Let $\omega_1, \omega_2$ and $\omega_3$ be the angular speed of the second hand, minute hand and hou

Question

Let $\omega_1, \omega_2$ and $\omega_3$ 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

Professor PC Thomas Classes · Answer

(a) : Angular velocity of second hand, $\omega_1=\frac{2 \pi}{1} \mathrm{rad} /$ minute Angular velocity of minute, $\omega_2=\frac{2 \pi}{60} \mathrm{rad} /$ minute Angular velocity of hour hand, $\omega_3=\frac{2 \pi}{12 	imes 60} \mathrm{rad} /$ minute Angular distance of second's hand in one minute  \ $ \begin{aligned} & x_1=360^{\circ} / 	ext { minute } \ & x_2=\frac{360^{\circ}}{60}=6^{\circ} / 	ext { minute } \ & x_3=\frac{360^{\circ}}{60 	imes 12}=0.5^{\circ} / 	ext { minute } \end{aligned} $ \ By above values, $\frac{\omega_1}{x_1}=\frac{\omega_2}{x_2}=\frac{\omega_3}{x_3}=\frac{2 \pi}{360^{\circ}}=k$ where, $k$ is a constant.