Identify the increasing order of the angular velocities of the following
1. Earth rotating about its own axis
2. Hour's hand of a clock
3. Second's hand of a clock
4. Flywheel of radius 2 m making 300 rpm

Options

A. a, b, c, d
B. b, c, d, a
C. c, d, a, b
D. d, a, b, c

Answer

To determine the increasing order of angular velocities for the given scenarios, we need to calculate the angular velocity (ω) for each case. The formula for angular velocity is:

\[ \omega = \frac{2\pi}{T} \]

where \( T \) is the time period in hours.

### Step 1: Calculate the angular velocity of the Earth rotating about its own axis.
- The time period \( T \) for the Earth to complete one rotation is 24 hours.

\[ \omega_{\text{Earth}} = \frac{2\pi}{24} \text{ rad/hour} \]

### Step 2: Calculate the angular velocity of the hour hand of a clock.
- The time period \( T \) for the hour hand to complete one rotation is 12 hours.

\[ \omega_{\text{Hour Hand}} = \frac{2\pi}{12} \text{ rad/hour} \]

### Step 3: Calculate the angular velocity of the second hand of a clock.
- The time period \( T \) for the second hand to complete one rotation is 60 seconds, which is \( \frac{1}{60} \) hours.

\[ \omega_{\text{Second Hand}} = \frac{2\pi}{\frac{1}{60}} = 120\pi \text{ rad/hour} \]

### Step 4: Calculate the angular velocity of the flywheel making 300 rpm.
- The flywheel makes 300 revolutions per minute (rpm). To convert this to hours, we note that 300 rpm is equivalent to \( 300 \times 60 \) revolutions per hour.

The time period \( T \) for one revolution is:

\[ T = \frac{1}{300} \text{ minutes} = \frac{1}{300} \times \frac{60}{1} \text{ hours} = \frac{1}{5} \text{ hours} \]

Now, we can calculate the angular velocity:

\[ \omega_{\text{Flywheel}} = \frac{2\pi}{\frac{1}{5}} = 10\pi \text{ rad/hour} \]

### Step 5: Compare the angular velocities.
Now we have the angular velocities:
1. Earth: \( \frac{2\pi}{24} \text{ rad/hour} \)
2. Hour Hand: \( \frac{2\pi}{12} \text{ rad/hour} \)
3. Second Hand: \( 120\pi \text{ rad/hour} \)
4. Flywheel: \( 10\pi \text{ rad/hour} \)

### Step 6: Convert to a common format for comparison.
- Earth: \( \frac{2\pi}{24} = \frac{\pi}{12} \text{ rad/hour} \)
- Hour Hand: \( \frac{2\pi}{12} = \frac{\pi}{6} \text{ rad/hour} \)
- Second Hand: \( 120\pi \text{ rad/hour} \)
- Flywheel: \( 10\pi \text{ rad/hour} \)

### Step 7: Arrange in increasing order.
Now we can compare:
- \( \frac{\pi}{12} < \frac{\pi}{6} < 10\pi < 120\pi \)

Thus, the increasing order of angular velocities is:
1. Earth
2. Hour Hand
3. Flywheel
4. Second Hand

### Final Answer:
The increasing order of angular velocities is:
- Earth < Hour Hand < Flywheel < Second Hand