The dimensional formula of power is

Options

A. `ML^(2)T^(-1)`
B. `ML^(2)T^(-3)`
C. `MLT^(-2)`
D. `MLT^(-1)`

Answer

To find the dimensional formula of power, we start with the definition of power in terms of work done and time taken.

### Step 1: Define Power
Power (P) is defined as the rate of doing work. Mathematically, it can be expressed as:
\[ P = \frac{W}{t} \]
where \( W \) is the work done and \( t \) is the time taken.

### Step 2: Express Work Done
Work done (W) can be defined as the dot product of force (F) and displacement (s):
\[ W = F \cdot s \]

### Step 3: Express Force
Force can be expressed using Newton's second law:
\[ F = m \cdot a \]
where \( m \) is mass and \( a \) is acceleration.

### Step 4: Express Acceleration
Acceleration (a) can be expressed in terms of displacement and time:
\[ a = \frac{s}{t^2} \]
Thus, we can substitute this into the force equation:
\[ F = m \cdot \left(\frac{s}{t^2}\right) \]

### Step 5: Substitute Force into Work Done
Now substituting the expression for force into the work done equation:
\[ W = F \cdot s = \left(m \cdot \frac{s}{t^2}\right) \cdot s = m \cdot \frac{s^2}{t^2} \]

### Step 6: Substitute Work Done into Power Equation
Now we can substitute this expression for work done into the power equation:
\[ P = \frac{W}{t} = \frac{m \cdot \frac{s^2}{t^2}}{t} = m \cdot \frac{s^2}{t^3} \]

### Step 7: Write the Dimensional Formula
Now we can express the dimensions:
- Mass (m) has the dimension \( [M] \)
- Displacement (s) has the dimension \( [L] \)
- Time (t) has the dimension \( [T] \)

Thus, the dimensional formula for power is:
\[ P = [M][L^2][T^{-3}] \]

### Final Answer
The dimensional formula of power is:
\[ [M^1 L^2 T^{-3}] \]

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