Let and be the angular speed of the second hand, minute hand and hour hand of a smoothly running analog clock, respectively. If and are their respective angular distances in 1 minute then the factor which remains constant () is: [2024]

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Solution

Q.
Let $ω_{1}, ω_{2}$ and $ω_{3}$ be the angular speed of the second hand, minute hand and hour hand of a smoothly running analog clock, respectively. If $x_{1}, x_{2}$ and $x_{3}$ are their respective angular distances in 1 minute then the factor which remains constant ( $k$ ) is: [2024]

1 $\frac{ω_{1}}{x_{1}} = \frac{ω_{2}}{x_{2}} = \frac{ω_{3}}{x_{3}} = k$

2 $ω_{1} x_{1} = ω_{2} x_{2} = ω_{3} x_{3} = k$

3 $ω_{1} x_{1}^{2} = ω_{2} x_{2}^{2} = ω_{3} x_{3}^{2} = k$

4 $ω_{1}^{2} x_{1} = ω_{1}^{2} x_{2} = ω_{1}^{2} x_{3} = k$

Ans.
(1)

Angular velocity of second hand,

$ω_{1} = \frac{2 π}{1} rad/minute$

Angular velocity of minute hand, $ω_{2} = \frac{2 π}{60}$ rad/minute

Angular velocity of hour hand, $ω_{3} = \frac{2 π}{12 \times 60}$ rad/minute

Angular distance of second's hand in one minute

$x_{1} = 360 ° / minute$

$x_{2} = \frac{360 °}{60} = 6 ° / minute$

$x_{3} = \frac{360 °}{60 \times 12} = 0.5 ° / minute$

By above values, $\frac{ω_{1}}{x_{1}} = \frac{ω_{2}}{x_{2}} = \frac{ω_{3}}{x_{3}} = \frac{2 π}{360 °} = k$

where, $k$ is a constant.

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