(a) What angle in radians is subtended by an arc 1.50 m long on the circumference of a circle of radius 2.50 m? What is this angle in degrees? (b) An arc 14.0 cm long on the circumference of a circle subtends an angle of 128°. What is the radius of the circle? (c) The angle between two radii of a circle with radius 1.50 m is 0.700 rad. What length of arc is intercepted on the circumference of the circle by the two radii?

The first arc length is${\mathrm{l}}_{\mathrm{a}}=1.5\mathrm{m}$ .

The first radius is ${\mathrm{r}}_{\mathrm{a}}=2.5\mathrm{m}$.

Angular velocity is the velocity by which a particle is moving in a circular path, and change in angular velocity with respect to time is angular acceleration.

The equation of arc length is given by,

$l=\alpha .r$

Here, l is arc length, is angle, r and is radius.

Find the angle for the first case as follows.

$$\begin{gathered} {\mathrm{l}}_{\mathrm{a}}={\mathrm{\alpha }}_{\mathrm{a}}.{\mathrm{r}}_{\mathrm{a}} \\ {\mathrm{\alpha }}_{\mathrm{a}}=\frac{{\mathrm{l}}_{\mathrm{a}}}{{\mathrm{r}}_{\mathrm{a}}} \\ =\frac{1.5\mathrm{m}}{2.5\mathrm{m}} \\ =0.6\mathrm{rad} \end{gathered}$$

Calculate the angle in degree.

$$\begin{gathered} {\mathrm{\alpha }}_{\mathrm{a}}=\left(0.6\mathrm{rad}\right)\left(\frac{180°}{\mathrm{\pi }\mathrm{rad}}\right) \\ =34.38° \end{gathered}$$

Therefore, the angle in radian and in degree is 0.6 rad, and$34.38°$ , respectively.

The second arc length is ${\mathrm{l}}_{\mathrm{b}}=0.14\mathrm{m}$.

The angle is ${\mathrm{\alpha }}_{\mathrm{b}}=128°$.

From equation (1), find the radius.

$$\begin{gathered} {\mathrm{r}}_{\mathrm{b}}=\frac{{\mathrm{l}}_{\mathrm{b}}}{{\mathrm{\alpha }}_{\mathrm{b}}} \\ =\frac{0.14\mathrm{m}}{128°} \\ =\frac{0.14\mathrm{m}}{{\displaystyle \frac{32}{45}}\mathrm{\pi }\mathrm{rad}} \\ =0.06267\mathrm{m} \end{gathered}$$

Therefore, the radius is 0.06267 m .

The radius is ${\mathrm{r}}_{\mathrm{c}}=1.50\mathrm{m}$.

The angle is 0.700 rad.

From equation (1), find the arc length.

$$\begin{gathered} {\mathrm{l}}_{\mathrm{c}}={\mathrm{\alpha }}_{\mathrm{c}}.{\mathrm{r}}_{\mathrm{c}} \\ =\left(1.5\mathrm{m}\right)\left(0.700\mathrm{rad}\right) \\ =1.05\mathrm{m} \end{gathered}$$

Therefore, the arc length is 1.05 m.