The relative angular speed of the minute hand and second hand of a clock is

Options

A. `(29pi)/(1800) rad//s`
B. `(39pi)/(1800) rad //s`
C. `(49 pi)/(1800 ) rad //s`
D. `(59 pi )/(1800) rad//s`

Explanation

To find the relative angular speed of the minute hand and the second hand of a clock, we can follow these steps:

### Step 1: Understand Angular Speed
The angular speed (ω) of an object in circular motion is defined as the angle it covers in a unit time. For a clock:
- The second hand completes one full rotation (2π radians) in 60 seconds.
- The minute hand completes one full rotation (2π radians) in 3600 seconds (60 minutes).

### Step 2: Calculate Angular Speeds
We can calculate the angular speeds of both hands:
- **Angular speed of the second hand (ω_second)**:
\[ \omega_{\text{second}} = \frac{2\pi \text{ radians}}{60 \text{ seconds}} = \frac{\pi}{30} \text{ radians/second} \]

- **Angular speed of the minute hand (ω_minute)**:
\[ \omega_{\text{minute}} = \frac{2\pi \text{ radians}}{3600 \text{ seconds}} = \frac{\pi}{1800} \text{ radians/second} \]

### Step 3: Determine Relative Angular Speed
The relative angular speed of the second hand with respect to the minute hand can be calculated using the formula:
\[ \omega_{\text{relative}} = \omega_{\text{second}} - \omega_{\text{minute}} \]

Substituting the values we calculated:
\[ \omega_{\text{relative}} = \frac{\pi}{30} - \frac{\pi}{1800} \]

### Step 4: Find a Common Denominator
To subtract these fractions, we need a common denominator. The least common multiple of 30 and 1800 is 1800. Thus, we can rewrite the first term:
\[ \frac{\pi}{30} = \frac{60\pi}{1800} \]
Now we can perform the subtraction:
\[ \omega_{\text{relative}} = \frac{60\pi}{1800} - \frac{\pi}{1800} = \frac{60\pi - \pi}{1800} = \frac{59\pi}{1800} \text{ radians/second} \]

### Step 5: Final Answer
The relative angular speed of the minute hand and the second hand of a clock is:
\[ \frac{59\pi}{1800} \text{ radians/second} \]

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