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Chapter 4: Problem 29

Weight on Earth is proportional to mass. On the Moon, too, weight is proportional to mass, but the constant of proportionality is different on the Moon than it is on Earth. Why?

Short Answer

Expert verified
Weight differs on Earth and Moon due to different gravitational accelerations (9.8 m/s虏 vs 1.6 m/s虏).

Step by step solution

01

- Understand Proportionality

Weight is directly proportional to mass, which means weight can be expressed as a constant multiplied by the mass: \(W = k \times m\).
02

- Identify Constants of Proportionality

The constant of proportionality is different for Earth and the Moon. On Earth, this constant is the gravitational acceleration \(g_E\), and on the Moon, it is \(g_M\). Thus, we have: \( W_E = g_E \times m \) for Earth and \( W_M = g_M \times m \) for the Moon.
03

- Compare Gravitational Acceleration

Gravitational acceleration \(g\) is different for Earth and the Moon due to their different masses and radii. \(g_E\) on Earth is approximately 9.8 m/s虏, and \(g_M\) on the Moon is approximately 1.6 m/s虏.
04

- Relate Weight to Gravitational Acceleration

The difference in gravitational acceleration means that the weight of an object will be less on the Moon compared to on Earth if the object's mass remains the same.

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Key Concepts

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

weight
Weight is the force exerted on an object due to gravity. It's calculated with the formula: \( W = m \times g \). Here, \(W\) stands for weight, \(m\) for mass, and \(g\) for gravitational acceleration.

This means your weight depends on your mass and the gravity acting on you.

For instance, even if your mass stays constant, your weight changes depending on whether you're on Earth or the Moon.
mass
Mass is the amount of matter in an object. It's measured in kilograms (kg) and doesn't change whether you're on Earth, the Moon, or floating in space.

Mass reflects the quantity of material in an object, not the force it experiences due to gravity.

In a weight vs. mass scenario, you might weigh less on the Moon, but your mass stays the same.
proportionality
Proportionality means that one quantity changes in direct correspondence to another.

In terms of weight and mass, weight is directly proportional to mass: \( W = k \times m \). Therefore, if mass doubles, weight also doubles assuming the constant \( k \) stays the same.

Gravitational acceleration is the constant of proportionality. On Earth, this constant is approximately 9.8 \( m/s^2 \), while on the Moon, it's about 1.6 \( m/s^2 \).
Moon
The Moon's gravitational acceleration is much weaker compared to Earth's.

This means that objects weigh less on the Moon. With \( g_M \approx 1.6 \, m/s^2 \), an object on the Moon experiences less gravitational force.

For example, if an object weighs 60 N on Earth, it would weigh about 10 N on the Moon.

The mass of the object is still the same, but due to the lower gravitational pull, its weight decreases significantly.
Earth
On Earth, the gravitational acceleration \( g_E \) is 9.8 \( m/s^2 \).

This relatively high gravitational force means objects weigh more here compared to the Moon.

For a mass \( m \), the weight on Earth would be \( W_E = 9.8 \times m \).

So, if you have a mass of 10 kg, you weigh about 98 N on Earth.

Understanding these differences helps explain why astronauts can hop around easily on the Moon but feel much heavier on Earth.

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Most popular questions from this chapter

Go to NASA's "Apollo 15 Hammer-Feather Drop" Web page (http://nssdc.gsfc.nasa.gov/planetary/lunar/apollo_15_ feather_drop.html) and watch the video from Apollo 15 of astronaut David Scott dropping the hammer and falcon feather on the Moon. (You might find a better version on YouTube.) What did this experiment show? What would happen if you tried this on Earth with a feather and a hammer? Would it work? What would you see? Suppose instead you dropped the hammer and a big nail. How would they fall? How fast do things fall on the Moon compared to on Earth?

If you could live in a house at the center of Earth, you could float from room to room as though you were living in the International Space Station. Explain why this statement is true.

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As described in Math Tools \(4.4,\) tidal force is proportional to the masses of the two objects and is inversely proportional to the cube of the distance between them. Some astrologers claim that your destiny is determined by the "influence" of the planets that are rising above the horizon at the moment of your birth. Compare the tidal force of Jupiter (mass \(1.9 \times 10^{27} \mathrm{kg}\); distance \(7.8 \times 10^{11}\) meters with that of the doctor in attendance at your birth (mass \(80 \mathrm{kg}, \text { distance } 1 \text { meter })\)

Both Kepler's laws and Newton's laws tell us something about the motion of the planets, but there is a fundamental difference between them. What is that difference?
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