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 (M) 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
Assume that mass (M) can be expressed in terms of pressure (P), velocity of light (C), and acceleration due to gravity (g) as:
\[ M \propto P^x C^y g^z \]

### Step 2: Write the dimensional formulas
The dimensional formulas for the quantities are:
- Pressure (P) has the dimensional formula:
\[ [P] = M L^{-1} T^{-2} \]
- Velocity of light (C) has the dimensional formula:
\[ [C] = L T^{-1} \]
- Acceleration due to gravity (g) has the dimensional formula:
\[ [g] = L T^{-2} \]

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

### Step 4: Expand the equation
Expanding this, we get:
\[ [M] = M^x L^{-x} T^{-2x} L^y T^{-y} L^z T^{-2z} \]
Combining the terms, we have:
\[ [M] = M^x L^{-x + y + z} T^{-2x - y - 2z} \]

### Step 5: Set up equations for dimensional consistency
For dimensional consistency, we equate the powers of M, L, and T:
1. For M:
\[ x = 1 \]
2. For L:
\[ -x + y + z = 0 \]
3. For T:
\[ -2x - y - 2z = 0 \]

### Step 6: Solve the equations
From the first equation, we have:
\[ x = 1 \]

Substituting \( x = 1 \) into the second equation:
\[ -1 + y + z = 0 \]
This simplifies to:
\[ y + z = 1 \quad \text{(Equation 1)} \]

Now substituting \( x = 1 \) into the third equation:
\[ -2(1) - y - 2z = 0 \]
This simplifies to:
\[ -2 - y - 2z = 0 \]
or
\[ y + 2z = -2 \quad \text{(Equation 2)} \]

### Step 7: Solve the system of equations
Now we can solve the two equations:
1. \( y + z = 1 \)
2. \( y + 2z = -2 \)

From Equation 1, we can express \( y \) in terms of \( z \):
\[ y = 1 - z \]

Substituting this into Equation 2:
\[ (1 - z) + 2z = -2 \]
This simplifies to:
\[ 1 + z = -2 \]
Thus,
\[ z = -3 \]

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

### Step 8: Final values
We have found:
- \( x = 1 \)
- \( y = 4 \)
- \( z = -3 \)

### Step 9: Write the dimensional formula for mass
Thus, the dimensional formula for mass (M) in terms of P, C, and g is:
\[ M = P^1 C^4 g^{-3} \]

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

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