Textbook Question
Use the formula v = r ω to find the value of the missing variable.
v = 9 m per sec , r = 5 m
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3. Unit Circle
Defining the Unit Circle
Problem 31
Textbook Question
The formula ω = θ/t can be rewritten as θ = ωt. Substituting ωt for θ converts s = rθ to s = rωt. Use the formula s = rωt to find the value of the missing variable.
s = 3π/4 km, r = 2 km, t = 4 sec
Verified step by step guidance1
Identify the given variables and the formula to use. Here, you have the arc length \(s = \frac{3\pi}{4}\) km, radius \(r = 2\) km, and time \(t = 4\) sec. The formula relating these is \(s = r \omega t\), where \(\omega\) is the angular velocity.
Write down the formula explicitly: \(s = r \omega t\). Since \(s\), \(r\), and \(t\) are known, you need to solve for \(\omega\).
Rearrange the formula to isolate \(\omega\): \(\omega = \frac{s}{r t}\).
Substitute the known values into the rearranged formula: \(\omega = \frac{\frac{3\pi}{4}}{2 \times 4}\).
Simplify the expression step-by-step to find the value of \(\omega\). Remember to keep the answer in terms of \(\pi\) unless asked otherwise.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Angular Displacement and Angular Velocity
Angular displacement (θ) measures the angle through which an object rotates, usually in radians. Angular velocity (ω) is the rate of change of angular displacement over time, expressed as ω = θ/t. Understanding this relationship allows conversion between angular and linear quantities.
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Arc Length Formula
The arc length (s) of a circle segment is the distance traveled along the circumference and is given by s = rθ, where r is the radius and θ is the angular displacement in radians. This formula connects linear distance with angular motion.
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Relating Linear and Angular Quantities
By substituting θ = ωt into s = rθ, we get s = rωt, linking linear distance (s), radius (r), angular velocity (ω), and time (t). This formula is essential for finding any missing variable when others are known in rotational motion problems.
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