Question

A particle moves from a point $(-2 \hat{i} + 5 \hat{j})$ to $( 4 \hat{j} + 3 \hat{k})$ when a force of $(4 \hat{i} + 3 \hat{j} ) N$ is applied. How much work has been done by the force ?

• A. 8 J
• B. 11 J
• C. 5 J
• D. 2 J

Answer 1

The Correct Option is C

Work Calculation Explanation 

We are asked to calculate the work done by a force \(\vec{F}\) over a displacement \(\vec{dS}\). The formula for work is:

\(W = \vec{F} \cdot \vec{dS}\)

Step 1: Given Values

The force \(\vec{F}\) and displacement \(\vec{dS}\) are given as:

• \(\vec{F} = 4 \hat{i} + 3 \hat{j}\)
• \(\vec{dS} = (4 \hat{j} + 3 \hat{k}) - (-2 \hat{i} + 5 \hat{j})\)

Step 2: Simplify Displacement

Now, let's simplify the displacement vector \(\vec{dS}\):

\(\vec{dS} = (4 \hat{j} + 3 \hat{k}) - (-2 \hat{i} + 5 \hat{j})\)

First, distribute the negative sign:

\(\vec{dS} = 4 \hat{j} + 3 \hat{k} + 2 \hat{i} - 5 \hat{j}\)

Now, simplify the terms:

\(\vec{dS} = 2 \hat{i} - \hat{j} + 3 \hat{k}\)

Step 3: Dot Product Calculation

Next, we calculate the dot product of the force and displacement vectors:

\(W = (4 \hat{i} + 3 \hat{j}) \cdot (2 \hat{i} - \hat{j} + 3 \hat{k})\)

Step 4: Perform the Dot Product

To compute the dot product, we multiply corresponding components and sum them:

• For the \(\hat{i}\)-component: \(4 \times 2 = 8\)
• For the \(\hat{j}\)-component: \(3 \times -1 = -3\)
• For the \(\hat{k}\)-component: \(0 \times 3 = 0\) (since there's no \(\hat{k}\)-component in \(\vec{F}\))

So, the dot product is:

\(W = 8 - 3 + 0 = 5 \, \text{J}\)

Conclusion:

The work done is 5 Joules (J).

Answer 2

Given: 
- Initial position: $\vec{r}_1 = -2 \hat{i} + 5 \hat{j}$
- Final position: $\vec{r}_2 = 4 \hat{j} + 3 \hat{k}$
- Force applied: $\vec{F} = 4 \hat{i} + 3 \hat{j}$

Step 1: Find displacement vector
\(\vec{d} = \vec{r}_2 - \vec{r}_1 = (4 \hat{j} + 3 \hat{k}) - (-2 \hat{i} + 5 \hat{j}) = 2 \hat{i} -1 \hat{j} + 3 \hat{k}\)

Step 2: Work done = dot product of force and displacement
\(W = \vec{F} \cdot \vec{d} = (4 \hat{i} + 3 \hat{j}) \cdot (2 \hat{i} -1 \hat{j} + 3 \hat{k})\)
\(W = 4 \cdot 2 + 3 \cdot (-1) + 0 \cdot 3 = 8 - 3 + 0 = \boxed{5\,J}\)

Final Answer:
\(\boxed{5\,J}\)