Question

A particle moves along a circle of radius $10m$. Determine the distance and magnitude of displacement after
(a) $0.5$ revolution
(b) $2$ revolutions
(c) $2.5$ revolutions

Answer 1

Step 1: Calculate the distance and magnitude of displacement after $\mathbf{0}\mathbf{.}\mathbf{5}$ the revolution
Given, the radius of circle, $r=10m$.

After $0.5$ revolution, the particle will cover a semi-circular path.

$\mathbf{\therefore }\mathbf{}\mathit{D}\mathit{i}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}\mathbf{=}\mathbf{}\mathit{c}\mathit{i}\mathit{r}\mathit{c}\mathit{u}\mathit{m}\mathit{f}\mathit{e}\mathit{r}\mathit{e}\mathit{n}\mathit{c}\mathit{e}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{s}\mathit{e}\mathit{m}\mathit{i}\mathit{c}\mathit{i}\mathit{r}\mathit{c}\mathit{l}\mathit{e}$

$$\begin{gathered} =\mathrm{\pi r} \\ =\mathrm{\pi }\times 10 \\ =10\mathrm{\pi }m \end{gathered}$$

$\mathit{D}\mathit{i}\mathit{s}\mathit{p}\mathit{l}\mathit{a}\mathit{c}\mathit{e}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathbf{}\mathbf{=}\mathbf{}\mathit{d}\mathit{i}\mathit{a}\mathit{m}\mathit{e}\mathit{t}\mathit{e}\mathit{r}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{c}\mathit{i}\mathit{r}\mathit{c}\mathit{l}\mathit{e}$

$$\begin{gathered} =2r \\ =2\times 10 \\ =20m \end{gathered}$$
Step 2: Calculate the distance and magnitude of displacement after $\mathbf{2}$ revolutions

After $2$ revolutions, the particle will cover two complete revolutions and it comes to starting point.
$\mathbf{\therefore }\mathbf{}\mathit{D}\mathit{i}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}\mathbf{=}\mathbf{2}\mathbf{}\mathbf{\times }\mathbf{}\mathbf{(}\mathbf{c}\mathbf{i}\mathbf{r}\mathbf{c}\mathbf{u}\mathbf{m}\mathbf{f}\mathbf{e}\mathbf{r}\mathbf{e}\mathbf{n}\mathbf{c}\mathbf{e}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{c}\mathbf{i}\mathbf{r}\mathbf{c}\mathbf{l}\mathbf{e}\mathbf{)}$

$$\begin{gathered} =2\times 2\mathrm{\pi r} \\ =4\mathrm{\pi r} \\ =4\times \mathrm{\pi }\times 10 \\ =40\mathrm{\pi }m \end{gathered}$$
$\mathit{D}\mathit{i}\mathit{s}\mathit{p}\mathit{l}\mathit{a}\mathit{c}\mathit{e}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathbf{=}\mathbf{0}\mathbf{}$(because the initial and final position after $2$ revolutions is the same)

Step 3: Calculate the distance and magnitude of displacement after $\mathbf{2}\mathbf{.}\mathbf{5}$ revolutions
After $2.5$ revolutions, the particle will cover two complete revolutions and then one semi-circular path.
$\therefore \mathbf{D}\mathbf{i}\mathbf{s}\mathbf{t}\mathbf{a}\mathbf{n}\mathbf{c}\mathbf{e}\mathbf{=}\mathbf{}\mathbf{2}\mathbf{}\mathbf{\times }\mathbf{}\mathbf{(}\mathbf{c}\mathbf{i}\mathbf{r}\mathbf{c}\mathbf{u}\mathbf{m}\mathbf{f}\mathbf{e}\mathbf{r}\mathbf{e}\mathbf{n}\mathbf{c}\mathbf{e}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{c}\mathbf{i}\mathbf{r}\mathbf{c}\mathbf{l}\mathbf{e}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{}\mathbf{c}\mathbf{i}\mathbf{r}\mathbf{c}\mathbf{u}\mathbf{m}\mathbf{f}\mathbf{e}\mathbf{r}\mathbf{e}\mathbf{n}\mathbf{c}\mathbf{e}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{s}\mathbf{e}\mathbf{m}\mathbf{i}\mathbf{c}\mathbf{i}\mathbf{r}\mathbf{c}\mathbf{l}\mathbf{e}$

$$\begin{gathered} =2\times 2\mathrm{\pi r}+\mathrm{\pi r} \\ =5\mathrm{\pi r} \\ =5\times \mathrm{\pi }\times 10 \\ =50\mathrm{\pi }\mathrm{m} \end{gathered}$$
$\mathit{D}\mathit{i}\mathit{s}\mathit{p}\mathit{l}\mathit{a}\mathit{c}\mathit{e}\mathit{m}\mathit{e}\mathit{n}\mathit{t}\mathbf{=}\mathbf{}\mathit{d}\mathit{i}\mathit{a}\mathit{m}\mathit{e}\mathit{t}\mathit{e}\mathit{r}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{c}\mathit{i}\mathit{r}\mathit{c}\mathit{l}\mathit{e}$ (because after $2$ revolutions it comes to its initial position and for $0.5$ revolution diameter is displacement)

$$\begin{gathered} =2r \\ =2\times 10 \\ =20m \end{gathered}$$
Final Answer:

(a) The distance and magnitude of displacement after $0.5$ revolution is $10\mathrm{\pi }m$ and $20m$.

(b) The distance and magnitude of displacement after $2$ revolution is $40\mathrm{\pi }m$ and $0m$.

(c) The distance and magnitude of displacement after $2.5$ revolution is $50\mathrm{\pi }m$ and $20m$.