Taking `"force" F, "length" L and "time" T` to be the fundamental equations , find the dimensions of
(a) density (b) pressure (c ) momentum and (d) energy

Explanation

To find the dimensions of density, pressure, momentum, and energy using force (F), length (L), and time (T) as fundamental quantities, we can follow these steps:

### Step 1: Find the dimension of Density
Density is defined as mass per unit volume. The formula for density (ρ) is:
\[ \rho = \frac{m}{V} \]
where \( m \) is mass and \( V \) is volume.

1. **Find the dimension of mass (m)**:
From Newton's second law, we know:
\[ F = m \cdot a \]
where \( a \) (acceleration) can be expressed as:
\[ a = \frac{L}{T^2} \]
Therefore, we can rearrange the equation to find mass:
\[ m = \frac{F}{a} = \frac{F}{\frac{L}{T^2}} = \frac{F \cdot T^2}{L} \]

2. **Substituting the dimension of mass into the density formula**:
The dimension of volume (V) is:
\[ V = L^3 \]
Thus, the dimension of density becomes:
\[ [\rho] = \frac{[m]}{[V]} = \frac{\frac{F \cdot T^2}{L}}{L^3} = \frac{F \cdot T^2}{L^4} \]

### Step 2: Find the dimension of Pressure
Pressure is defined as force per unit area. The formula for pressure (P) is:
\[ P = \frac{F}{A} \]
where \( A \) is area.

1. **Find the dimension of area (A)**:
The dimension of area is:
\[ A = L^2 \]

2. **Substituting the dimension of force into the pressure formula**:
Thus, the dimension of pressure becomes:
\[ [P] = \frac{[F]}{[A]} = \frac{F}{L^2} \]

### Step 3: Find the dimension of Momentum
Momentum (p) is defined as the product of mass and velocity. The formula for momentum is:
\[ p = m \cdot v \]
where \( v \) (velocity) can be expressed as:
\[ v = \frac{L}{T} \]

1. **Substituting the dimension of mass and velocity into the momentum formula**:
Thus, the dimension of momentum becomes:
\[ [p] = [m] \cdot [v] = \left(\frac{F \cdot T^2}{L}\right) \cdot \left(\frac{L}{T}\right) = \frac{F \cdot T}{L} \]

### Step 4: Find the dimension of Energy
Energy (E) is defined as the work done, which is the product of force and displacement. The formula for energy is:
\[ E = F \cdot d \]
where \( d \) is displacement.

1. **Substituting the dimension of force and displacement into the energy formula**:
Thus, the dimension of energy becomes:
\[ [E] = [F] \cdot [d] = F \cdot L \]

### Summary of Dimensions:
- Density: \([ \rho ] = F \cdot T^2 \cdot L^{-4}\)
- Pressure: \([ P ] = F \cdot L^{-2}\)
- Momentum: \([ p ] = F \cdot T \cdot L^{-1}\)
- Energy: \([ E ] = F \cdot L\)