Problem 79 - Algebra and Trigonometry

Question

Fan A ceiling fan with 16 -in. blades rotates at 45 \(\mathrm{rpm}\) . (a) Find the angular speed of the fan in rad/min. (b) Find the linear speed of the tips of the blades in in./min.

Step-by-Step Solution

Answer

(a) 90\pi rad/min (b) 1440\pi in/min

Step 1: Convert RPM to Angular Speed

The fan rotates at 45 revolutions per minute (RPM). To find the angular speed in rad/min, recall that one revolution is equal to \(2\pi\) radians. Therefore, we multiply the number of revolutions per minute by \(2\pi\). \[\text{Angular speed} = 45 \times 2\pi = 90\pi \text{ rad/min}\]

Step 2: Calculate Linear Speed

The linear speed of the tip of the blade is related to the angular speed by the formula: \[v = r\omega\]where \(v\) is linear speed, \(r\) is the radius (length of blade, 16 inches), and \(\omega\) is the angular speed. Substituting the known values:\[v = 16 \times 90\pi = 1440\pi \text{ inches/min}\]

Key Concepts

Understanding Angular Speed

Angular speed is a measure of how quickly an object rotates or revolves relative to another point, typically around a fixed axis. When we talk about angular speed, we are referring to the amount of rotation that occurs in a specific time period. For understanding, imagine the blades of a ceiling fan rotating around the fan's central point.

The unit most commonly used for angular speed in trigonometry is radians per unit time, such as radians per minute (rad/min). This is because the radian is a natural measure of angle based on the radius of a circle. One complete revolution around a circle is equivalent to an angle of \(2\pi\) radians.

To convert revolutions per minute (RPM) to radians per minute, we use the conversion factor that 1 revolution equals \(2\pi\) radians. So, if a fan spins at 45 RPM, its angular speed in rad/min is calculated by multiplying \(45\) by \(2\pi\), resulting in an angular speed of \(90\pi\) rad/min. This tells us how fast the fan "spins around" in terms of radians.

Exploring Linear Speed

Linear speed refers to how fast a point on the edge of a rotating object is moving along its path. Unlike angular speed, which measures rotation rate, linear speed measures how fast something is moving along a line. A good example is the tip of a blade of a rotating ceiling fan.

To find linear speed, we use the formula \(v = r\omega\), where \(v\) is the linear speed, \(r\) is the radius, and \(\omega\) is the angular speed. The radius in the case of the fan is the length of the blade from the center to the tip, which is 16 inches.

Substituting the radius and angular speed, we calculate the linear speed as \(16 \times 90\pi = 1440\pi\) inches per minute. This result gives us the speed at which the blade tips of the fan are moving through the air.

Radians per Minute: A Fundamental Unit

Radians per minute (rad/min) is a unit of angular speed that is fundamental in trigonometry and rotational motion. When dealing with rotating objects, using radians provides a clear understanding of how far a point has rotated along a circular path in a given time frame.

Why radians? A radian is based on the radius of the circle, making it a natural and suitable way to express angles that result from circular motion. Since one revolution equals \(2\pi\) radians, using radians makes calculations associated with circular or rotational motion more mathematically manageable.

In the given problem about the fan, noting that the angular speed is \(90\pi\) rad/min helps correlate the physical motion of the fan to a mathematical representation. Recognizing how radians per minute work helps deepen understanding of how rotations translate into real-time movements.