Question

Two forces are acting on a body: \[ \vec{F}_1 = 3\hat{i} -5\hat{j} +2\hat{k} \] \[ \vec{F}_2 = 8\hat{i} +2\hat{j} -3\hat{k} \] The displacement of the body is \(25\,\text{m\) along the direction \(3\hat{i}-4\hat{j}\). Find the work done.}

• A. \(225\,\text{J}\)
• B. \(200\,\text{J}\)
• C. \(125\,\text{J}\)
• D. \(325\,\text{J}\)

Answer 1

The Correct Option is A

Concept:
Work done by a force is given by the dot product of force and displacement. \[ W = \vec{F}\cdot\vec{S} \] If multiple forces act simultaneously, the resultant force is \[ \vec{F}_{net}=\vec{F}_1+\vec{F}_2 \] Step 1: Find the resultant force. \[ \vec{F}_{net} = (3+8)\hat{i}+(-5+2)\hat{j}+(2-3)\hat{k} \] \[ \vec{F}_{net}=11\hat{i}-3\hat{j}-\hat{k} \] Step 2: Find the displacement vector. Direction vector: \[ 3\hat{i}-4\hat{j} \] Magnitude of this direction: \[ \sqrt{3^2+4^2}=5 \] Unit vector: \[ \frac{3\hat{i}-4\hat{j}}{5} \] Since displacement magnitude is \(25\,m\), \[ \vec{S}=25\left(\frac{3\hat{i}-4\hat{j}}{5}\right) \] \[ \vec{S}=15\hat{i}-20\hat{j} \] Step 3: Calculate the work done. \[ W=\vec{F}_{net}\cdot\vec{S} \] \[ =(11\hat{i}-3\hat{j}-\hat{k})\cdot(15\hat{i}-20\hat{j}) \] \[ W=11\times15 + (-3)(-20) + (-1)(0) \] \[ W=165+60 \] \[ W=225\,\text{J} \]