Question 246: Work, Energy and Power & Circular Motion

Question

A small particle moves to position $$5 \hat{i}-2 \hat{j}+\hat{k}$$ from its initial position $$2 \hat{i}+3 \hat{j}-4 \hat{k}$$ under the action of force $$5 \hat{i}+2 \hat{j}+7 \hat{k} \mathrm{~N}$$. The value of work done will be __________ J.

Accepted Answer

The given expression calculates the work done by a force vector $\vec{F} = 5\hat{i} + 2\hat{j} + 7\hat{k}$ when it acts on an object that moves from an initial position vector $\vec{r}_i = 2\hat{i} + 3\hat{j} - 4\hat{k}$ to a final position vector $\vec{r}_f = 5\hat{i} - 2\hat{j} + \hat{k}$.
To find the work done, we use the dot product of the force and displacement vectors :
$$
\begin{aligned}
& W=\vec{F} \cdot\left(\vec{r}_f-\vec{r}_{\mathrm{i}}ight) \\
& =(5 \hat{i}+2 \hat{j}+7 \hat{k}) \cdot((5 \hat{i}-2 \hat{j}+\hat{k})-(2 \hat{i}+3 \hat{j}-4 \hat{k})) \\
& =(5 \hat{i}+2 \hat{j}+7 \hat{k}) \cdot(3 \hat{i}-5 \hat{j}+5 \hat{k}) \\
& =15-10+35 \\
& =40 \mathrm{~J}
\end{aligned}
$$

Work, Energy and Power & Circular Motion

A small particle moves to position $5 \hat{i}-2 \hat{j}+\hat{k} from its initial position 2 \hat{i}+3 \hat{j}-4 \hat{k} under the action of force 5 \hat{i}+2 \hat{j}+7 \hat{k} \mathrm{~N}$. The value of work done will be __________ J.

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