The ratio of angular velocity of rotation of minute hand of a clock with the angular velocity of rotation of the earth about its own axis is

The correct option is 24.

To find the ratio of the angular velocity of the minute hand of a clock to the angular velocity of the Earth's rotation, we can start by using the definition of angular velocity.

Angular velocity ($\omega$) is given by the formula:

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

where $T$ is the time period of one complete rotation.

First, let us determine the time period for the minute hand of a clock ($T_{\text{minute}}$). The minute hand completes one full rotation of 360 degrees (or $2\pi$ radians) in 1 hour.

$T_{\text{minute}}=1\text{hour}$

Therefore, the angular velocity of the minute hand (${\omega }_{\text{minute}}$) is:

${\omega }_{\text{minute}}=\frac{2\pi }{1\text{hour}}$

Next, let us determine the time period for the Earth's rotation about its own axis ($T_{\text{Earth}}$). The Earth completes one full rotation about its axis in 24 hours.

$T_{\text{Earth}}=24\text{hours}$

Therefore, the angular velocity of the Earth (${\omega }_{\text{Earth}}$) is:

${\omega }_{\text{Earth}}=\frac{2\pi }{24\text{hours}}$

Now, we find the ratio of the angular velocity of the minute hand to the angular velocity of the Earth:

$\text{Ratio}=\frac{{\omega }_{\text{minute}}}{{\omega }_{\text{Earth}}}=\frac{\left(\frac{2\pi }{1\text{hour}}\right)}{\left(\frac{2\pi }{24\text{hours}}\right)}$

Simplifying this fraction by canceling the common $2\pi$ term in both the numerator and the denominator, we get:

$\text{Ratio}=\frac{24\text{hours}}{1\text{hour}}=24$

Thus, the ratio of the angular velocity of the minute hand of a clock to the angular velocity of rotation of the Earth about its own axis is 24.