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# What is angular displacement in radian of a second hand of a clock in $10\sec$ ?

Last updated date: 06th Sep 2024
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Hint: In order to solve this question we need to understand what is time and how it has been measured? Time is an eternal quantity which keeps growing and always moving forward. It has been to categorize events, so that we can study more of it. By the Standard International Unit, time has been in microseconds to years, decades etc. Earth spins around its own axis in almost $24hr$ so our clock has been designed in a circular manner so that every hour can be categorized in minutes and seconds. Conversion for which is given as, $1\min = 60\sec$ and $1hr = 60\min$.

Angular displacement is defined as the angle by which a body rotates in some time or it can be defined as it is the ratio of arc length to the radius of circle in which it has been calculated. It is denoted by $\theta$ and can be measured in two units, first is degree which is represented as $^\circ$ and second is radian.

$\theta (rad) = \dfrac{\pi }{{180}} \times \theta ^\circ$
So $360^\circ$ is in radian, $\dfrac{\pi }{{180}} \times 360$ = $2\pi$ (radians)
Since we know, a second covers a full circle ( $360^\circ$ ) in $60\sec$.
Since for $60\sec$ angular displacement is $2\pi$ radian.
So for $1\sec$ angular displacement is $\dfrac{{2\pi }}{{60}}$ radian.
Hence for $10\sec$ angular displacement is $\dfrac{{2\pi }}{{60}} \times 10$ radian.
Angular displacement by second hand in $10\sec$ is, $\theta = \dfrac{\pi }{3}$ radian.