The product of energy and time is called action. Therefore dimensional formula for action is same as that for

Options

A. (a)power
B. (b)angular energy
C. (c)force `xx` velocity
D. (d)impulse `xx` distance

Answer

To find the dimensional formula for action, which is defined as the product of energy and time, we can follow these steps:

### Step 1: Write the formula for action
Action (S) is defined as:
\[ S = \text{Energy} \times \text{Time} \]

### Step 2: Determine the dimensional formula for energy
The dimensional formula for energy (E) is given by:
\[ [E] = [M][L^2][T^{-2}] \]
where:
- \( M \) is mass,
- \( L \) is length,
- \( T \) is time.

### Step 3: Determine the dimensional formula for time
The dimensional formula for time (T) is:
\[ [T] = [T^1] \]

### Step 4: Combine the dimensional formulas
Now, we can combine the dimensional formulas for energy and time:
\[ [S] = [E] \times [T] \]
Substituting the formulas we have:
\[ [S] = [M][L^2][T^{-2}] \times [T^1] \]

### Step 5: Simplify the expression
Now, we simplify the expression:
\[ [S] = [M][L^2][T^{-2 + 1}] = [M][L^2][T^{-1}] \]

### Step 6: Conclusion
Thus, the dimensional formula for action is:
\[ [S] = [M][L^2][T^{-1}] \]

### Step 7: Identify the equivalent physical quantity
Now, we need to find which physical quantity has the same dimensional formula. We can check the options provided in the question.

1. **Power**: \( [P] = [M][L^2][T^{-3}] \) (not the same)
2. **Angular Energy**: \( [AE] = [M][L][T^{-2}] \) (not the same)
3. **Force × Velocity**: \( [F][V] = [M][L^2][T^{-2}] \times [L][T^{-1}] = [M][L^2][T^{-3}] \) (not the same)
4. **Impulse × Distance**: \( [I][D] = [M][L][T^{-1}] \times [L] = [M][L^2][T^{-1}] \) (same)

Thus, the correct answer is that the dimensional formula for action is the same as that for impulse multiplied by distance.