Question

The length of the second hand of a clock is $4\;{\rm{cm}}$. The speed of the tip of the second hand is:
A. ${0.24\,cm/s}$
B. ${0.32\,cm/s}$
C. ${0.42\,cm/s}$
D. ${0.50\,cm/s}$

Answer 1

In the solution we will use the equation of circumference of circle and the equation of the time, speed and distance which represent the relation as the distance covered by the object is equal to the product of speed and time.

Complete step by step answer:
Given:
The length of the second hand of a clock is$4\;{\rm{cm}}$ means it works as a radius for the watch, $r = 4\;{\rm{cm}}$.
The circumference of the watch can be calculated by using the circumference formula ${\rm{Circumference}}\;{\rm{ = }}\;{\rm{2}}\pi {\rm{r}}$.
Substitute the value of $r = 4\;{\rm{cm}}$ in ${\rm{Circumference}}\;{\rm{ = }}\;{\rm{2}}\pi {\rm{r}}$ to find the value of circumference of the watch.
${\rm{Circumference}}\;{\rm{ = }}\;{\rm{2}}\pi {\rm{r}}\\\ \Rightarrow{\rm{ = 2}}\left( {\dfrac{{22}}{7}} \right)\left( 4 \right)\\\ \Rightarrow \dfrac{{176}}{7}\\\ \Rightarrow \approx 25.14\;{\rm{cm}}$
The time taken by the second hand of the watch to complete one round is $t = 60\;\sec$.
Substitute the value ${\rm{Circumference}} \approx 25.14\;{\rm{cm}}$ and the value of time $t = 60\;\sec$ in the formula ${\rm{Speed}}\;{\rm{ = }}\;\dfrac{{{\rm{Circumference}}}}{{{\rm{time}}}}$ to obtain the value of speed of the tip of the second hand.
$Speed = \dfrac{Circumference}{time}$
$\Rightarrow$ $\dfrac{25.14}{60}$
$\Rightarrow$ $\approx 0.42\,cm$

Therefore, the option (C) is the correct answer that is $0.42{{{\rm{cm}}} {\left/ { {{{\rm{cm}}} {\rm{s}}}} \right. } {\rm{s}}}$.

Note: Remember that the tip of the second hand is on the circumference of the clock so the speed of the tip of the second hand is equal to the circumferential speed. The speed of the different hands of the clock are different because the speed depends on the time taken to complete one complete revolution of the clock by that respective hand of the clock. For example the time taken to complete one revolution by the second hand is $60\;{\rm{sec}}$ and for the minute hand is $3600\;{\rm{sec}}$ whereas for the hour hand is equal to the $43200\;\sec$.