On earth, the value of G = 6.67 × 10 -11 Nm 2kg -2 . What is the value on moon, where acceleration due to gravity is nearly one - sixth than that of earth?
6.67 × 10 -11 Nm 2kg -2
Understanding the Gravitational Constant G
The question asks for the value of the gravitational constant (G) on the Moon, given its value on Earth and the relationship between acceleration due to gravity on the Earth and the Moon.
Let's break down the key concepts involved:
- Gravitational Constant (G): This is a fundamental constant in physics. It appears in Newton's law of universal gravitation, which describes the force of attraction between any two objects with mass. The formula for the gravitational force (F) between two objects with masses $m_1$ and $m_2$ separated by a distance $r$ is given by:
$$F = G \frac{m_1 m_2}{r^2}$$
- Acceleration due to gravity (g): This is the acceleration experienced by an object due to the gravitational pull of a celestial body like Earth or the Moon. Its value depends on the mass and radius of the celestial body. The acceleration due to gravity 'g' on the surface of a planet with mass M and radius R is given by:
$$g = \frac{GM}{R^2}$$
The question states that the acceleration due to gravity on the Moon is nearly one-sixth than that of Earth ($g_{moon} \approx \frac{1}{6} g_{earth}$). This fact is correct, but it relates to 'g', not 'G'.
The crucial point to understand is the difference between 'G' and 'g'.
- 'g' (acceleration due to gravity) is not a universal constant. It varies depending on the celestial body (its mass and radius) and even slightly on location on that body.
- 'G' (the gravitational constant) is a universal constant. This means its value is the same everywhere in the universe, regardless of the location, the masses of the objects involved, or the distance between them.
The value of G on Earth is given as $6.67 \times 10^{-11} \text{ Nm}^2\text{kg}^{-2}$. Since G is a universal constant, its value does not change when you go from Earth to the Moon, or anywhere else in the universe.
Therefore, the value of the gravitational constant G on the Moon is exactly the same as its value on Earth.
The value of G on the Moon is $6.67 \times 10^{-11} \text{ Nm}^2\text{kg}^{-2}$.
Let's compare this with the given options:
- Option 1: $40.01 \times 10^{-11} \text{ Nm}^2\text{kg}^{-2}$
- Option 2: $1.11 \times 10^{-11} \text{ Nm}^2\text{kg}^{-2}$
- Option 3: $6.67 \times 10^{-11} \text{ Nm}^2\text{kg}^{-2}$
- Option 4: $2.67 \times 10^{-11} \text{ Nm}^2\text{kg}^{-2}$
Option 3 matches the calculated value based on the understanding that G is a universal constant.
Key Differences: Gravitational Constant (G) vs. Acceleration due to gravity (g)
It is important not to confuse G and g. Here's a summary:
| Feature | Gravitational Constant (G) | Acceleration due to gravity (g) |
|---|---|---|
| Nature | Universal Constant | Depends on celestial body (variable) |
| Value on Earth | $6.67 \times 10^{-11} \text{ Nm}^2\text{kg}^{-2}$ | Approximately $9.8 \text{ m/s}^2$ |
| Value on Moon | $6.67 \times 10^{-11} \text{ Nm}^2\text{kg}^{-2}$ (Same as Earth) | Approximately $1.62 \text{ m/s}^2$ (About 1/6th of Earth's g) |
| Involved in | Newton's Law of Universal Gravitation ($F = G \frac{m_1 m_2}{r^2}$) | Force on an object due to gravity ($F = mg$) or motion under gravity |
The fact that $g_{moon}$ is different from $g_{earth}$ is due to the Moon having a different mass and radius compared to Earth. This does not affect the fundamental constant G.
Therefore, the value of G remains constant.
Revision Table: Gravitational Concepts
| Concept | Symbol | Nature | Approximate Value (Earth) |
|---|---|---|---|
| Gravitational Constant | G | Universal Constant | $6.67 \times 10^{-11} \text{ Nm}^2\text{kg}^{-2}$ |
| Acceleration due to gravity | g | Variable (depends on location) | $9.8 \text{ m/s}^2$ |
Additional Information on the Gravitational Constant
The value of the gravitational constant G was first experimentally determined by Henry Cavendish in 1798 using a torsion balance. His experiment provided the first accurate measurement of G, which in turn allowed the calculation of the Earth's mass.
The constant G is one of the fundamental constants of nature. Its precise value is important in many areas of physics and astronomy, including calculating the mass of planets and stars, understanding the motion of celestial bodies, and studying cosmology.
Despite being a fundamental constant, G is one of the least precisely known fundamental constants, mainly due to the weakness of the gravitational force between laboratory-sized objects.
Understanding that G is universal is key to solving problems involving gravity in different locations in space.
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