The dimensional formula of angular momentum is equal to that of the

Options

A. force x time
B. power x time
C. work x time
D. momentum x time

Answer

To find the dimensional formula of angular momentum and determine which of the given options it is equal to, we can follow these steps:

### Step 1: Understand the Definition of Angular Momentum
Angular momentum (L) is defined as the product of the moment of inertia (I) and angular velocity (ω). It can also be expressed as the cross product of the position vector (r) and linear momentum (p).

### Step 2: Write the Formula for Angular Momentum
The formula for angular momentum can be expressed as:
\[ L = r \times p \]
Where:
- \( r \) is the position vector (distance),
- \( p \) is the linear momentum.

### Step 3: Determine the Dimensions of Linear Momentum
Linear momentum (p) is given by:
\[ p = m \cdot v \]
Where:
- \( m \) is mass (M),
- \( v \) is velocity (L/T).

Thus, the dimensional formula for linear momentum is:
\[ [p] = M \cdot \frac{L}{T} = MLT^{-1} \]

### Step 4: Determine the Dimensions of Angular Momentum
Now, substituting the dimensions of linear momentum into the angular momentum formula:
\[ L = r \cdot p \]
The dimension of distance (r) is L. Therefore:
\[ [L] = [r] \cdot [p] = L \cdot (MLT^{-1}) = ML^2T^{-1} \]

### Step 5: Compare with Given Options
Now we need to check which of the given options has the same dimensional formula as \( ML^2T^{-1} \).

1. **Option A: Force × Time**
- Dimension of Force: \( [F] = MLT^{-2} \)
- Therefore, \( [F \cdot t] = MLT^{-2} \cdot T = ML T^{-1} \) (Not equal to \( ML^2T^{-1} \))

2. **Option B: Power × Time**
- Power is defined as work done per unit time: \( [P] = \frac{[W]}{T} \)
- Dimension of Work: \( [W] = F \cdot d = MLT^{-2} \cdot L = ML^2T^{-2} \)
- Therefore, \( [P] = \frac{ML^2T^{-2}}{T} = ML^2T^{-3} \)
- Thus, \( [P \cdot t] = ML^2T^{-3} \cdot T = ML^2T^{-2} \) (Not equal to \( ML^2T^{-1} \))

3. **Option C: Work × Time**
- From above, \( [W] = ML^2T^{-2} \)
- Therefore, \( [W \cdot t] = ML^2T^{-2} \cdot T = ML^2T^{-1} \) (This matches with \( ML^2T^{-1} \))

4. **Option D: Momentum × Time**
- Dimension of Momentum: \( [p] = MLT^{-1} \)
- Therefore, \( [p \cdot t] = MLT^{-1} \cdot T = ML \) (Not equal to \( ML^2T^{-1} \))

### Conclusion
The dimensional formula of angular momentum \( ML^2T^{-1} \) is equal to that of **Option C: Work × Time**.