Deriving the Dimensional Formula of Power

The question asks us to find the dimensional formula of power, given the dimensional formula for energy. Power is a fundamental concept in physics, and understanding its dimensions helps us relate it to other physical quantities.

Understanding Power and Energy Dimensions

Power is defined as the rate at which work is done or energy is transferred or converted. Mathematically, it can be expressed as:

Power = Energy / Time

We are given the dimensional formula for energy:

$[Energy] = ML^2T^{-2}$

Where:

• M represents Mass
• L represents Length
• T represents Time

The dimensional formula for time is simply:

$[Time] = T$

Calculating the Dimensional Formula for Power

To find the dimensional formula for power, we can use the relationship: Power = Energy / Time.

We substitute the dimensions of energy and time into this relationship:

$[Power] = \frac{[Energy]}{[Time]}$

$[Power] = \frac{ML^2T^{-2}}{T}$

Now, we simplify the expression by applying the rules of exponents. When dividing terms with the same base, we subtract the exponents. Here, the exponent for T in the numerator is -2, and the exponent for T in the denominator is 1.

$[Power] = ML^2T^{-2-1}$

$[Power] = ML^2T^{-3}$

Conclusion on Power's Dimensions

Therefore, the dimensional formula for power is $ML^2T^{-3}$. This formula indicates how power scales with mass, length, and time.

Matching with Provided Options

Comparing our derived formula $ML^2T^{-3}$ with the given options:

• Option 1: $MLT^{-2}$
• Option 2: $ML^2T^{-2}$ (This is the dimension of Energy)
• Option 3: $ML^2T^{-3}$
• Option 4: $ML^{-1}T^{-2}$

Our calculated dimensional formula for power, $ML^2T^{-3}$, matches Option 3.