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## Q. 3.4

A force of 4 N has the direction π/3. What is the work done in moving an object from the point (1, 2) to the point (5, 4), where distances are measured in meters?

## Verified Solution

A unit vector with direction $\pi / 3$ is given by $\mathbf{u}=(\cos \pi / 3) \mathbf{i}+(\sin \pi / 3) \mathbf{j}=(1 / 2) \mathbf{i}+(\sqrt{3} / 2) \mathbf{j}$. Thus $\mathbf{F}=4 \mathbf{u}=2 \mathbf{i}+2 \sqrt{3} \mathbf{j}$. The displacement vector $\mathbf{d}$ is given by $(5-1) \mathbf{i}+(4-2) \mathbf{j}=4 \mathbf{i}+2 \mathbf{j}$. Thus

$W=\mathbf{F} \cdot \mathbf{d}=(2 \mathbf{i}+2 \sqrt{3} \mathbf{j}) \cdot(4 \mathbf{i}+2 \mathbf{j})=(8+4 \sqrt{3}) \approx 14.93 \mathrm{~N} \cdot \mathrm{m}$

The component of $\mathbf{F}$ in the direction of motion is sketched in Figure 5 .