Question
How would your present weight change if Earth had twice the mass that it does now?
It would be 1/4 of your present weight.
It would be cut in half.
It would be four times your present weight.
It would double.
It would be 1/4 of your present weight.
It would be cut in half.
It would be four times your present weight.
It would double.
Answers
Answered by
GPT-4o mini
Your weight on the surface of the Earth depends on the gravitational force exerted by the Earth, which is determined by both the mass of the Earth and the distance from its center. The relationship between weight (W), mass of the object (m), and gravitational acceleration (g) can be expressed with the equation:
\[ W = m \cdot g \]
If the mass of the Earth were to double, the gravitational acceleration at the surface would also double. This is because gravitational acceleration is directly proportional to the mass of the Earth:
\[ g' = \frac{G \cdot (2M)}{R^2} = 2 \cdot \left(\frac{G \cdot M}{R^2}\right) = 2g \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.
If \( g \) doubles, your weight would also double:
\[ W' = m \cdot g' = m \cdot (2g) = 2 \cdot (m \cdot g) = 2W \]
Therefore, if the Earth had twice the mass that it does now, your weight would double. Thus, the correct answer is:
**It would double.**
\[ W = m \cdot g \]
If the mass of the Earth were to double, the gravitational acceleration at the surface would also double. This is because gravitational acceleration is directly proportional to the mass of the Earth:
\[ g' = \frac{G \cdot (2M)}{R^2} = 2 \cdot \left(\frac{G \cdot M}{R^2}\right) = 2g \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.
If \( g \) doubles, your weight would also double:
\[ W' = m \cdot g' = m \cdot (2g) = 2 \cdot (m \cdot g) = 2W \]
Therefore, if the Earth had twice the mass that it does now, your weight would double. Thus, the correct answer is:
**It would double.**
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