A motor is rotating at a constant angular velocity of 500 rpm. The angular displacement per second is

Options

A. `(3)/(50 pi)` rad
B. `(3 pi)/(50)` rad
C. `(25 pi)/(3)` rad
D. `(50 pi)/(3)` rad

Explanation

To find the angular displacement per second when a motor is rotating at a constant angular velocity of 500 rpm, we can follow these steps:

### Step 1: Convert RPM to Revolutions per Second
The angular velocity (ω) is given in revolutions per minute (rpm). We need to convert this to revolutions per second (rps).

\[ \text{Revolutions per second} = \frac{500 \text{ rpm}}{60} = \frac{500}{60} \text{ rps} \]

### Step 2: Simplify the Fraction
Now, simplify the fraction:

\[ \frac{500}{60} = \frac{50}{6} \text{ rps} \]

### Step 3: Calculate Angular Displacement in Revolutions per Second
In one second, the motor completes \( \frac{50}{6} \) revolutions.

### Step 4: Convert Revolutions to Radians
We know that 1 revolution is equal to \( 2\pi \) radians. Therefore, the angular displacement (θ) in radians per second is:

\[ \theta = \left(\frac{50}{6} \text{ revolutions}\right) \times (2\pi \text{ radians/revolution}) \]

### Step 5: Calculate Angular Displacement
Now, calculate θ:

\[ \theta = \frac{50}{6} \times 2\pi = \frac{100\pi}{6} = \frac{50\pi}{3} \text{ radians} \]

### Conclusion
Thus, the angular displacement per second is:

\[ \theta = \frac{50\pi}{3} \text{ radians} \]

### Final Answer
The correct option is option 4: \( \frac{50\pi}{3} \) radians.

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