Find the angular speed ω for each of the following.


a wind turbine with blades turning at a rate of 15 revolutions per minute

Verified step by step guidance

1. Understand that angular speed \( \omega \) is the rate at which an object rotates or revolves relative to another point, expressed in radians per second.
2. Recall that one complete revolution is equal to \( 2\pi \) radians.
3. Convert the given rate of revolutions per minute to revolutions per second by dividing by 60, since there are 60 seconds in a minute.
4. Multiply the number of revolutions per second by \( 2\pi \) to convert the revolutions to radians per second.
5. The resulting value is the angular speed \( \omega \) in radians per second.

Key Concepts

Angular Speed

Angular speed, denoted by ω, measures how quickly an object rotates around a central point, typically expressed in radians per second. It is calculated by the formula ω = θ/t, where θ is the angle in radians and t is the time in seconds. Understanding angular speed is crucial for converting between different units of rotational motion, such as revolutions per minute (RPM) to radians per second.

Conversion of Units

To solve problems involving angular speed, it is often necessary to convert units. For instance, converting revolutions per minute to radians per second involves using the fact that one revolution equals 2π radians. The conversion formula is ω (rad/s) = RPM × (2π rad/1 rev) × (1 min/60 s), which allows for a seamless transition between different measurement systems.

Revolutions per Minute (RPM)

Revolutions per minute (RPM) is a unit of rotational speed that indicates how many complete turns an object makes in one minute. In the context of a wind turbine, knowing the RPM helps in determining the efficiency and performance of the turbine. It is essential to understand how to interpret and manipulate this unit to find the corresponding angular speed in radians per second.