Problem 81 - Algebra and Trigonometry

Question

Gravity The gravitational force \(F\) exerted by the earth on an object having a mass of 100 \(\mathrm{kg}\) is given by the equation $$ F=\frac{4,000,000}{d^{2}} $$ where \(d\) is the distance (in \(\mathrm{km} )\) of the object from the center of the earth, and the force \(F\) is measured in newtons (N). For what distances will the gravitational force exerted by the earth on this object be between 0.0004 \(\mathrm{N}\) and 0.01 \(\mathrm{N} ?\)

Step-by-Step Solution

Answer

The distance \(d\) must be between 20,000 km and 100,000 km.

Step 1: Understand the Problem

You need to find the distance \(d\) such that the gravitational force \(F\) on the 100 kg object is between 0.0004 N and 0.01 N, using the equation \(F=\frac{4,000,000}{d^2}\).

Step 2: Set Up the Inequality

You have two inequalities to solve: \(0.0004 < \frac{4,000,000}{d^2} < 0.01\). This means finding \(d\) values that satisfy both conditions simultaneously.

Step 3: Solve the First Inequality

For \(\frac{4,000,000}{d^2} > 0.0004\), solve for \(d\): \[\frac{4,000,000}{d^2} > 0.0004\] Multiply both sides by \(d^2\) and divide by 0.0004: \[4,000,000 > 0.0004d^2\] \[d^2 < \frac{4,000,000}{0.0004}\] \[d^2 < 10,000,000,000\] Take the square root: \[d < 100,000\]

Step 4: Solve the Second Inequality

For \(\frac{4,000,000}{d^2} < 0.01\), solve for \(d\): \[\frac{4,000,000}{d^2} < 0.01\] Multiply both sides by \(d^2\) and divide by 0.01: \[4,000,000 < 0.01d^2\] \[d^2 > 400,000,000\] Take the square root: \[d > 20,000\]

Step 5: Combine the Inequality Results

Combine the results from both inequalities: \(20,000 < d < 100,000\). This is the range of distances for which the gravitational force is between 0.0004 N and 0.01 N.

Key Concepts

Distance from Earth

When discussing gravitational force, one pivotal factor is the **distance from Earth**. This is specifically the distance between the object in question and the center of the Earth. In this exercise, the distance is denoted as \(d\) and is measured in kilometers (km).

Why is this distance so critical? It's because the gravitational force decreases with the square of the distance. So, as an object moves further away from Earth, the gravitational pull weakens significantly. This principle leads us to what's known as an inverse square law, where gravity lessens quickly as distance increases.

In our problem, the distance \(d\) is crucial for determining when the gravitational pull falls between certain limits, specifically 0.0004 N and 0.01 N. As we will see, solving such problems requires understanding this distance concept well.

Force Equation

The concept of gravitational force is often encapsulated by a **force equation**. For this exercise, the equation given is: \[ F = \frac{4,000,000}{d^2} \] where \(F\) is the gravitational force measured in newtons (N), and \(d\) is the distance to the Earth's center in kilometers.

Let's break down what this equation implies:
- **Numerator (4,000,000):** Represents a constant that embodies the product of the earth's gravitational constant and the object's mass (100 kg in this case). - **Denominator \(d^2\):** As distance increases, the value of \(d^2\) also increases, thereby reducing the fraction value and forcing \(F\) to become smaller. This highlights how gravitational force relies on distance, a point that will be further explored as we look at inequalities.

Inequality Solution

To find a range of distances where the gravitational force remains between two values, we use an **inequality solution**. Our task here involves satisfying:\[ 0.0004 < \frac{4,000,000}{d^2} < 0.01 \]

To tackle this, we break it down into two separate inequalities:
- **First Inequality:** - Solve \( \frac{4,000,000}{d^2} > 0.0004 \) to find the upper boundary.
- Result: \(d < 100,000\) km- **Second Inequality:** - Solve \( \frac{4,000,000}{d^2} < 0.01 \) to find the lower boundary. - Result: \(d > 20,000\) kmBy combining these results, we conclude the distance must be between \(20,000\) km and \(100,000\) km for the force to be within the desired range. This solution emphasizes how math can be used to define specific physical conditions.

Mass and Gravity

Understanding the interplay of **mass and gravity** is crucial in conversations about gravitational forces. The mass of the object determines the amount of gravitational pull it feels, which in this exercise is given as 100 kg.

Mass affects gravitational force in that:

• A larger mass will result in a greater gravitational force.
• But even this force decreases rapidly with distance (as indicated in the force equation).

Gravity is a force of attraction that exists between all masses, but on Earth, it enjoys special attention since it defines weight. By keeping the mass constant (as in our example of 100 kg), one can explore how distance affects the force, eliminating variable complexity and focusing purely on distance's role.