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A pendulum of 14cm long oscillates through an angle of ${{18}^{\circ }}$ . Find the length (in cm) of the path described by its extremities.

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Last updated date: 17th Jun 2024
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Answer
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Hint: Now we know that the pendulum moves in a circular motion. Hence we can say that the angle is the angle of arc and the length of arc is the radius of the circular motion. Hence we are given with the angle and radius of the circular motion. Now we know that the formula for length of an arc with a given angle and radius is given by $2\pi r\times \dfrac{\theta }{360}$ . Hence we have the length of the required arc.

Complete step by step answer:
Now let us consider the pendulum.
We know that the motion of a pendulum is somewhat circular hence we can say that the extremity of the pendulum will make a circular arc.
Now we are given that the angle of the circular arc will be ${{18}^{\circ }}$ and the radius of the arc is 14cm.
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Now we know that the circumference of the arc is given by $2\pi r\times \dfrac{\theta }{360}$ where $\theta $ is the angle made by the arc and r is the radius.
Now we have the radius is 14cm and the angle is ${{18}^{\circ }}$ .
Now substituting the values of angle and radius we get,
The length of arc is
$\begin{align}
  & \Rightarrow l=2\pi \left( 14 \right)\times \dfrac{18}{360} \\
 & \Rightarrow l=28\pi \dfrac{1}{20} \\
 & \Rightarrow l=\dfrac{28\pi }{20} \\
\end{align}$
Hence the length of the arc is $\dfrac{28\pi }{20}$ .

Note: Now note that if the angle is in radians then the length of arc is given by angle × radius. Hence if the angle is given by degree then we can convert the angle into radians and then multiply the angle with radius. Hence we will have the length of the arc.