Textbook Question
Find the angular speed ω for each of the following.
a wind turbine with blades turning at a rate of 15 revolutions per minute
Ch. 3 - Radian Measure and The Unit Circle
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 4, Problem 3.39
Find the linear speed v for each of the following.
the tip of the minute hand of a clock, if the hand is 7 cm long
Verified step by step guidance1
Identify the radius of the circular path traced by the tip of the minute hand. Since the minute hand is 7 cm long, the radius \(r\) is 7 cm.
Determine the angular speed \(\omega\) of the minute hand. The minute hand completes one full revolution (360 degrees or \(2\pi\) radians) in 60 minutes, so calculate \(\omega\) in radians per minute using the formula \(\omega = \frac{2\pi}{T}\), where \(T\) is the period (60 minutes).
Recall the relationship between linear speed \(v\) and angular speed \(\omega\): \(v = r \times \omega\).
Substitute the values of \(r\) and \(\omega\) into the formula \(v = r \times \omega\) to express the linear speed of the tip of the minute hand.
Interpret the result as the linear speed in centimeters per minute, which represents how fast the tip of the minute hand moves along its circular path.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Angular Velocity
Angular velocity measures how fast an object rotates or revolves relative to a fixed point, expressed in radians per second. For a clock's minute hand, it completes one full rotation (2π radians) every 3600 seconds, which helps determine the rate of rotation needed to find linear speed.
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Relationship Between Linear and Angular Velocity
Linear speed (v) at a point on a rotating object is related to angular velocity (ω) by the formula v = ωr, where r is the radius or distance from the rotation axis. This relationship allows conversion from rotational speed to the actual speed along the circular path.
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Radius of Rotation
The radius of rotation is the distance from the center of rotation to the point of interest. In this problem, the length of the minute hand (7 cm) serves as the radius, which is essential for calculating the linear speed of the tip of the hand.
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Related Practice
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1
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