Question

Find the angle in radians through which a pendulum swings if its length is 75cm and describes an arc of length
[i] 10cm [ii] 15cm [iii] 21cm

Answer 1

Hint: Use the fact that the angle subtended by an arc of length r in a circle of radius r is equal to 1 radian. Hence determine the angle subtended by the arcs of length 10cm, 15cm and 21cm in a circle of radius 75cm.

Complete step-by-step answer:
Circular System of measurement of an angle: In this system of measurement, an angle is measured in radians. The measure of an angle subtended by an arc of length equal to the radius in a circle is equal to 1 radian.

[i] We have AC = 75cm. arc(BC) = 10cm.
The angle subtended by an arc of length 75cm = 1 radian (by definition).
Hence, the angle subtended by an arc of length 1cm $=\dfrac{1}{75}$radians.
Hence, the angle subtended by an arc of length 10cm $=\dfrac{10}{75}=\dfrac{2}{15}$ radians.
[ii] We have AC = 75cm. arc(BC) = 15cm.
The angle subtended by an arc of length 75cm = 1 radian (by definition).
Hence, the angle subtended by an arc of length 1cm $=\dfrac{1}{75}$radians.
Hence, the angle subtended by an arc of length 15cm $=\dfrac{15}{75}=\dfrac{3}{15}$ radians.
[iii] We have AC = 75cm. arc(BC) = 21cm.
The angle subtended by an arc of length 75cm = 1 radian (by definition).
Hence, the angle subtended by an arc of length 1cm $=\dfrac{1}{75}$radians.
Hence, the angle subtended by an arc of length 21cm $=\dfrac{21}{75}=\dfrac{7}{25}$ radians.

Note: Alternatively, we have
If x is the measure of an angle in radians, l the length of the arc and r the radius, then
$x=\dfrac{l}{r}$
Hence we have
[i] l = 10, r = 75
$\Rightarrow x=\dfrac{10}{75}=\dfrac{2}{15}$
[ii] l = 15, r = 75
$\Rightarrow x=\dfrac{15}{75}=\dfrac{3}{15}$
[iii] l = 21, r = 75
$\Rightarrow x=\dfrac{21}{75}=\dfrac{7}{25}$