If force (F), length(L) and time (T) as chosen as the fundamental quantities, then what would be the dimensional formula for the density?

Answer

To find the dimensional formula for density when force (F), length (L), and time (T) are chosen as fundamental quantities, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Definition of Density**:
Density (D) is defined as mass (M) per unit volume (V). Mathematically, it is expressed as:
\[ D = \frac{M}{V} \]
The volume (V) can be expressed in terms of length (L) as:
\[ V = L^3 \]
Therefore, the dimensional formula for density can be rewritten as:
\[ D = \frac{M}{L^3} \]

2. **Express Mass in Terms of Force**:
Force (F) is defined as mass (M) times acceleration (A). The dimensional formula for acceleration is:
\[ A = \frac{L}{T^2} \]
Thus, the dimensional formula for force can be expressed as:
\[ F = M \cdot A = M \cdot \frac{L}{T^2} \implies M = \frac{F \cdot T^2}{L} \]

3. **Substituting Mass in the Density Formula**:
Now substituting the expression for mass (M) into the density formula:
\[ D = \frac{M}{L^3} = \frac{\frac{F \cdot T^2}{L}}{L^3} = \frac{F \cdot T^2}{L^4} \]

4. **Finding the Dimensional Formula for Density**:
Rearranging gives:
\[ D = F \cdot L^{-4} \cdot T^2 \]
Now we need to express the dimensional formula for force (F):
\[ F = M \cdot L \cdot T^{-2} \]
Thus, substituting this into the density formula:
\[ D = (M \cdot L \cdot T^{-2}) \cdot L^{-4} \cdot T^2 \]

5. **Combining the Dimensions**:
This simplifies to:
\[ D = M^1 \cdot L^{1-4} \cdot T^{-2 + 2} = M^1 \cdot L^{-3} \cdot T^0 \]

6. **Final Dimensional Formula for Density**:
Therefore, the final dimensional formula for density is:
\[ D = M^1 \cdot L^{-3} \cdot T^0 \]
This can be expressed as:
\[ [D] = M^1 L^{-3} \]

### Summary:
The dimensional formula for density when force, length, and time are the fundamental quantities is:
\[ [D] = M^1 L^{-3} \]