A boy sitting on a merry-round make 200 revolutions per minute . Calculate the angular speed of the merry-go-round.

Explanation

To solve the problem of calculating the angular speed of a merry-go-round given that a boy makes 200 revolutions per minute, we can follow these steps:

### Step 1: Understand the relationship between revolutions and radians
1 revolution is equal to \(2\pi\) radians. Therefore, if the boy makes 200 revolutions, the total angle in radians can be calculated as:
\[ \text{Total angle in radians} = \text{Number of revolutions} \times 2\pi = 200 \times 2\pi \]

### Step 2: Convert revolutions per minute to radians per minute
Now, we can calculate the total angle in radians:
\[ \text{Total angle in radians} = 200 \times 2\pi = 400\pi \text{ radians} \]

### Step 3: Convert time from minutes to seconds
Since the angular speed is typically expressed in radians per second, we need to convert the time from minutes to seconds. There are 60 seconds in a minute, so:
\[ \text{Time in seconds} = 1 \text{ minute} = 60 \text{ seconds} \]

### Step 4: Calculate angular speed
Angular speed (\(\omega\)) is defined as the total angle traveled divided by the time taken. Therefore:
\[ \omega = \frac{\text{Total angle in radians}}{\text{Time in seconds}} = \frac{400\pi \text{ radians}}{60 \text{ seconds}} \]

### Step 5: Simplify the expression
Now we simplify the expression:
\[ \omega = \frac{400\pi}{60} = \frac{20\pi}{3} \text{ radians per second} \]

### Final Answer
The angular speed of the merry-go-round is:
\[ \omega = \frac{20\pi}{3} \text{ radians per second} \]

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