An object is displaced from position vector `r_1=(2hati+3hatj)` m to `r_2=(4hati+6hatj)` m under a force `F=(3x^2hati+2yhatj)N` Find the work done by this force.

Explanation

To find the work done by the force \( F = (3x^2 \hat{i} + 2y \hat{j}) \) when an object is displaced from position vector \( \mathbf{r_1} = (2 \hat{i} + 3 \hat{j}) \) m to \( \mathbf{r_2} = (4 \hat{i} + 6 \hat{j}) \) m, we can follow these steps:

### Step 1: Define the displacement vector
The displacement vector \( \mathbf{dr} \) can be defined as:
\[ \mathbf{dr} = d\mathbf{r} = dx \hat{i} + dy \hat{j} \]

### Step 2: Write the work done formula
The work done \( W \) by the force during the displacement from \( \mathbf{r_1} \) to \( \mathbf{r_2} \) is given by the line integral:
\[ W = \int_{\mathbf{r_1}}^{\mathbf{r_2}} \mathbf{F} \cdot d\mathbf{r} \]
Substituting the expressions for \( \mathbf{F} \) and \( d\mathbf{r} \):
\[ W = \int_{\mathbf{r_1}}^{\mathbf{r_2}} (3x^2 \hat{i} + 2y \hat{j}) \cdot (dx \hat{i} + dy \hat{j}) \]

### Step 3: Calculate the dot product
Calculating the dot product:
\[ \mathbf{F} \cdot d\mathbf{r} = (3x^2 \hat{i} + 2y \hat{j}) \cdot (dx \hat{i} + dy \hat{j}) = 3x^2 dx + 2y dy \]
Thus, the work done becomes:
\[ W = \int_{\mathbf{r_1}}^{\mathbf{r_2}} (3x^2 dx + 2y dy) \]

### Step 4: Set up the limits of integration
The limits of integration for \( x \) will be from \( x_1 = 2 \) to \( x_2 = 4 \), and for \( y \) from \( y_1 = 3 \) to \( y_2 = 6 \).

### Step 5: Integrate with respect to \( x \) and \( y \)
We can separate the integral:
\[ W = \int_{2}^{4} 3x^2 dx + \int_{3}^{6} 2y dy \]

Calculating the first integral:
\[ \int 3x^2 dx = x^3 \quad \text{(from 2 to 4)} \]
Evaluating:
\[ [4^3 - 2^3] = [64 - 8] = 56 \]

Calculating the second integral:
\[ \int 2y dy = y^2 \quad \text{(from 3 to 6)} \]
Evaluating:
\[ [6^2 - 3^2] = [36 - 9] = 27 \]

### Step 6: Add the results of the integrals
Combining both results:
\[ W = 56 + 27 = 83 \text{ Joules} \]

### Final Answer
The work done by the force is \( W = 83 \) Joules.

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