Problem 65 - Elementary Algebra

Question

If two objects with masses 50 kilograms and 100 kilograms are \(1 / 2\) meter apart, then they produce approximately \(1.34 \times 10-6\) newtons (N) of force. Calculate the gravitational constant.

Step-by-Step Solution

Answer

The gravitational constant is \( 6.7 \times 10^{-11} \).

Step 1: Understand the Gravitational Force Formula

The gravitational force between two objects can be calculated using the formula \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \), where \( F \) is the force between the objects, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the objects, and \( r \) is the distance between the centers of the two masses.

Step 2: Substitute Known Values

Substitute the known values into the formula: mass \( m_1 = 50 \) kg, mass \( m_2 = 100 \) kg, distance \( r = 0.5 \) meters, and force \( F = 1.34 \times 10^{-6} \) N. This gives the equation: \( 1.34 \times 10^{-6} = \frac{G \cdot 50 \cdot 100}{(0.5)^2} \).

Step 3: Simplify the Equation

Simplify the equation \( 1.34 \times 10^{-6} = \frac{G \cdot 5000}{0.25} \). The denominator \( (0.5)^2 = 0.25 \), therefore we simplify it to \( 1.34 \times 10^{-6} = 20000G \).

Step 4: Solve for Gravitational Constant \( G \)

To find \( G \), rearrange the equation to \( G = \frac{1.34 \times 10^{-6}}{20000} \). Compute \( G \) to get \( G = 6.7 \times 10^{-11} \).

Step 5: Confirm the Units

Ensure that the units are consistent. Since force is in newtons, mass in kilograms, and distance in meters, \( G \) should have units of \( m^3 \, kg^{-1} \, s^{-2} \), which matches the standard unit for the gravitational constant.

Key Concepts

Gravitational Force Formula

The gravitational force formula is a powerful tool in physics that explains how two objects attract each other. It’s expressed as:

• \( F = \frac{G \cdot m_1 \cdot m_2}{r^2} \)

This formula shows that the gravitational force (\( F \)) depends on three things:

• The gravitational constant (\( G \)), a special number that remains the same across the universe.
• The masses of the two objects \( m_1 \) and \( m_2 \).
• The distance \( r \) between them, squared.

When using this formula, remember it describes a central force directed along the line joining the centers of the two objects. Gravitational force decreases rapidly as the distance between the objects increases.

Masses and Distance

In the gravitational force formula, the masses of the objects and the distance between them play crucial roles. Here’s how:

• Masses: Both \( m_1 \) and \( m_2 \) represent the masses in kilograms. Greater mass means a stronger gravitational pull. If one object is much more massive, it will exert a stronger gravitational attraction.
• Distance: Represented by \( r \), it is the space between the centers of the two masses. It’s important to note that distance affects the force inversely squared. This means that as distance increases, the gravitational force decreases significantly. It shows how even small increases in distance can lead to a large drop in gravitational force.

In practical terms, these variables illustrate why planets don’t crash into each other despite the gravitational forces at play due to their relatively large distances.

Scientific Notation

Scientific notation is a way to write very large or small numbers succinctly. It’s essential when dealing with units of measurement in physics. Here’s how it works:

• A number is written as a product of a number (usually between 1 and 10) and a power of ten.
• For example, \( 1.34 \times 10^{-6} \) stands for 0.00000134.

Using scientific notation helps simplify calculations involving the gravitational force formula, which often includes very small values like the gravitational constant (\( G = 6.7 \times 10^{-11} \)). It’s a concise way to represent significant figures without writing lots of zeros.

Units of Measurement

Understanding units of measurement is crucial when calculating gravitational force. Here are the units you need to pay attention to:

• Mass: Measured in kilograms (kg), mass quantifies how much matter is in an object.
• Distance: Measured in meters (m), distance explains how far apart the two objects are.
• Force: Measured in newtons (N), force is the interaction that causes a change in an object’s motion.

For the gravitational constant \( G \), the units are \( m^3 \, kg^{-1} \, s^{-2} \). This set of units ensures that when you calculate gravitational force, the result is accurate and meaningful in the context of these standard physical quantities.