Question

If $\omega_1$ is the angular velocity of the hour hand of a clock and $\omega_2$ is the angular velocity of the earth's rotation about its axis, then the ratio $\omega_1 : \omega_2$ is

• A. $1 : 2$
• B. $2 : 3$
• C. $3 : 2$
• D. $2 : 1$

Hint:

Think about it conceptually: angular velocity measures how fast something spins. The hour hand goes around twice in 24 hours ($2$ full circles), while the Earth goes around only once in 24 hours ($1$ full circle). Since the hour hand completes twice as many revolutions in the same amount of time, its angular speed must be exactly double that of the Earth!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the numerical ratio between two distinct steady periodic rotations: the angular velocity of a clock's hour hand ($\omega_1$) and the angular velocity of the Earth spinning around its own geographic axis ($\omega_2$).

Step 2: Key Formula or Approach:
Angular velocity ($\omega$) is related to the total time period ($T$) required to complete one full revolution ($2\pi$ radians) by the formula: $$\omega = \frac{2\pi}{T}$$ Therefore, the ratio of two angular velocities is inversely proportional to the ratio of their respective time periods: $$\frac{\omega_1}{\omega_2} = \frac{T_2}{T_1}$$

Step 3: Detailed Explanation:
Let's determine the time periods for both rotations in identical units (hours): 1. Hour hand of a clock ($T_1$): The hour hand takes exactly 12 hours to complete one full cycle around the dial face. $$T_1 = 12\ \text{hours}$$ 2. Earth's axial rotation ($T_2$): The Earth takes exactly 1 day to complete one full rotation about its axis. $$T_2 = 24\ \text{hours}$$ Now, substitute these time periods into our inverse ratio formula: $$\frac{\omega_1}{\omega_2} = \frac{24\ \text{hours}}{12\ \text{hours}} = \frac{2}{1} = 2 : 1$$

Step 4: Final Answer:
The ratio $\omega_1 : \omega_2$ is $2 : 1$, which corresponds to option (D).