A particle of mass 1 kg is moving along x - axis and a force F is also acting along x -axis in such a way that its displacement is varying as : `x=3t^(2)`. Find work done by force F when it will move 2m.

Options

A. 12 J
B. 16 J
C. 32 J
D. 42 J

Answer

To solve the problem step by step, we will follow the principles of kinematics and dynamics to find the work done by the force \( F \).

### Step 1: Understand the displacement function
The displacement of the particle is given by the equation:
\[ x = 3t^2 \]
This indicates how the position of the particle changes with time \( t \).

### Step 2: Find the velocity
To find the velocity \( v \), we differentiate the displacement \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d(3t^2)}{dt} = 6t \]

### Step 3: Find the acceleration
Next, we differentiate the velocity \( v \) to find the acceleration \( a \):
\[ a = \frac{dv}{dt} = \frac{d(6t)}{dt} = 6 \, \text{m/s}^2 \]
This indicates that the acceleration is constant.

### Step 4: Calculate the force using Newton's second law
Using Newton's second law, the force \( F \) acting on the particle can be calculated as:
\[ F = m \cdot a \]
Given that the mass \( m = 1 \, \text{kg} \):
\[ F = 1 \cdot 6 = 6 \, \text{N} \]

### Step 5: Determine the work done by the force
The work done \( W \) by the force when the particle moves a displacement \( s = 2 \, \text{m} \) is given by the formula:
\[ W = F \cdot s \]
Since both the force and displacement are in the same direction (along the x-axis), we can directly multiply:
\[ W = 6 \, \text{N} \cdot 2 \, \text{m} = 12 \, \text{J} \]

### Conclusion
The work done by the force \( F \) when the particle moves 2 meters is:
\[ \boxed{12 \, \text{J}} \]