Let's solve the problem step-by-step:
1. Understanding the Problem:
- We have a mass [tex]\( m = 10 \, \text{Kg} \)[/tex] moving in a circular path.
- The radius of the circular path is [tex]\( r = 5 \, \text{cm} \)[/tex].
- The angular distance covered by the mass is [tex]\( 30^\circ \)[/tex].
2. Converting Units:
- Convert the radius from centimeters to meters because the standard unit of length in the International System of Units (SI) is meters.
[tex]\[
r = 5 \, \text{cm} = 5 \div 100 = 0.05 \, \text{m}
\][/tex]
- Convert the angle from degrees to radians since the standard unit of angular measure in the SI system is the radian.
[tex]\[
\text{Angle in radians} = 30^\circ \times \left(\frac{\pi}{180}\right) = \frac{30 \pi}{180} = \frac{\pi}{6} \approx 0.5236 \, \text{radians}
\][/tex]
3. Calculating the Arc Length:
- The formula for the length of the arc [tex]\( s \)[/tex] of a circle subtended by an angle [tex]\( \theta \)[/tex] (in radians) at the center is:
[tex]\[
s = r \cdot \theta
\][/tex]
Given:
[tex]\[
r = 0.05 \, \text{m}
\][/tex]
[tex]\[
\theta = 0.5236 \, \text{radians}
\][/tex]
Plugging in the values:
[tex]\[
s = 0.05 \times 0.5236 = 0.02618 \, \text{m}
\][/tex]
4. Finding the Closest Option:
- From the given options, we need to determine which one is closest to our calculated arc length.
[tex]\[
\text{A. } 2.62 \times 10^{-2} \, \text{rad} = 0.0262 \, \text{m}
\][/tex]
[tex]\[
\text{B. } 20 \, \text{rad}
\][/tex]
[tex]\[
\text{C. } 2.5 \times 10^{-1} \, \text{rad} = 0.25 \, \text{m}
\][/tex]
[tex]\[
\text{D. } 0.03 \, \text{rad}
\][/tex]
Comparing the values, the closest one to [tex]\( 0.02618 \, \text{m} \)[/tex] is:
A. [tex]\( 2.62 \times 10^{-2} \, \text{rad} = 0.0262 \, \text{m} \)[/tex]
Therefore, the length of the arc [tex]\( AB \)[/tex] is approximately [tex]\( 0.0262 \, \text{m} \)[/tex]. The correct answer among the options provided is:
(A) [tex]\( 2.62 \times 10^{-2} \, \text{rad} \)[/tex]
