If velocity, force and time are taken as the fundamental quantities, them using dimensional analysis choose the correct dimensional formula for mass among the following. [K is a dimensionless constant]

Options

A. `Q=Kv^(-1)FT`
B. `Q=Kv^(3)FT^(2)`
C. `Q=2Kv^(-2)FT`
D. `Q=3Kv^(2)Ft`

Answer

To find the dimensional formula for mass using velocity, force, and time as fundamental quantities, we can follow these steps:

### Step 1: Define the Dimensional Formula
Assume the dimensional formula for mass \( Q \) can be expressed in terms of velocity \( V \), force \( F \), and time \( T \) as:
\[ Q = K \cdot V^a \cdot F^b \cdot T^c \]
where \( K \) is a dimensionless constant, and \( a, b, c \) are the powers we need to determine.

### Step 2: Write the Dimensional Formulas
The dimensional formulas for the quantities involved are:
- Velocity \( V \): \( [V] = L^1 T^{-1} \)
- Force \( F \): \( [F] = M^1 L^1 T^{-2} \)
- Time \( T \): \( [T] = T^1 \)

### Step 3: Substitute the Dimensional Formulas
Substituting the dimensional formulas into our equation gives:
\[ Q = K \cdot (L^1 T^{-1})^a \cdot (M^1 L^1 T^{-2})^b \cdot (T^1)^c \]

### Step 4: Combine the Dimensions
Now, combining the dimensions, we have:
\[ Q = K \cdot M^b \cdot L^{a+b} \cdot T^{-a - 2b + c} \]

### Step 5: Equate to the Dimensional Formula for Mass
The dimensional formula for mass is:
\[ [M] = M^1 L^0 T^0 \]
This gives us the following equations by equating the powers of \( M \), \( L \), and \( T \):
1. \( b = 1 \) (for \( M \))
2. \( a + b = 0 \) (for \( L \))
3. \( -a - 2b + c = 0 \) (for \( T \))

### Step 6: Solve the Equations
From equation (1), we have:
\[ b = 1 \]
Substituting \( b = 1 \) into equation (2):
\[ a + 1 = 0 \implies a = -1 \]
Substituting \( a = -1 \) and \( b = 1 \) into equation (3):
\[ -(-1) - 2(1) + c = 0 \implies 1 - 2 + c = 0 \implies c = 1 \]

### Step 7: Write the Dimensional Formula for Mass
Now substituting the values of \( a \), \( b \), and \( c \) back into the original equation for \( Q \):
\[ Q = K \cdot V^{-1} \cdot F^1 \cdot T^1 \]
Thus, the dimensional formula for mass in terms of velocity, force, and time is:
\[ Q = K \cdot V^{-1} \cdot F \cdot T \]

### Final Answer
The correct dimensional formula for mass is:
\[ Q = K \cdot V^{-1} \cdot F \cdot T \]