If Earth's mass is doubled without changing its size, your weight would

Options

A. increase because gravitational force increases
B. decrease because gravitational force increases
C. increase because gravitational force decreases
D. not change because you are still on Earth.

Answer

To solve the question, "If Earth's mass is doubled without changing its size, your weight would...", we need to understand the relationship between mass, weight, and gravity. Here’s a step-by-step breakdown:

### Step 1: Understand the Concept of Weight
Weight is defined as the force exerted on an object due to gravity. It is calculated using the formula:
\[ W = m \cdot g \]
where:
- \( W \) is the weight,
- \( m \) is the mass of the object,
- \( g \) is the acceleration due to gravity.

### Step 2: Identify the Acceleration Due to Gravity
The acceleration due to gravity on the surface of a planet is given by:
\[ g = \frac{G \cdot M}{r^2} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( r \) is the radius of the planet.

### Step 3: Initial Weight Calculation
Let’s denote the initial mass of the Earth as \( M \) and the radius as \( r \). The initial weight of a person (with mass \( m \)) on Earth can be expressed as:
\[ W_1 = m \cdot g = m \cdot \frac{G \cdot M}{r^2} \]

### Step 4: Doubling the Mass of the Earth
If the mass of the Earth is doubled (i.e., \( M \) becomes \( 2M \)), the new acceleration due to gravity becomes:
\[ g' = \frac{G \cdot (2M)}{r^2} = 2 \cdot \frac{G \cdot M}{r^2} = 2g \]

### Step 5: New Weight Calculation
Now, the new weight \( W_2 \) of the person on the Earth with doubled mass is:
\[ W_2 = m \cdot g' = m \cdot (2g) = 2 \cdot (m \cdot g) = 2W_1 \]
This shows that the new weight is double the initial weight.

### Conclusion
Thus, if Earth's mass is doubled without changing its size, your weight would also be doubled.

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