Question

In one dimension, the magnitude of the gravitational force of attraction between a particle of mass \(m_{1}\) and one of mass \(m_{2}\) is given by $$F_{x}(x)=G \frac{m_{1} m_{2}}{x^{2}}$$ where \(G\) is a constant and \(x\) is the distance between the particles. (a) What is the potential energy function \(U(x) ?\) Assume that \(U(x) \rightarrow 0\) as \(x \rightarrow \infty,(b)\) How much work is required to increase the separation of the particles from \(x=x_{1}\) to \(x=\) \(x_{1}+d ?\)

Short answer

The potential energy function is \(U(x)=-\frac{Gm_1m_2}{x}\). And the work done to increase the separation of particles from \(x=x_1\) to \(x=x_1+d\) is \(W = U(x_1+d) - U(x_1) = \[-\frac{Gm_1m_2}{x_1+d} + \frac{Gm_1m_2}{x_1}\]\)

Step-by-step solution

Identifying the Potential Energy Function

From the physics of gravitation, we know that the potential energy in a gravity field is given by \(-\frac{Gm_1m_2}{r}\). Since, gravity is always attractive and the force of gravity does work on the particles to decrease the separation. Therefore, the potential energy is negative. Also, it is customary to choose the potential energy to be 0 at infinite separation. So, in this case the potential energy function will be \(U(x)=-\frac{Gm_1m_2}{x}\)

Calculating Work Done

To determine the work done, we need to calculate the potential energy difference between two positions. Work is defined as the change in kinetic energy and also we know from the conservation laws of physics that total energy (kinetic + potential) in a closed system is conserved. Hence, work done to move the particles from \(x = x_1\) to \(x = x_1+d\) is: \(W = U(x_1+d) - U(x_1) = \[-\frac{Gm_1m_2}{x_1+d} + \frac{Gm_1m_2}{x_1}\]\)

Key concepts

Gravitational Force

Gravitational force is an invisible force that acts between two masses in the universe. It's one of the four fundamental forces in physics and can be explained using a very simple formula. This force is responsible for the attraction between objects with mass, like the Earth and the Moon.

The formula for gravitational force is given by Newton's formula:

• \( F = G \frac{m_1 m_2}{r^2} \)
  Here, \( F \) represents the gravitational force.
• \( G \) is the gravitational constant, a very small number, which tells us how weak gravitational attraction is compared to other forces.
• \( m_1 \) and \( m_2 \) are the masses of the objects.
• And \( r \) is the distance between the centers of the two masses.

Why Gravitational Force is Important:
- It is the reason why we stay grounded on Earth and why the planets orbit the Sun.
- While it is weaker than electromagnetic and nuclear forces, it acts over much larger distances and thus dominates at astronomical scales.

Work Done

Work is a concept that helps us understand how energy is transferred from one system to another. When you apply a force to move an object, it results in work being done on that object.

In the context of gravitational forces, work done can be understood through the change in potential energy between two points. When two masses, say planets, are moved apart (or together), work is done due to gravitational force.

Here's how to calculate work done if two masses change their separation:

• The formula for work done, \( W \), is the difference in potential energy between two positions.
• \( W = U(x_2) - U(x_1) \)
• Where \( U(x) = -\frac{Gm_1m_2}{x} \)
• This equation tells us that the work done depends on initial and final distances \( x_1 \) and \( x_2 \), respectively.

Why Calculating Work Done Matters:- It's a tangible way of quantifying how much energy is needed to move objects under gravitational attraction.
- It helps in understanding things like satellite launches where massive amounts of work are required to overcome Earth's gravity.

Newton's Law of Gravitation

Isaac Newton proposed the Law of Gravitation in the late 17th century. It revolutionized the way we understand the universe and how bodies interact with each other.

Newton's Law of Gravitation states that every point mass attracts every other point mass with a force along the line intersecting both points. This force is proportional to the product of their masses and inversely proportional to the square of the distance between them.

The governing equation is:

• \( F = G \frac{m_1 m_2}{r^2} \)
• Here, \( G \) is known as the gravitational constant and is approximately \( 6.674 \, \times 10^{-11} \, \text{Nm}^2/\text{kg}^2 \).
• \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between the centers of the two masses.

Implications of Newton's Law:
- This law allows us to predict planetary motions and explains the tides, orbits, and even phenomena like black holes.
- It sets the foundation for classical mechanics and later influenced Einstein's theory of general relativity.
- The law helps engineers and scientists in fields like astrophysics and aerospace, providing a groundwork for calculations about both the Earth and the cosmos.