Textbook Question
CONCEPT PREVIEW Find the radius of each circle.
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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 3.41
Find the linear speed v for each of the following.
a point on the edge of a flywheel of radius 2 m, rotating 42 times per min
Verified step by step guidance1
Identify the given values: the radius of the flywheel \(r = 2\) meters, and the rotational speed \(f = 42\) revolutions per minute (rpm).
Recall the formula for linear speed \(v\) of a point on the edge of a rotating object: \(v = r \times \omega\), where \(\omega\) is the angular velocity in radians per unit time.
Convert the rotational speed from revolutions per minute to angular velocity in radians per second using the conversion: \(\omega = 2\pi \times \frac{f}{60}\), because one revolution equals \(2\pi\) radians and there are 60 seconds in a minute.
Substitute the values into the angular velocity formula: \(\omega = 2\pi \times \frac{42}{60}\) radians per second.
Finally, calculate the linear speed by multiplying the radius by the angular velocity: \(v = 2 \times \omega\). This will give the linear speed in meters per second.
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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, typically expressed in radians per second. It relates the number of rotations per unit time to the angle covered, where one full rotation equals 2π radians. Converting rotations per minute (rpm) to radians per second is essential for calculations involving rotational motion.
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Linear Speed in Circular Motion
Linear speed (v) of a point on a rotating object is the distance traveled per unit time along the circular path. It is related to angular velocity (ω) by the formula v = ωr, where r is the radius. This relationship connects rotational motion to linear motion at the edge of the rotating body.
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Unit Conversion and Application
Accurate unit conversion is crucial, especially converting rpm to radians per second and ensuring radius units are consistent. Applying the correct formula with proper units allows for precise calculation of linear speed, which is vital in practical problems involving rotating objects like flywheels.
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Related Practice
Textbook Question
Find a calculator approximation to four decimal places for each circular function value.
sec 7.3159
Textbook Question
Give an expression that generates all angles coterminal with an angle of π/6 radian. Let n represent any integer.
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Textbook Question
Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.
cos s = 0.9250
Textbook Question
Find the linear speed v for each of the following.
the tip of a propeller 3 m long, rotating 500 times per min (Hint: r = 1.5 m)
Textbook Question
Convert each degree measure to radians. Leave answers as multiples of π .
45°