Question

A particle starts from the point $(0\mathrm{m},8\mathrm{m})$ and moves with a uniform velocity of $3\mathrm{m}/\mathrm{s}$. After $5$ seconds, the angular velocity of the particle about the origin will be

Answer 1

Step 1: Given data

Point $=(0\mathrm{m},8\mathrm{m})$

Velocity $\left(\mathrm{v}\right)=3\mathrm{m}/\mathrm{s}$

Time $\left(\mathrm{t}\right)=5$ seconds.

Step 2: Formula used

The angular velocity

$\mathrm{\omega }=\mathrm{v}\times \mathrm{r}$

$\mathrm{\omega }=$Angular velocity

$\mathrm{v}=$Velocity

$\mathrm{r}=$radius

Step 3: Determine the value of $\mathrm{x}$

Consider a figure

After $5$ seconds, the particle is at point B and has traveled a linear distance
$\mathrm{x}=\mathrm{v}\times \mathrm{t}$

$\mathrm{x}=5\times 3\mathrm{m}$

$\mathrm{x}=15\mathrm{m}$

Step 4: Determine the angular velocity

Now by using Pythagoras theorem we find the distance of $\mathrm{OB}$
$\mathrm{OB}=\sqrt{8^2+{15}^2}$

$\mathrm{OB}=17\mathrm{m}$
It is from the origin $\mathrm{O}$.
Therefore, the angular velocity about the origin.
$\mathrm{\omega }=\mathrm{v}\times \mathrm{r}$
(Where $\mathrm{v}$ is the velocity, $\mathrm{r}$ is the radius, and $\mathrm{\omega }$ its angular velocity.)

We find the value of $\mathrm{sin\theta }$ from a right-angled triangle shown in the above figure by using a trigonometric ratio.
$\mathrm{sin\theta }=\frac{\mathrm{P}}{\overline{\mathrm{H}}}$
( $\mathrm{P}$ is the perpendicular, and $\mathrm{H}$ is the Hypotenuse.)
$\mathrm{sin\theta }=\frac{8}{17}\left(\mathrm{By}\mathrm{using}\mathrm{trigonometric}\mathrm{ratio}\right)$
When the body rotates about origin then we find out the angular velocity
$\mathrm{\omega }=\frac{\mathrm{vsin\theta }}{\mathrm{r}}$

$\mathrm{\omega }=\frac{3\times 8}{{\mathrm{r}}^2}$

$\mathrm{\omega }=\frac{24}{{17}^2}$

$\mathrm{\omega }=\frac{24}{289}\mathrm{rad}/\mathrm{s}$

$\mathrm{\omega }=\frac{24}{289}\mathrm{rad}/\mathrm{s}$

Therefore, the angular velocity of the particle about the origin will be $\frac{24}{289}\mathrm{rad}/\mathrm{s}$.