Identify the increasing order of the angular velocities of the following 1. earth rotating about its own…
Identify the increasing order of the angular velocities of the following
1. earth rotating about its own axis
2. hour's hand of a clock
3. second's hand of a clock
4. flywheel of radius $2 \mathrm{~m}$ making $300\mathrm{rpm}$
1,2,3,4
3,4,1,2
3,4,1,2
4,1,2,3
Solution
(1) Angular velocity of earth,
$\begin{aligned}
\omega_1 & =\frac{2 \pi}{T}=\frac{2 \pi}{24 \times 60 \times 60} \\
& =\frac{2 \pi}{86400} \mathrm{rad} / \mathrm{s}
\end{aligned}$
(2) Angular velocity of hour's hand of a clock,
$\begin{aligned}
\omega_2 & =\frac{2 \pi}{T} \\
& =\frac{2 \pi}{12 \times 60 \times 60}=\frac{2 \pi}{43200} \mathrm{rad} / \mathrm{s}
\end{aligned}$
(3) Angular velocity of second's hand of a clock,
$\omega_3=\frac{2 \pi}{T}=\frac{2 \pi}{1 \times 60}=\frac{2 \pi}{60} \mathrm{rad} / \mathrm{s}$
(4) Angular velocity of flywheel,
$\begin{aligned}
\omega_4 & =2 \pi n=2 \pi \times \frac{300}{60} \\
& =2 \pi \times 5 \mathrm{rad} / \mathrm{s}=10 \pi \mathrm{rad} / \mathrm{s}
\end{aligned}$
So, the increasing order is
$1 < 2 < 3 < 4$