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

Understand that angular speed \( \omega \) is the rate of change of the angle with respect to time, usually measured in radians per second (rad/s).

Note that the problem gives the rotational speed in revolutions per minute (rpm), so the first step is to convert revolutions to radians. Recall that one revolution corresponds to \( 2\pi \) radians.

Convert the given speed from revolutions per minute to radians per minute by multiplying the number of revolutions by \( 2\pi \): \[ \omega = 15 \times 2\pi \quad \text{radians per minute} \]

Next, convert the angular speed from radians per minute to radians per second by dividing by 60 (since there are 60 seconds in a minute): \[ \omega = \frac{15 \times 2\pi}{60} \quad \text{radians per second} \]

This expression now represents the angular speed \( \omega \) in radians per second. You can simplify this expression further if needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angular Speed (ω)

Angular speed measures how fast an object rotates or revolves relative to a fixed point, expressed in radians per second or revolutions per minute. It quantifies the rate of change of the angular position of a rotating body.

Conversion between Revolutions and Radians

One complete revolution corresponds to 2π radians. To convert angular speed from revolutions per minute (rpm) to radians per second, multiply by 2π and divide by 60, since there are 60 seconds in a minute.

Units and Dimensional Analysis

Understanding and converting units correctly is essential in trigonometry and physics problems. Here, converting rpm to radians per second ensures the angular speed is in standard SI units, facilitating further calculations or comparisons.

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