If pressure P, velocity of light c and acceleration due to gravity g are chosen as fundamental units, then dimensional formula of mass is

Options

A. `Pc^(3)g^(-4)`
B. `Pc^(-4) g^(3)`
C. `Pc^(4) g^(-3)`
D. `Pc^(4)g^(3)`

Answer

To find the dimensional formula of mass when pressure (P), velocity of light (c), and acceleration due to gravity (g) are chosen as fundamental units, we can follow these steps:

### Step 1: Write the relationship for mass
Assume that mass (M) can be expressed in terms of the fundamental quantities P, c, and g:
\[ M \propto P^x \cdot c^y \cdot g^z \]

### Step 2: Write the dimensional formulas for each quantity
- The dimensional formula for pressure (P) is:
\[ [P] = \frac{F}{A} = \frac{MLT^{-2}}{L^2} = M L^{-1} T^{-2} \]

- The dimensional formula for velocity of light (c) is:
\[ [c] = \frac{L}{T} \]

- The dimensional formula for acceleration due to gravity (g) is:
\[ [g] = \frac{L}{T^2} \]

### Step 3: Substitute the dimensional formulas into the mass equation
Substituting the dimensional formulas into our mass equation gives:
\[ [M] = [P]^x \cdot [c]^y \cdot [g]^z \]
\[ [M] = (M L^{-1} T^{-2})^x \cdot (L T^{-1})^y \cdot (L T^{-2})^z \]

### Step 4: Expand the equation
Expanding the right side:
\[ [M] = M^x \cdot L^{-x} \cdot T^{-2x} \cdot L^y \cdot T^{-y} \cdot L^z \cdot T^{-2z} \]
Combining the terms:
\[ [M] = M^x \cdot L^{-x + y + z} \cdot T^{-2x - y - 2z} \]

### Step 5: Compare dimensions
Now, we compare the dimensions on both sides:
- For mass (M): \( x = 1 \)
- For length (L): \( -x + y + z = 0 \)
- For time (T): \( -2x - y - 2z = 0 \)

### Step 6: Solve the equations
From \( x = 1 \):
1. Substitute \( x = 1 \) into the length equation:
\[ -1 + y + z = 0 \]
\[ y + z = 1 \] (Equation 1)

2. Substitute \( x = 1 \) into the time equation:
\[ -2(1) - y - 2z = 0 \]
\[ -2 - y - 2z = 0 \]
\[ y + 2z = -2 \] (Equation 2)

### Step 7: Solve the system of equations
From Equation 1:
\[ y = 1 - z \]

Substituting into Equation 2:
\[ (1 - z) + 2z = -2 \]
\[ 1 + z = -2 \]
\[ z = -3 \]

Now substituting \( z = -3 \) back into Equation 1:
\[ y + (-3) = 1 \]
\[ y = 4 \]

### Step 8: Write the final dimensional formula
Now we have:
- \( x = 1 \)
- \( y = 4 \)
- \( z = -3 \)

Thus, the dimensional formula for mass is:
\[ M = P^1 \cdot c^4 \cdot g^{-3} \]

### Final Answer
The dimensional formula of mass is:
\[ M = P \cdot c^4 \cdot g^{-3} \]