Understanding the Universal Gravitational Constant G

The question asks about the nature of the quantity \(G\) in the formula for gravitational force, which is given by Newton's Law of Gravitation:

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

In this formula:

• \(F\) is the magnitude of the gravitational force between the two objects.
• \(m_1\) and \(m_2\) are the masses of the two objects.
• \(r\) is the distance between the centers of the two objects.
• \(G\) is the universal gravitational constant.

Let's analyze each statement provided in the options to determine which one is true for the quantity \(G\).

Analyzing the Statements about Quantity G

Here is a breakdown of each statement:

1. The quantity G depends on the local value of g, acceleration due to gravity

   This statement is incorrect. The acceleration due to gravity (\(g\)) on the surface of a planet depends on the planet's mass (\(M\)) and radius (\(R\)), and also on the universal gravitational constant \(G\), according to the formula \(g = \frac{GM}{R^2}\). While \(g\) depends on \(G\), \(G\) itself does not depend on \(g\). \(G\) is a fundamental constant of nature.

2. The quantity G is greater at the surface of the Earth

   This statement is incorrect. The quantity \(G\) is a constant value throughout the universe, regardless of location, including the surface of the Earth. It does not change with position or environment.

3. The quantity G is used only when earth is one of the two masses

   This statement is incorrect. The universal gravitational constant \(G\) is used to calculate the gravitational force between any two objects with mass, whether they are planets, stars, objects on Earth, or any other massive bodies. It is not specific to calculations involving Earth.

4. The quantity G is a universal constant

   This statement is correct. The quantity \(G\) is known as the universal gravitational constant. It has a fixed value that is the same everywhere in the universe, regardless of the nature, size, or location of the masses involved, or the medium between them. Its experimentally determined value is approximately \(6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2\).

Conclusion on the Nature of Quantity G

Based on the analysis of each statement, the only true statement about the quantity \(G\) in Newton's Law of Gravitation is that it is a universal constant. This means its value is fixed and does not change based on factors like location, surrounding environment, or the specific masses involved.

Revision Table: Key Facts about the Universal Gravitational Constant G

Additional Information: Gravitation Concepts

Understanding the universal gravitational constant \(G\) is crucial when studying gravitation. Here are some related points:

• Newton's Law of Gravitation: This fundamental law describes the attractive force between any two objects with mass. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• Difference between G and g:
  • \(G\) is the universal gravitational constant, a fixed value applicable throughout the universe.
  • \(g\) is the acceleration due to gravity, which is the acceleration experienced by an object due to the gravitational pull of a massive body (like a planet). The value of \(g\) varies depending on the mass and size of the planet, altitude, and local geological features. For Earth's surface, \(g\) is approximately \(9.8 \, \text{m/s}^2\), but it is not constant across the Earth's surface or at different heights.
• Importance of G: The constant \(G\) allows us to calculate the magnitude of the gravitational force if we know the masses and the distance between them. It is a fundamental constant like the speed of light or Planck's constant.