# Step-by-Step Solution ## 1. **Given Data** - Rotational speed of shaft, \( N = 450 \) rpm (revolutions per minute) ## 2. **Required** - Find the angular velocity \( \omega \) in radians per second (rad/s). ## 3. **Formula for Angular Velocity** Angular velocity \( \omega \) (in rad/s) is related to rotational speed \( N \) (in rpm) by: \[ \omega = 2\pi N / 60 \] where: - \( N \) is in revolutions per minute (rpm) - \( 2\pi \) radians = 1 revolution - 60 seconds = 1 minute ## 4. **Substitute the Values** \[ \omega = \frac{2\pi \times 450}{60} \] ## 5. **Simplify** \[ \omega = \frac{2\pi \times 450}{60} = \frac{2 \times 3.1416 \times 450}{60} \] Calculate numerator: \[ 2 \times 3.1416 = 6.2832 \] \[ 6.2832 \times 450 = 2827.44 \] Now divide by 60: \[ \omega = \frac{2827.44}{60} = 47.124 \] ## 6. **Final Answer** \[ \boxed{47.12 \ \text{rad/s}} \] (Rounded to two decimal places.) --- **Summary:** A shaft rotating at 450 rpm has an angular velocity of **47.12 rad/s**.