Question

Calculate the angular velocity and linear velocity of a tip of minute hand of length 10cm.

Answer 1

In a clock there are three hand hour hand, minute hand, and second hand each hand is different time period. minute hand takes one hour to complete one rotation, second hand takes one minute to complete one rotation and hour hand takes $12$ hour to complete one rotation, in this question minute hand is given minute hand takes one hour to complete one rotation.

Complete step by step answer:
Step I:
Angular velocity is defined as the speed of rotation of an object. It shows how an object changes its position with respect to time.
Expression of angular velocity, $\omega = \dfrac{\theta }{t}$ …… (1)
Step II:
The linear velocity shows the rate of change of displacement with time if the object is moving in a straight line.
Expression of linear velocity, $v = \omega \times r$ ……. (2)
Where,
r is the radius of tip of minute hand
t is the time in seconds
$\theta$ is the angular distance travelled by minute hand in radians
Step III:
Given values:
Length of a minute hand is $10cm = 0.1m$
To find the value of $\theta$
Minute hand takes a one hour to complete one rotation
1 rotation is equal to ${360^ \circ }$
Convert degree into radian, multiply {360^ \circ }$$$$ \times $$$$\dfrac{\pi }{{{{180}^ \circ }}}
$\theta$ = ${360^ \circ }$ = $2\pi$( radians )
Also evaluate the value of time in seconds
Minute hand takes a one hour to complete one rotation
One minute = $60\sec$
Time in seconds = One hour $= 60\min$=$60 \times 60 = 3600\sec$
Step IV:
To find the angular velocity of a tip of minute hand
Using eq. (1)
$\omega = \dfrac{\theta }{t}$
$\Rightarrow \omega = \dfrac{{2 \times 3.14}}{{3600}}$
$\Rightarrow \omega = 1.745 \times {10^{ - 3}}rad/\sec$
Step V:
To find the linear velocity at the tip of minute hand
Using eq. (2)
V = $r \times \omega$
Length of minute hand = radius of the minute hand = 0.10m
$\Rightarrow v = 0.1 \times 0.1745 \times {10^{ - 3}}m/s$
$\Rightarrow v = 1.745 \times {10^{ - 4}}m/s$

$\therefore$ The angular velocity of the minute hand is $1.745 \times {10^{ - 3}}rad/\sec$
The linear velocity of the minute hand is $1.745 \times {10^{ - 4}}m/s$

Note:
It is to be noted that the term angular velocity and linear velocity are different terms. Since angular motion is always moving in a circular motion, a force is always required to keep it in circular motion. But a linear velocity does not require any external force to keep the object moving along a straight path.