Question

(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?

Short answer

1. The angle in radian and in degree is 0.6 rad, and $34.38°$, respectively.
2. The radius is 0.06267 m.
3. The arc length is 1.05 m.

Step-by-step solution

Identification of given data

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

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

Concept/Significance of angular velocity and acceleration

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.

Determine the angle in radians and degree

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.

Determine the radius of the circle(b)

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 .

Determine the length of arc(c)

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.