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Chapter 13: Problem 17

(a) What will an object weigh on the Moon's surface if it weighs \(100 \mathrm{~N}\) on Earth's surface? (b) How many Earth radii must this same object be from the center of Earth if it is to weigh the same as it does on the Moon?

Short Answer

Expert verified
(a) 16.63 N; (b) 2.45 Earth radii.

Step by step solution

01

Understanding Weight on the Moon

On Earth, the weight of an object is computed using the formula: \[ W_e = m imes g_e \]where \( W_e \) is the weight on Earth (100 N), \( m \) is the mass, and \( g_e \) is the acceleration due to gravity on Earth (approximately 9.81 m/s虏). On the Moon, the weight \( W_m \) is given by:\[ W_m = m imes g_m \]where \( g_m \) is the moon鈥檚 gravitational acceleration (approximately 1.63 m/s虏).
02

Calculating Mass of the Object

To find the mass of the object, rearrange the Earth weight formula:\[ m = \frac{W_e}{g_e} \]Substitute the given values:\[ m = \frac{100\, \text{N}}{9.81\, \text{m/s}^2} \approx 10.20 \text{ kg} \]
03

Calculating Weight on the Moon

Use the mass from Step 2 to calculate the weight on the Moon:\[ W_m = m \times g_m \]Plug in the values:\[ W_m = 10.20 \text{ kg} \times 1.63 \text{ m/s}^2 \approx 16.63 \text{ N} \]
04

Understanding Reduced Weight Condition at a Distance from Earth's Center

The weight \( W \' \) of an object at a distance \( r \) from the center of Earth can be described by the formula:\[ W\' = \frac{G \times M_e \times m}{r^2} \]where \( G \) is the gravitational constant, and \( M_e \) is Earth's mass. We want \( W\' \) to be equal to the Moon weight \( W_m \).
05

Setting Up Equation for Reduced Weight at Increased Earth's Radius

From Step 4, the object's weight at a distance \( r \) is equal to:\[ W_m = m \times \frac{g_e}{(r/R)^2} \]Solving \( W\' = W_m \) gives:\[ W_m = \frac{W_e}{(r/R)^2} \]
06

Calculating Distance in Earth Radii from Earth's Center

Use algebra to solve for \( r/R \):\[ 16.63 = \frac{100}{(r/R)^2} \]\[ (r/R)^2 = \frac{100}{16.63} \approx 6.013 \]\[ r/R \approx \sqrt{6.013} \approx 2.45 \]
07

Conclusion on Distance from Earth's Center

Therefore, to weigh the same as on the Moon, the object must be approximately 2.45 Earth radii from the center of the Earth.

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Key Concepts

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

Weight on the Moon
When you're measuring weight on Earth, you鈥檙e actually calculating the gravitational force acting on an object. On the Moon, gravity isn鈥檛 as strong, which makes everything weigh less. Gravity on the Moon is about 1/6th of Earth's gravity.
This means if you weigh 100 N on Earth, you鈥檙e going to weigh quite a bit less on the Moon.
Using the formula:
  • Earth weight: \( W_e = m \times g_e \)
  • Moon weight: \( W_m = m \times g_m \)
You need to know the mass of the object so that you can use the Moon鈥檚 gravity value, \( g_m = 1.63 \, \text{m/s}^2 \), to find its weight on the Moon. Applying these formulas, we calculated the object would weigh roughly 16.63 N on the Moon.
Acceleration due to Gravity
Gravitational acceleration tells you how fast objects will accelerate towards the surface of a celestial body due to gravity.
On Earth, this is approximately 9.81 m/s虏, meaning objects will speed up at this rate when falling. On the Moon, gravitational acceleration is much weaker at about 1.63 m/s虏. This difference explains why you would weigh less on the Moon.
For example, by using the formula \( m = \frac{W_e}{g_e} \), we could determine the mass of an object based on its Earth weight, allowing us to then find how much it will weigh under Moon鈥檚 weaker gravity at \( g_m \). The step by step approach ensures you understand the changes induced by different gravity strengths.
Earth Radii Distance
Determining the equivalent weight an object would have if it were positioned at a certain distance in Earth radii can be an intriguing problem.
The concept of Earth radii simplifies understanding changes in gravitational effects as you move objects further from Earth's center. When an object鈥檚 gravitational pull is lessened, it may weigh as though it is on the Moon even while still within the Earth's gravitational pull.
You can use the formula:
  • Effective weight at distance: \( W' = \frac{G \times M_e \times m}{r^2} \)
Given that we matched this with Moon's weight (16.63 N) using Earth's radius as a base unit, we discovered the object needed to be located approximately 2.45 Earth radii away from the center of the Earth to weigh the same as on the Moon. This illustrates how distance can affect weight due to changes in gravitational pull.

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Most popular questions from this chapter

(a) What is the escape speed on a spherical asteroid whose radius is \(500 \mathrm{~km}\) and whose gravitational acceleration at the surface is \(3.0 \mathrm{~m} / \mathrm{s}^{2} ?\) (b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of \(1000 \mathrm{~m} / \mathrm{s} ?\) (c) With what speed will an object hit the asteroid if it is dropped from \(1000 \mathrm{~km}\) above the surface?B

A satellite orbits a planet of unknown mass in a circle of radius \(2.0 \times 10^{7} \mathrm{~m} .\) The magnitude of the gravitational force on the satellite from the planet is \(F=80 \mathrm{~N}\). (a) What is the kinetic energy of the satellite in this orbit? (b) What would \(F\) be if the orbit radius were increased to \(3.0 \times 10^{7} \mathrm{~m} ?\)

In 1956, Frank Lloyd Wright proposed the construction of a mile-high building in Chicago. Suppose the building had been constructed. Ignoring Earth's rotation, find the change in your weight if you were to ride an elevator from the street level, where you weigh \(600 \mathrm{~N},\) to the top of the building.

A mass \(M\) is split into two parts, \(m\) and \(M-m,\) which are then separated by a certain distance. What ratio m/M maximizes the magnitude of the gravitational force between the parts?

A \(20 \mathrm{~kg}\) satellite has a circular orbit with a period of \(2.4 \mathrm{~h}\) and a radius of \(8.0 \times 10^{6} \mathrm{~m}\) around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is \(8.0 \mathrm{~m} / \mathrm{s}^{2},\) what is the radius of the planet?
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