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Page 272: Practice Problems
The angular speed of the CD is $omega = 44 mathrm{~rad/s}$. The tangential speed of the CD is $v = 1.25 mathrm{~m/s}$.
$textbf{Required: }$
Finding the distance from the axis of rotation (radius).
As the textbook mentions, the tangential speed of a rotating object is given by
$$
begin{align*}
v &= r ~ omega \
end{align*}
$$
Rearrange and solve for the distance from the axis of rotation:
$$
begin{align*}
r &= dfrac{ v }{ omega } \
&= dfrac{ 1.25 mathrm{~m/s} }{ 44 mathrm{~rad/s} } \
&= 0.028 mathrm{~m} \
&= 2.841 mathrm{~cm}
end{align*}
$$
So, the distance from the axis of rotation (radius) is $2.841 mathrm{~cm}$.
The length of the hour’s hand is $r = 8.2 mathrm{~cm}$. The time taken to cover one revolution by the hour hand is $Delta t = 24 mathrm{~hr}$.
$textbf{Required: }$
Finding the tangential speed of the tip of the hour’s hand.
As the textbook mentions, the tangential speed of a rotating object is given by
$$
begin{align*}
v &= r ~ omega \
end{align*}
$$
In order to evaluate the angular speed, we use the following relation:
$$
begin{align*}
omega &= dfrac{ 2 pi }{ T } \
end{align*}
$$
Substituting from the previous calculations, then we get
$$
begin{align*}
v &= r ~ omega \
&= r ~ dfrac{ 2 pi }{ T } \
&= 8.2 times 10^{-2} mathrm{~m} times dfrac{ 2 pi }{ 24 times 60 times 60 mathrm{~s} } \
&= 5.963 times 10^{-6} mathrm{~m/s}
end{align*}
$$
So, the tangential speed of the tip of the hour’s hand is $5.963 times 10^{-6} mathrm{~m/s}$.