Dimension of force is

Options

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

Answer

To find the dimension of force, we can follow these steps:

### Step 1: Understand the Definition of Force
Force is defined by Newton's second law of motion, which states that force (F) is equal to mass (m) multiplied by acceleration (a):
\[ F = m \cdot a \]

### Step 2: Identify the Dimensions of Mass
The dimension of mass (m) is represented as:
\[ [m] = M \]
where \( M \) is the dimension symbol for mass.

### Step 3: Understand the Definition of Acceleration
Acceleration (a) is defined as the change in velocity (v) per unit time (t). Mathematically, it can be expressed as:
\[ a = \frac{\Delta v}{\Delta t} \]
Velocity (v) is defined as displacement (s) per unit time (t):
\[ v = \frac{s}{t} \]
Thus, the dimension of velocity is:
\[ [v] = \frac{[s]}{[t]} = \frac{L}{T} \]
where \( L \) is the dimension symbol for length and \( T \) is the dimension symbol for time.

### Step 4: Determine the Dimension of Acceleration
Since acceleration is the change in velocity per unit time, we can express its dimension as:
\[ [a] = \frac{[v]}{[t]} = \frac{L/T}{T} = \frac{L}{T^2} \]
So, the dimension of acceleration is:
\[ [a] = LT^{-2} \]

### Step 5: Combine the Dimensions to Find the Dimension of Force
Now, we can substitute the dimensions of mass and acceleration back into the equation for force:
\[ [F] = [m] \cdot [a] = M \cdot (LT^{-2}) \]
Thus, the dimension of force is:
\[ [F] = MLT^{-2} \]

### Conclusion
The dimension of force is:
\[ [F] = MLT^{-2} \]