The minute hand of a clock is R = 1.2 m long and the hour…

The minute hand of a clock is $R = 1.2\ \mathrm{m}$ long and the hour hand is $r = 1\ \mathrm{m}$ long. What are the angular velocities of the hands, and what are the speeds of their tips?

Explanation of Method

Angular velocities of clock hands and speeds of their tips.

Step 1: Angular velocities

A hand that makes one full turn in time $T$ has $\omega = 2\pi/T$. The minute hand turns once per hour, the hour hand once per 12 hours: $\omega_m = \frac{2\pi}{3600\ \mathrm{s}} \approx 1.75 \cdot 10^{-3}\ \mathrm{rad/s}$.

\[ \omega_h = \frac{2\pi}{43\,200\ \mathrm{s}} \approx 1.45 \cdot 10^{-4}\ \mathrm{rad/s}. \]

Step 2: Speeds of the tips

A tip at distance $\rho$ from the axis moves with $v = \omega\rho$: $v_m = \omega_m R = 1.75 \cdot 10^{-3} \cdot 1.2 \approx 2.09 \cdot 10^{-3}\ \mathrm{m/s}$.

\[ v_h = \omega_h r \approx 1.45 \cdot 10^{-4}\ \mathrm{m/s}. \]

\[\begin{gathered} \omega_m \approx 1.75 \cdot 10^{-3}\ \mathrm{rad/s}, \quad v_m \approx 2.09\ \mathrm{mm/s} \\ \omega_h \approx 1.45 \cdot 10^{-4}\ \mathrm{rad/s}, \quad v_h \approx 0.145\ \mathrm{mm/s} \end{gathered}\]

Check with ratios

$ \dfrac{\omega_m}{\omega_h} = \dfrac{43\,200}{3600} = 12, $

$ \dfrac{v_m}{v_h} = 12 \cdot \dfrac{1.2}{1} = 14.4\ \checkmark $

$v_m$ $v_h$ $R = 1.2\ \mathrm{m}$ $r = 1\ \mathrm{m}$ $\text{minute hand: once per hour}$ $v_m \approx 2.09\ \mathrm{mm/s}$ $\text{hour hand: once per 12 h}$ $v_h \approx 0.145\ \mathrm{mm/s}$ $\omega_m = 12\,\omega_h$ $v_m \approx 14.4\,v_h$

Both tips move along circles, so their velocities are tangent to the clock face. The minute hand turns 12 times as fast and is longer, so its tip is 14.4 times faster. (Arrow lengths are not to scale.)

Result

\[\boxed{\,\begin{gathered} \omega_m \approx 1.75 \cdot 10^{-3}\ \mathrm{rad/s}, \quad v_m \approx 2.09\ \mathrm{mm/s} \\ \omega_h \approx 1.45 \cdot 10^{-4}\ \mathrm{rad/s}, \quad v_h \approx 0.145\ \mathrm{mm/s} \end{gathered}\,}\]