Ch. 3 - Radian Measure and The Unit Circle

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 3 - Radian Measure and The Unit Circle

Problem 25

Chapter 4, Problem 25

Use the formula v = r ω to find the value of the missing variable.

v = 12 m per sec, ω = 3π/2 radians per sec

Verified step by step guidance

Identify the given variables and the formula: the formula is \(v = r \times \omega\), where \(v\) is the linear velocity, \(r\) is the radius, and \(\omega\) is the angular velocity.

From the problem, you have \(v = 12\) m/s and \(\omega = \frac{3\pi}{2}\) radians per second. The missing variable is \(r\) (the radius).

Rearrange the formula to solve for \(r\): \(r = \frac{v}{\omega}\).

Substitute the known values into the rearranged formula: \(r = \frac{12}{\frac{3\pi}{2}}\).

Simplify the expression by dividing 12 by \(\frac{3\pi}{2}\) to find the value of \(r\).

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Key Concepts

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

Linear Velocity (v)

Linear velocity refers to the speed at which a point on a rotating object moves along its circular path. It is measured in units like meters per second (m/s) and depends on both the angular velocity and the radius of the rotation.

Angular Velocity (ω)

Angular velocity measures how fast an object rotates or spins, expressed in radians per second (rad/s). It represents the rate of change of the angular displacement and is crucial for relating rotational motion to linear motion.

Relationship Between Linear and Angular Velocity (v = rω)

The formula v = rω connects linear velocity (v), radius (r), and angular velocity (ω). It shows that linear velocity is the product of the radius of the circular path and the angular velocity, allowing calculation of any one variable if the other two are known.

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