Questions: An astronaut weighs 1000 N on the surface of the Earth. She moves so that she is 3 radii from the center of the Earth. What is her new weight? O 34 N O 250 N O 111 N O 102 N

Solution Steps

The astronaut's weight on the surface of the Earth is given as 1000 N. We need to determine her weight when she is at a distance of 3 Earth radii from the center of the Earth.

The weight of an object due to gravity is given by: \[ F = \frac{G M m}{r^2} \] where:

• \( F \) is the gravitational force (weight),
• \( G \) is the gravitational constant,
• \( M \) is the mass of the Earth,
• \( m \) is the mass of the astronaut,
• \( r \) is the distance from the center of the Earth.

At the surface of the Earth, the distance \( r \) is equal to the Earth's radius \( R \). Therefore, the astronaut's weight \( W_{\text{surface}} \) is: \[ W_{\text{surface}} = \frac{G M m}{R^2} = 1000 \, \text{N} \]

When the astronaut is at a distance of 3 Earth radii, the new distance \( r' \) is \( 3R \). The new weight \( W_{\text{new}} \) is: \[ W_{\text{new}} = \frac{G M m}{(3R)^2} = \frac{G M m}{9R^2} \]

The ratio of the new weight to the weight at the surface is: \[ \frac{W_{\text{new}}}{W_{\text{surface}}} = \frac{\frac{G M m}{9R^2}}{\frac{G M m}{R^2}} = \frac{1}{9} \]

Using the ratio, the new weight is: \[ W_{\text{new}} = \frac{1}{9} \times 1000 \, \text{N} = 111.1111 \, \text{N} \]

Rounding to four significant digits: \[ W_{\text{new}} = 111.1 \, \text{N} \]

Final Answer