The dimensional formula for potential energy is

Options

A. `M^(2)L^(2)T^(-2)`
B. `M^(1)L^(-2)T^(-2)`
C. `M^(1)L^(2)T^(-2)`
D. `M^(1)L^(2)T^(-3)`

Answer

To find the dimensional formula for potential energy, we will start by using the formula for gravitational potential energy, which is given by:

\[ PE = mgh \]

where:
- \( PE \) is the potential energy,
- \( m \) is the mass,
- \( g \) is the acceleration due to gravity,
- \( h \) is the height.

### Step 1: Identify the dimensions of each quantity
1. **Mass (m)**: The dimensional formula for mass is represented as \([M]\).
2. **Acceleration due to gravity (g)**: The acceleration is defined as the change in velocity per unit time. The dimensional formula for velocity is \([L T^{-1}]\), so for acceleration, we have:
\[ g = \frac{[L T^{-1}]}{[T]} = [L T^{-2}] \]
3. **Height (h)**: The dimensional formula for height is simply \([L]\).

### Step 2: Substitute the dimensions into the potential energy formula
Now we can substitute the dimensional formulas into the potential energy formula:

\[ PE = mgh \]

Substituting the dimensions we found:

\[ [PE] = [M] \cdot [L T^{-2}] \cdot [L] \]

### Step 3: Simplify the expression
Now we will simplify the expression:

\[ [PE] = [M] \cdot [L^2 T^{-2}] \]

Thus, the dimensional formula for potential energy becomes:

\[ [PE] = [M^1 L^2 T^{-2}] \]

### Final Answer
The dimensional formula for potential energy is:

\[ [M^1 L^2 T^{-2}] \]