The Energy `(E)` angular momentum `(L)` and universal gravitational constant `(G)` are chosen as fundamental quantities. The dimensions of universal gravitational constant in the dimensional formula of Planks constant `(h)` is

Options

A. zero
B. `-1`
C. `(5)/(3)`
D. 1

Answer

To find the dimensions of the universal gravitational constant \( G \) in the dimensional formula of Planck's constant \( h \), we will follow these steps:

### Step 1: Write the dimensions of energy \( E \)
Energy is defined as the work done, which can be expressed as force times displacement. The dimension of force is given by:
\[ \text{Force} = \text{mass} \times \text{acceleration} = [M][L][T^{-2}] \]
Thus, the dimension of energy \( E \) is:
\[ [E] = [F][L] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]

### Step 2: Write the dimensions of angular momentum \( L \)
The angular momentum \( L \) is defined as the product of moment of inertia and angular velocity. Its dimension is:
\[ [L] = [M][L^2][T^{-1}] \]

### Step 3: Write the dimensions of the universal gravitational constant \( G \)
The universal gravitational constant \( G \) is defined in the context of Newton's law of gravitation:
\[ F = \frac{G m_1 m_2}{r^2} \]
From this, we can derive the dimensions of \( G \):
\[ [G] = \frac{[F][L^2]}{[M^2]} = \frac{[M][L][T^{-2}][L^2]}{[M^2]} = [M^{-1}][L^3][T^{-2}] \]

### Step 4: Write the dimensions of Planck's constant \( h \)
Planck's constant relates energy and frequency:
\[ E = hf \implies h = \frac{E}{f} \]
The dimension of frequency \( f \) is \( [T^{-1}] \). Therefore, the dimension of \( h \) is:
\[ [h] = \frac{[E]}{[T^{-1}]} = [M][L^2][T^{-2}] \cdot [T] = [M][L^2][T^{-1}] \]

### Step 5: Express \( h \) in terms of \( G \) and \( L \)
We can express the dimensions of \( h \) in terms of \( G \) and \( L \):
\[ [h] = [M^a][L^b][G^c] \]
Substituting the dimensions we found:
\[ [M][L^2][T^{-1}] = [M^a][L^b][M^{-1}][L^3][T^{-2}]^c \]
This leads to:
\[ [M^{1-a-c}][L^{2-b+3c}][T^{-1+2c}] \]

### Step 6: Set up the equations for the dimensions
Now we can set up the equations based on the dimensions:
1. \( 1 - a - c = 0 \) (for mass)
2. \( 2 - b + 3c = 0 \) (for length)
3. \( -1 + 2c = 0 \) (for time)

### Step 7: Solve the equations
From the third equation:
\[ 2c = 1 \implies c = \frac{1}{2} \]
Substituting \( c \) into the first equation:
\[ 1 - a - \frac{1}{2} = 0 \implies a = \frac{1}{2} \]
Substituting \( c \) into the second equation:
\[ 2 - b + 3 \cdot \frac{1}{2} = 0 \implies 2 - b + \frac{3}{2} = 0 \implies b = \frac{7}{2} \]

### Step 8: Conclusion
Thus, the dimensions of the universal gravitational constant \( G \) in the dimensional formula of Planck's constant \( h \) is:
\[ c = 0 \]