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Chapter 4: Problem 3

If the earth were twice as massive, but still the same size, how would our weight differ?

Short Answer

Expert verified
If the Earth were twice as massive, your weight would double.

Step by step solution

01

Understand the relationship between mass, gravity, and weight

Weight on Earth is determined by the gravitational force which is a function of the mass of the Earth and the individual's mass. The formula for gravitational force (which determines weight) is: \[ F = G \frac{m_1 m_2}{r^2} \]Here, \( F \) is the force (weight), \( G \) is the gravitational constant, \( m_1 \) is the mass of the Earth, \( m_2 \) is the mass of the individual, and \( r \) is the radius of the Earth.
02

Identify the given changes

The problem states that the Earth's mass is doubled, which means \( m_1 \) becomes \(2m\). The radius \( r \) remains the same.
03

Calculate the new weight

Using the modified mass in the gravitational force formula, substitute \( 2m \) for \( m_1 \):\[ F' = G \frac{(2m)m_2}{r^2} \]This simplifies to:\[ F' = 2 \times G \frac{m m_2}{r^2} \]Which shows that the new weight (\( F' \)) is twice the original weight.
04

Compare the new weight with the original weight

The original weight \( F \) can be written as:\[ F = G \frac{m m_2}{r^2} \]So, the new weight \( F' \) is:\[ F' = 2F \]Therefore, if the Earth were twice as massive, you would weigh twice as much as you do now.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

mass and weight relationship
Understanding the relationship between mass and weight is crucial. Mass measures the amount of matter in an object and is typically measured in kilograms (kg). It remains constant regardless of location. Weight, however, is the force exerted by gravity on that mass. It is dependent on both the mass of the object and the gravitational pull of the planet. Thus, while mass stays the same, your weight can change depending on where you are in the universe. For instance, you would weigh less on the Moon because the Moon鈥檚 gravitational force is weaker than Earth鈥檚.
gravitational constant
The gravitational constant, represented by the symbol \(G\), is a fundamental constant in physics. Its value is \[ 6.67430 \times 10^{-11} \frac{m^3}{kg \times s^2} \]. This very small number indicates the strength of gravity and appears in the equation for gravitational force. Despite its small size, \(G\) plays a huge role in calculating the gravitational pull between two objects. It shows up in Newton's law of universal gravitation and is essential for determining how much objects attract one another in the universe.
force equation in physics
The force equation in physics we often consider for gravity is: \[ F = G \frac{m_1 m_2}{r^2} \]. This equation calculates the gravitational force between two masses (\

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