Problem 47 - Algebra 2

Question

According to the Law of Universal Gravitation, the attractive force \(F\) in Newtons between any two bodies in the universe is directly proportional to the product of the masses \(m_{1}\) and \(m_{2}\) in kilograms of the two bodies and the product of the masses \(m_{1}\) and \(m_{2}\) in kilograms of the two bodies and inversely proportional to the square of the distance \(d\) in meters between the bodies. That is, \(F=G \frac{m_{1} m_{2}}{d_{2}} . G\) is the universal gravitational constant. Its value is \(6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}\). Find the gravitational force exerted on each other by two 1000-kilogram iron balls a distance of 0.1 meter apart.

Step-by-Step Solution

Answer

The gravitational force is \( 6.67 \times 10^{-3} \; \text{Newtons}.\)

Step 1: Understand the Formula

The Universal Law of Gravitation formula is given by:\[ F = G \frac{m_1 m_2}{d^2} \]where:- \( F \) is the gravitational force between two objects,- \( G \) is the gravitational constant \( 6.67 \times 10^{-11} \text{Nm}^2/\text{kg}^2 \),- \( m_1 \) and \( m_2 \) are the masses of the two objects, and- \( d \) is the distance between the objects.

Step 2: Substitute Values into the Formula

Let's substitute the given values into the formula:- \( m_1 = 1000 \; \text{kg} \)- \( m_2 = 1000 \; \text{kg} \)- \( d = 0.1 \; \text{meters} \)- \( G = 6.67 \times 10^{-11} \; \text{Nm}^2/\text{kg}^2 \)The formula becomes:\[ F = 6.67 \times 10^{-11} \frac{1000 \times 1000}{(0.1)^2} \]

Step 3: Calculate the Denominator

First, calculate the denominator, which is the square of the distance:\[ d^2 = (0.1)^2 = 0.01 \; \text{m}^2 \]

Step 4: Calculate the Numerator

Now calculate the numerator, which is the product of the masses:\[ m_1 m_2 = 1000 \times 1000 = 1000000 \; \text{kg}^2 \]

Step 5: Compute the Gravitational Force

Now, plug the values of the numerator and denominator back into the formula:\[ F = 6.67 \times 10^{-11} \times \frac{1000000}{0.01} \]Calculate the value:\[ F = 6.67 \times 10^{-11} \times 100000000 = 6.67 \times 10^{-3} \; \text{N} \]

Step 6: Conclusion

The gravitational force exerted on each other by the two 1000-kilogram iron balls that are 0.1 meters apart is \( 6.67 \times 10^{-3} \; \text{Newtons} \).

Key Concepts

Gravitational Constant

In the realm of physics, the gravitational constant, denoted as \( G \), plays a crucial role. It is a fundamental constant in the Universal Law of Gravitation. The constant \( G \) provides a scale for the strength of gravity in the universe. Its value is famously small, recorded as \( 6.67 \times 10^{-11} \text{Nm}^2/\text{kg}^2 \). This tiny number highlights why the gravitational force is quite weak compared to other fundamental forces, like electromagnetism.

• The gravitational constant is key in the formula \( F = G \frac{m_1 m_2}{d^2} \), which calculates the gravitational force \( F \) between two masses.
• It reveals how much force two 1-kilogram masses would exert on each other if they were 1 meter apart.

This constant is a bridge that links mass, distance, and force within the gravitational framework. Without \( G \), we wouldn't be able to quantify or predict gravitational attraction between objects.

Distance and Force Relationship

One of the fascinating aspects of universal gravitation is the relationship between distance and force. Gravity behaves in a very specific way: it's inversely proportional to the square of the distance between two masses. In simpler terms, as the distance between two objects (\( d \)) increases, the gravitational force (\( F \)) decreases, and it does so quickly.
The force drops off according to the square of the distance: if you double the distance, the force will become one-fourth its original value:

• This means \( F \propto \frac{1}{d^2} \).
• For example, if two objects are 0.1 meters apart, reducing their distance to 0.05 meters increases the gravitational attraction.

Understanding this relationship helps us comprehend why celestial bodies, like planets, can exert significant gravitational pull despite the vast spaces between them.

Mass Product in Gravitation

Mass is a central player in the gravitational dance. The gravitational force between two objects is directly proportional to the product of their masses \( m_1 \times m_2 \). This means the heavier the objects, the stronger the gravitational pull between them.
Consider this:

• Larger masses exert greater gravitational force, impacting how planets orbit the sun and moons orbit planets.
• The relationship is given by \( F \propto m_1 m_2 \). So, if you double one mass, the force doubles.

Understanding the role of mass product in gravity helps explain why Earth has more gravitational pull than the Moon. It's simply because Earth is much more massive.
This concept is essential for calculating gravitational forces in both small-scale phenomena (like falling objects) and large-scale phenomena (like planetary motion).