Question

A particle moves along a circular track of $6\,m$ radius such that the arc of the circular track covered subtends an angle of ${30^ \circ }$ at the centre. The distance covered by the body is:
(A) $\pi \,m$
(B) $13\pi \,m$
(C) $4\pi \,m$
(D) $6\pi \,m$

Answer 1

Hint: In this problem the solution can be determined by using the Length of arc formula. To find the length of the arc, the radius of the arc and the centre angle is required. Both the required terms are given in the problem, substitute the given data in the formula, the length of the arc can be determined.

Formulae Used:
Length of the arc is equal to the product of the radius of the arc and the centre angle of the arc.
$l = r \times \theta $
Where, $r$ is the radius of the arc, $l$ is the length of the arc and $\theta $ is the centre angle.

Complete step-by-step solution:
Given that,
The radius of the arc, $r = 6\,m$
The centre angle of the arc, $\theta = {30^ \circ }$
By using the length of the arc formula,
$l = r \times \theta \,...................\left( 1 \right)$
By substituting the radius of the arc and the centre angle of the arc in the above equation (1), then
$l = 6\,m \times {30^ \circ }$
In the above equation, the multiplication is not possible, so the centre angle of the arc is converted to the radian unit, then, ${30^ \circ } = \dfrac{\pi }{6}$,
$l = 6 \times \dfrac{\pi }{6}$
By cancelling the same terms in the numerator and the denominator, then the above equation is written as,
$l = \pi \,m$
Thus, the above equation shows the length of the arc.
Hence, the option (A) is the correct answer.

Note:- The centre angle of the arc is given in degree but here we substitute the centre angle of the arc in the radian for the further calculation. And substituting the values in the length of the arc formula, the length of the arc can be determined.