The diamensions of time in energy are

Options

A. 0
B. -2
C. 2
D. 1

Answer

To find the dimensions of time in energy, we will follow these steps:

### Step 1: Understand the relationship between energy and work
Energy is defined as the capacity to do work. Therefore, the dimensions of energy are the same as the dimensions of work done.

### Step 2: Write the formula for work done
The work done (W) is given by the formula:
\[ W = \text{Force} \times \text{Displacement} \]

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

The dimensions of mass (m) are:
\[ [M] \]

The dimensions of acceleration (a) are:
\[ [L T^{-2}] \]
where \( L \) is length and \( T \) is time.

Thus, the dimensions of force (F) are:
\[ [F] = [M][L T^{-2}] = [M L T^{-2}] \]

### Step 4: Find the dimensions of displacement
Displacement (d) has the dimensions of length:
\[ [d] = [L] \]

### Step 5: Combine the dimensions to find the dimensions of work
Now, substituting the dimensions of force and displacement into the work formula:
\[ [W] = [F] \times [d] = [M L T^{-2}] \times [L] \]
\[ [W] = [M L^2 T^{-2}] \]

### Step 6: Conclusion about the dimensions of energy
Since energy has the same dimensions as work, we conclude:
\[ [E] = [M L^2 T^{-2}] \]

### Step 7: Isolate the dimension of time
From the expression for energy, we can isolate the dimension of time (T):
The dimension of time in energy is:
\[ [T^{-2}] \]

Thus, the dimension of time in energy is:
\[ T^{-2} \]

### Final Answer
The dimensions of time in energy are \( T^{-2} \).

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