Question

The minute hand of a clock is 8cm long. Calculate the linear speed of an ant at this time.

Answer 1

Hints :Recall that when an object rotates in circular motion, it is said to have some angular velocity. It is defined as the rate of velocity at which the particle is rotating at a particular point. The angular position of the object changes with time. But when the object changes its displacement while moving along a straight path it is said to have some linear velocity.

Complete step by step solution:
Step I:
The angular velocity shows as to how fast an object can move or change its position with respect to the centre of origin. The formula for angular velocity is given by

$\omega = \dfrac{{2\pi }}{T}$

Where $\omega$is the angular velocity
$T$is the time taken by the minute hand of the clock
The time taken by the minute hand to reach back to its starting point will be one minute. Or

$T = 60 \times 60 = 3600\sec$

Step II:
Substituting all the values and solving,

$\omega = \dfrac{{2 \times 3.14}}{{3600}}$
$\omega = 1.74 \times {10^{ - 3}}rad/\sec$

Step III:
The linear velocity of the ant at this time is given by

$v = \omega L$

Where $v$is the linear velocity
$L$is the length
Given $L = 8cm = 0.08m$
Substituting the value in the equation and solving,

$v = 1.74 \times {10^{ - 3}} \times 0.08$
$v = 1.39 \times {10^{ - 4}}m/s$

Step IV:
Therefore, the linear speed of ant at this time is $1.39 \times {10^{ - 4}}m/s$

Note: It is to be remembered that the linear velocity and the angular velocity have both magnitude and direction. So they are both vector quantities. Any object that is moving in a circular path will have both linear and angular velocities. If the object is rotating in clockwise direction then it will have negative angular velocity. But if the object is rotating in anticlockwise direction, then it will have positive angular velocity.