Among the following quantities , the quantity whose dimension is independent of mass and length is

Options

A. Energy
B. Thrust
C. Momentum
D. Angular velocity

Answer

To solve the problem of identifying the quantity whose dimension is independent of mass and length, we will analyze each of the given quantities step by step.

### Step 1: Identify the quantities
The quantities we need to analyze are:
1. Force
2. Energy
3. Thrust
4. Momentum
5. Angular velocity

### Step 2: Analyze each quantity

1. **Force (F)**:
- The formula for force is \( F = m \cdot a \), where \( m \) is mass and \( a \) is acceleration.
- The dimension of force is:
\[ [F] = [M][L][T^{-2}] = M L T^{-2} \]
- This dimension is dependent on mass (M) and length (L).

2. **Energy (E)**:
- The formula for energy is \( E = F \cdot d \), where \( d \) is displacement.
- The dimension of energy is:
\[ [E] = [F][L] = (M L T^{-2})[L] = M L^2 T^{-2} \]
- This dimension is also dependent on mass (M) and length (L).

3. **Thrust (T)**:
- Thrust is essentially a force, so it has the same dimension as force.
- Thus, the dimension of thrust is:
\[ [T] = M L T^{-2} \]
- This dimension is dependent on mass (M) and length (L).

4. **Momentum (p)**:
- The formula for momentum is \( p = m \cdot v \), where \( v \) is velocity.
- The dimension of momentum is:
\[ [p] = [M][L][T^{-1}] = M L T^{-1} \]
- This dimension is dependent on mass (M) and length (L).

5. **Angular Velocity (ω)**:
- Angular velocity is defined as the rate of change of angular displacement, which can be expressed as:
\[ \omega = \frac{v}{r} \]
where \( v \) is velocity and \( r \) is radius.
- The dimension of angular velocity is:
\[ [\omega] = \frac{[L][T^{-1}]}{[L]} = [T^{-1}] \]
- This dimension is independent of mass (M) and length (L).

### Step 3: Conclusion
Among the quantities analyzed, the only quantity whose dimension is independent of mass and length is **Angular Velocity (ω)**.

### Final Answer:
The quantity whose dimension is independent of mass and length is **Angular Velocity (ω)**.

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