Force acting on a particale is `(2hat(i)+3hat(j))N`. Work done by this force is zero, when the particle is moved on the line `3y+kx=5`. Here value of k is `(`Work done `W=vec(F).vec(d))`

Options

A. 2
B. 4
C. 6
D. 8

Answer

To solve the problem, we need to find the value of \( k \) such that the work done by the force \( \vec{F} = (2\hat{i} + 3\hat{j}) \, \text{N} \) is zero when the particle is moved along the line defined by the equation \( 3y + kx = 5 \).

### Step-by-Step Solution:

1. **Understand the Work Done Formula**:
The work done \( W \) by a force \( \vec{F} \) when a particle moves through a displacement \( \vec{d} \) is given by:
\[ W = \vec{F} \cdot \vec{d} \]
For the work done to be zero, we need:
\[ \vec{F} \cdot \vec{d} = 0 \]

2. **Identify the Displacement Vector**:
The displacement vector \( \vec{d} \) can be expressed in terms of its components:
\[ \vec{d} = dx \hat{i} + dy \hat{j} \]

3. **Differentiate the Line Equation**:
The line equation is given as:
\[ 3y + kx = 5 \]
To find the relationship between \( dx \) and \( dy \), we differentiate both sides:
\[ 3 \frac{dy}{dt} + k \frac{dx}{dt} = 0 \]
Rearranging gives:
\[ 3 dy = -k dx \quad \Rightarrow \quad dy = -\frac{k}{3} dx \]

4. **Substitute \( dy \) into the Displacement Vector**:
Substitute \( dy \) into the displacement vector:
\[ \vec{d} = dx \hat{i} + \left(-\frac{k}{3} dx\right) \hat{j} = dx \hat{i} - \frac{k}{3} dx \hat{j} \]
Factor out \( dx \):
\[ \vec{d} = dx \left(\hat{i} - \frac{k}{3} \hat{j}\right) \]

5. **Calculate the Dot Product**:
Now calculate the dot product \( \vec{F} \cdot \vec{d} \):
\[ \vec{F} = 2 \hat{i} + 3 \hat{j} \]
\[ \vec{F} \cdot \vec{d} = (2 \hat{i} + 3 \hat{j}) \cdot \left(dx \left(\hat{i} - \frac{k}{3} \hat{j}\right)\right) \]
This simplifies to:
\[ = dx \left(2 \cdot 1 + 3 \cdot \left(-\frac{k}{3}\right)\right) \]
\[ = dx \left(2 - k\right) \]

6. **Set the Dot Product to Zero**:
For the work done to be zero:
\[ dx (2 - k) = 0 \]
Since \( dx \neq 0 \) (the particle is moving), we have:
\[ 2 - k = 0 \quad \Rightarrow \quad k = 2 \]

### Conclusion:
The value of \( k \) is \( 2 \).