Question

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

Answer 1

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 $1hr = 60\min$.

Complete step by step answer:
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.

Conversion between degree and 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.

Note: It should be remembered that, to solve this problem we use the unitary method, in which we first calculate for one unit and later multiply the quantity of one unit to the value which we want to find out. Also angular displacement is mathematically defined as the angular speed with which the body is rotating per unit time.