Question

A body weight \(500 \mathrm{~N}\) on the surface of the earth. How much would it weight half way below the surface of earth (A) \(125 \mathrm{~N}\) (B) \(250 \mathrm{~N}\) (C) \(500 \mathrm{~N}\) (D) \(1000 \mathrm{~N}\)

Short answer

The weight of the body half-way below the surface of the Earth is approximately 250 N. The correct answer is (B) \(250 \mathrm{~N}\).

Step-by-step solution

Write down the gravitational force formula

The formula for the gravitational force between two objects is given by: \[F = G \frac{m_1 m_2}{r^2}\] Where: - F is the gravitational force - G is the gravitational constant (\(6.674 \times 10^{-11} \mathrm{Nm^2/kg^2}\)) - \(m_1\) and \(m_2\) are the masses of the two objects - r is the distance between the centers of the objects In this exercise, we will consider Earth as one object and the body with a weight of 500 N as the other object.

Determine the mass of the body

First, we need to find the mass of the body. To do this, we can use the formula for weight: \[W = mg\] Where: - W is the weight of the body (500 N) - m is the mass of the body (which we want to find) - g is the acceleration due to gravity on Earth's surface (\(9.81\mathrm{m/s^2}\)) Solving for m: \[m = \frac{W}{g}\]

Calculate the mass of the body

Using the values given, we can now calculate the mass of the body: \[m = \frac{500\mathrm{~N}}{9.81\mathrm{m/s^2}} \approx 51\mathrm{kg}\]

Find the new distance to the center of the Earth

Now, we need to find the new distance to the center of the Earth when the body is half-way below the surface. Let's call this new distance \(r'\). The radius of the Earth is approximately 6371 km. Therefore, the new distance to the center of the Earth is half of this value: \[r' = \frac{1}{2} (6371\mathrm{km}) = 3185.5\mathrm{km}\] Make sure to convert this value to meters: \[r' = 3185.5\mathrm{km} \times 1000\mathrm{m/km} = 3185500\mathrm{m}\]

Calculate the new gravitational force

Using the gravitational force formula, we can now calculate the new gravitational force acting on the body at half-way below the surface: \[F' = G \frac{m_1 m_2}{{r'}^2}\] Here, \(m_1\) is the body's mass (51 kg) and \(m_2\) is Earth's mass (\(5.972 \times 10^{24}\mathrm{kg}\)). Using the values, we get: \[F' \approx \frac{6.674 \times 10^{-11} \mathrm{Nm^2/kg^2} \times 51\mathrm{kg} \times 5.972 \times 10^{24}\mathrm{kg}}{{(3185500\mathrm{m})}^2}\] \[F' \approx 250\mathrm{~N}\]

Choose the correct answer

From our calculation, the weight of the body half-way below the surface of the Earth is approximately 250 N, thus the correct answer is: (B) \(250 \mathrm{~N}\)

Key concepts

Weight Calculation

When we talk about weight, we are referring to the gravitational force that Earth exerts on an object. This force is what gives us the sensation of weight when we stand on a scale. It's important to note that weight is different from mass; weight changes depending on where you are in the universe, but your mass stays constant.
To calculate weight, we use the formula:

• \( W = mg \)

Here, \( W \) is the weight, \( m \) is the mass, and \( g \) is the gravitational acceleration at Earth's surface which is approximately \( 9.81 \mathrm{m/s^2} \). By rearranging this formula, you can also calculate the mass if weight is known, as was done in the exercise to find the mass of the object as \( 51 \mathrm{kg} \).
This concept of weight calculation is pivotal in scenarios involving earth science and physics because it helps explain the forces working on objects on Earth and beyond.

Earth's Radius

The radius of Earth is a crucial factor when it comes to calculating gravitational force and determining various other geological and physical characteristics of our planet. It is approximately 6371 kilometers. Understanding Earth's radius helps us measure distances and understand how gravitational force varies with depth as well as altitude.
When dealing with problems that involve depth or height above Earth's surface, we use the radius in calculations to determine how gravity's influence changes. For instance, as in the exercise provided, half-way below the Earth's surface means halving this distance (radius), influencing the gravitational force experienced by an object.
Knowing Earth's radius and how it factors into calculations allows us to appreciate how gravity behaves differently inside the Earth compared to its surface.

Gravitational Constant

The gravitational constant, denoted as \( G \), is a key part of calculating the gravitational force between two masses. It's often referred to as Newton's constant because it is a fundamental component of his law of universal gravitation. Its value is \( 6.674 \times 10^{-11} \mathrm{Nm^2/kg^2} \).
The role of this constant is to mathematically link the masses involved and the distance between them in calculating the gravitational force according to the formula:

• \( F = G \frac{m_1 m_2}{r^2} \)

In practice, \( G \) allows us to compute the force of attraction between any two masses, which is invaluable in fields ranging from engineering to astrophysics. Understanding \( G \) helps us explore how objects interact under gravity, whether they are as large as planets or as small as individual atoms.

Distance to Center of Earth

The distance to the center of Earth is a significant factor in determining the gravitational effects felt at any point within or on the surface of our globe. In our exercise, this concept is vital when assessing how the gravitational force changes as you move half-way towards Earth's center.
Normally, this distance is the radius of the Earth (approximately 6371 kilometers), but as we move below the surface, this distance decreases, altering the gravitational force experienced by an object.
When the object in the exercise was moved half-way below the surface, the distance changed to around 3185.5 kilometers. This reduced distance increased the gravitational pull as per the formula \( F = G \frac{m_1 m_2}{r^2} \), highlighting how sensitive gravitational force is to changes in distance between the masses involved.