The angular velocity of a particle rotating in a circular orbit 100 times per minute is

Options

A. `1.66 rad//s`
B. `10.47 rad//s`
C. `10.47 deg//s`
D. `60 deg//s`

Answer

To find the angular velocity of a particle rotating in a circular orbit at 100 times per minute, we can follow these steps:

### Step 1: Understand the relationship between angular velocity and rotations
The angular velocity (ω) is defined as the angular displacement (θ) per unit time (t). In this case, the particle completes 100 rotations in one minute.

### Step 2: Convert rotations to radians
Since angular displacement in physics is typically measured in radians, we need to convert the number of rotations to radians.
- One complete rotation corresponds to \(2\pi\) radians.
- Therefore, for 100 rotations:
\[ \theta = 100 \times 2\pi \text{ radians} \]
\[ \theta = 200\pi \text{ radians} \]

### Step 3: Convert time from minutes to seconds
We also need to convert the time from minutes to seconds because we want the angular velocity in radians per second.
- 1 minute = 60 seconds.

### Step 4: Calculate angular velocity
Now we can calculate the angular velocity using the formula:
\[ \omega = \frac{\theta}{t} \]
Substituting the values we have:
\[ \omega = \frac{200\pi \text{ radians}}{60 \text{ seconds}} \]

### Step 5: Simplify the expression
We can simplify this expression:
\[ \omega = \frac{200\pi}{60} = \frac{10\pi}{3} \text{ radians per second} \]

### Step 6: Calculate the numerical value
To find a numerical approximation for \(\omega\):
\[ \omega \approx 10.47 \text{ radians per second} \]

### Final Answer
Thus, the angular velocity of the particle is approximately:
\[ \omega \approx 10.47 \text{ rad/s} \]