Question

Consider a second hand with length $20\,cm$.
$\left( a \right)$ Find the angular velocity of the second hand.
$\left( b \right)$ Find the magnitude of velocity of the tip of the second hand.
$\left( c \right)$ Find acceleration of the tip of the second hand.

Answer 1

Hint: When the hands of a clock move, it has all three motions that are rotational motion, periodic motion and circular motion. The motion of the hand of the clock is rotational and periodic but when the tip of the needle is considered it is circular and periodic motion.

Formulas used:
$\omega = \dfrac{\theta }{t}$
$\Rightarrow v = r \times \omega $
$\Rightarrow a = \dfrac{{{v^2}}}{r}$
Where, $\omega $ = Angular velocity, $\theta $ = Angular displacement, $t$ = Time, $v$ = Linear velocity, $r$ = Radius and $a$ = Acceleration.

Complete step by step answer:
$\left( a \right)$ Angular velocity of second hand.
$\omega = \dfrac{\theta }{t}$ ………….. $\left( 1 \right)$
Angular velocity $\omega $ = $?$
There are \[2\pi \] radians in one complete rotation, and that takes the second hand $60$ seconds to complete.
So, the rate of rotation that is angular velocity is equal to $\dfrac{{2\pi }}{{60}}$ = $\dfrac{\pi }{{30}}$$\dfrac{{rad}}{{\sec }}$
Therefore, Angular velocity of second hand $\omega $ = $0.105\,\dfrac{{rad}}{{\sec }}$

$\left( b \right)$ Magnitude of velocity of tip of second hand.
$v = r \times \omega $ ……… $\left( 2 \right)$
Substituting the given data in equation $\left( 2 \right)$
$v = 20 \times 0.1047$
On simplifying the above equation, we get
Therefore, the velocity of the tip of the second hand, $v = 2.094\,m{s^{ - 1}}$.

$\left( b \right)$ Acceleration of tip of second hand.
$a = \dfrac{{{v^2}}}{r}$ ………. $\left( 3 \right)$
Substituting the given data in equation $\left( 3 \right)$
$a = \dfrac{{{{\left( {2.094} \right)}^2}}}{{20}}$
On simplifying the above equation, we get
Therefore, the acceleration of tip of second hand,$a = 0.219\,m{s^{ - 2}}$.

Additional information: Angular velocity or rotational velocity also known as angular frequency vector is the vector measure of rotate rate, that is it refers to how fast an object rotates or revolves with respect to another point. That is how fast the angular orientation or position of an object changes with time.

Note: The angular velocity is independent of the clock size, however for larger clocks the linear velocity of the pointers at the end of the hands will be greater. The second hand goes through $2\pi $ radians in $1\min $.