distance: Now, if the distance also doubles, we need to adjust for that. The new distance becomes \( 2d \), so the gravitational force becomes: \[ F'' = G \frac{4m_1 m_2}{(2d)^2} = G \frac{4m_1 m_2}{4d^2} = G \frac{m_1 m_2}{d^2} \] This indicates the force decreases to the original gravitational force.
Putting it all together, we can summarize:
- The gravitational force increases by a factor of 4 due to the doubling of the masses.
- The gravitational force decreases by a factor of 4 due to the doubling of the distance.
When both effects are combined:
- Increasing by a factor of 4 (due to mass) and then decreasing by a factor of 4 (due to distance) results in: \[ \text{Overall effect} = 4 \times \frac{1}{4} = 1 \] This means the gravitational force remains constant.
Thus, the correct response is:
C The gravitational attraction remains constant.
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