Textbook Question
Use the formula ω = θ/t to find the value of the missing variable.
ω = 2π/3 radians per sec, t = 3 sec
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3. Unit Circle
Defining the Unit Circle
Problem 21
Textbook Question
Use the formula v = r ω to find the value of the missing variable.
r = 12 m , ω = 2π/3 radians per sec
Verified step by step guidance1
Identify the given variables and the formula: the radius \(r = 12\) meters, the angular velocity \(\omega = \frac{2\pi}{3}\) radians per second, and the formula relating linear velocity \(v\), radius \(r\), and angular velocity \(\omega\) is \(v = r \times \omega\).
Substitute the known values into the formula: replace \(r\) with 12 and \(\omega\) with \(\frac{2\pi}{3}\), so the equation becomes \(v = 12 \times \frac{2\pi}{3}\).
Simplify the multiplication by multiplying the constants and keeping \(\pi\) as is: calculate \(12 \times \frac{2}{3}\) first, then multiply the result by \(\pi\).
Express the simplified product as the linear velocity \(v\) in terms of \(\pi\), which represents the angular velocity converted to linear velocity.
Interpret the result as the linear speed of a point on the circumference of a circle with radius 12 meters rotating at \(\frac{2\pi}{3}\) radians 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 (ω)
Angular velocity measures how fast an object rotates or revolves, expressed in radians per second. It indicates the angle covered per unit time and is crucial for relating rotational motion to linear motion.
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Radius (r)
The radius is the distance from the center of a circular path to any point on its circumference. In rotational motion, it determines the path length traveled by a point on the rotating object and directly affects linear velocity.
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Linear Velocity (v) and the formula v = rω
Linear velocity is the speed of a point moving along a circular path, calculated by multiplying the radius by the angular velocity (v = rω). This formula connects rotational speed to the actual speed along the circle's edge.
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