What is the dimension of gravitational constant?

Understanding the Dimension of Gravitational Constant

The gravitational constant, usually denoted by \( G \), is a fundamental physical constant that appears in Newton's Law of Universal Gravitation. This law describes the force of gravitational attraction between two bodies.  

Newton's Law of Universal Gravitation

Newton's law states that the force (\( F \)) between two point masses (\( m_1 \) and \( m_2 \)) is directly proportional to the product of their masses and inversely proportional to the square of the distance (\( r \)) between their centers. Mathematically, this is expressed as:

\[ F = G \frac{m_1 m_2}{r^2} \]

Here, \( G \) is the constant of proportionality, known as the universal gravitational constant.

Deriving the Dimension of \( G \)

To find the dimension of \( G \), we can rearrange the formula to isolate \( G \):

\[ G = \frac{F r^2}{m_1 m_2} \]

Now, let's determine the dimensions of each quantity in this rearranged equation.

• Dimension of Force (\( F \)): Force is defined by Newton's second law (\( F = ma \), mass × acceleration). The dimension of mass is \( [\text{M}] \) and the dimension of acceleration is \( [\text{LT}^{-2}] \) (length per time squared). Therefore, the dimension of force is \( [\text{MLT}^{-2}] \).
• Dimension of Distance (\( r \)): Distance is a length, so its dimension is \( [\text{L}] \). Since it is \( r^2 \), its dimension is \( [\text{L}^2] \).
• Dimension of Mass (\( m_1 \) and \( m_2 \)): The dimension of mass is \( [\text{M}] \). Since we have the product \( m_1 m_2 \), its dimension is \( [\text{M} \times \text{M}] = [\text{M}^2] \).

Substituting Dimensions

Now, substitute these dimensions into the equation for \( G \):

\[ [G] = \frac{[\text{MLT}^{-2}] [\text{L}^2]}{[\text{M}^2]} \]

Simplifying the Dimensions

Let's simplify the expression for the dimension of \( G \):

\[ [G] = \frac{[\text{MLT}^{-2} \text{L}^2]}{[\text{M}^2]} = \frac{[\text{ML}^{1+2}\text{T}^{-2}]}{[\text{M}^2]} = \frac{[\text{ML}^3\text{T}^{-2}]}{[\text{M}^2]} \]

Now, bring the mass dimension from the denominator to the numerator by changing the sign of the exponent:

\[ [G] = [\text{M}^{1-2}\text{L}^3\text{T}^{-2}] = [\text{M}^{-1}\text{L}^3\text{T}^{-2}] \]

Thus, the dimension of the gravitational constant \( G \) is \( [\text{M}^{-1}\text{L}^3\text{T}^{-2}] \).

Comparison with Options

Let's compare the derived dimension \( [\text{M}^{-1}\text{L}^3\text{T}^{-2}] \) with the given options:

• Option 1: \( [\text{ML}^3\text{T}^{-2}] \)
• Option 2: \( [\text{M}^{-1}\text{L}^3\text{T}^{-2}] \)
• Option 3: \( [\text{M}^{2}\text{L}^{-2}\text{T}^{-2}] \)
• Option 4: \( [\text{M}^{2}\text{L}^{-1}\text{T}^{-2}] \)

The derived dimension matches Option 2.

Revision Table: Key Physical Quantities and their Dimensions

Additional Information: Importance of Dimensions

Dimensional analysis is a powerful tool in physics. It helps in:

• Checking the consistency of physical equations. If an equation is correct, the dimensions on both sides must be the same.
• Deriving relationships between physical quantities. Sometimes, the form of an equation can be predicted based on the dimensions of the quantities involved.
• Understanding the physical nature of quantities. Dimensions tell us what fundamental quantities (mass, length, time, etc.) make up a derived quantity.

The gravitational constant \( G \) is a universal constant, meaning its value is the same everywhere in the universe. Its value is approximately \( 6.674 \times 10^{-11} \, \text{N} (\text{m}/\text{kg})^2 \) or \( 6.674 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \). The units \( \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \) directly correspond to the dimensions \( [\text{L}^3 \text{M}^{-1} \text{T}^{-2}] \), which is \( [\text{M}^{-1}\text{L}^3\text{T}^{-2}] \) when written in the standard MLA format.

Understanding the Gravitational Constant Dimensional Formula

The question asks about the dimension of the gravitational constant. To find the gravitational constant dimensional formula, we need to use the formula for the gravitational force between two objects, as described by Newton's Law of Universal Gravitation.

Newton's Law of Gravitation and the Constant G

Newton's Law states that the force ($F$) between two point masses ($m_1$ and $m_2$) separated by a distance ($r$) is given by:

$$\mathrm{F} = \mathrm{G} \frac{\mathrm{m}_1 \mathrm{m}_2}{\mathrm{r}^2}$$

Here, G is the gravitational constant. Our goal is to find the dimensions of this constant G. We can rearrange the formula to isolate G: 

$$\mathrm{G} = \frac{\mathrm{F} \mathrm{r}^2}{\mathrm{m}_1 \mathrm{m}_2}$$

Determining the Dimensions

To find the dimension of the gravitational constant, we need to know the dimensions of the quantities on the right side of the equation:

• Force (F)
• Distance (r)
• Mass ($m_1$ and $m_2$)

The fundamental dimensions are Mass (M), Length (L), and Time (T).

Let's list the dimensions of each term:

• The dimension of Mass (m) is $[M^1]$. Since we have two masses ($m_1$ and $m_2$), their combined dimension in the denominator will be $[M^1] \times [M^1] = [M^2]$.
• The dimension of Distance (r) is $[L^1]$. Since the distance is squared ($r^2$), its dimension is $[L^1]^2 = [L^2]$.
• The dimension of Force (F) is derived from Newton's second law, $F = ma$ (mass × acceleration). The dimension of mass is $[M^1]$ and the dimension of acceleration is $[L^1 T^{-2}]$ (since acceleration is change in velocity over time, and velocity is change in displacement over time). Therefore, the dimension of Force is $[M^1] \times [L^1 T^{-2}] = [M^1 L^1 T^{-2}]$.

We can summarize these dimensions in a table:

Calculating the Dimensional Formula for Gravitational Constant

Now substitute these dimensions into the equation for G:

$$\mathrm{[G]} = \frac{\mathrm{[F]} \mathrm{[r^2]}}{\mathrm{[m_1]} \mathrm{[m_2]}}$$

$$\mathrm{[G]} = \frac{\mathrm{[M^1 L^1 T^{-2}]} \mathrm{[L^2]}}{\mathrm{[M^1]} \mathrm{[M^1]}}$$

Combine the terms in the numerator and denominator:

$$\mathrm{[G]} = \frac{\mathrm{[M^1 L^{1+2} T^{-2}]}}{\mathrm{[M^{1+1}]}}$$

$$\mathrm{[G]} = \frac{\mathrm{[M^1 L^3 T^{-2}]}}{\mathrm{[M^2]}}$$

Now, move the mass dimension from the denominator to the numerator by changing the sign of its exponent:

$$\mathrm{[G]} = \mathrm{[M^1 M^{-2} L^3 T^{-2}]}$$

Combine the Mass terms:

$$\mathrm{[G]} = \mathrm{[M^{1-2} L^3 T^{-2}]}$$

$$\mathrm{[G]} = \mathrm{[M^{-1} L^3 T^{-2}]}$$

This is the dimensional formula for gravitational constant.

Comparing with Options to find the Dimension of Gravitational Constant

Let's compare our derived gravitational constant dimensional formula with the given options:

• Option 1: $[M^1 L^3 T^{-2}]$ - This does not match.
• Option 2: $[M^{-1} L^3 T^{-2}]$ - This matches our derived dimensional formula of gravitational constant.
• Option 3: $[M^2 L^{-2} T^{-2}]$ - This does not match.
• Option 4: $[M^2 L^{-1} T^{-2}]$ - This does not match.

Therefore, the correct dimension of gravitational constant is $[M^{-1} L^3 T^{-2}]$. Understanding the dimensions of gravitational constant is crucial in physics.

The calculation confirms that the gravitational constant dimensional formula is $[M^{-1} L^3 T^{-2}]$. This approach of breaking down a physical quantity into its fundamental dimensions is how we determine the dimensions of gravitational constant or any other physical constant.