Textbook Question
Use the formula ω = θ/t to find the value of the missing variable.
ω = 0.91 radian per min, t = 8.1 min
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3. Unit Circle
Defining the Unit Circle
1:46 minutesProblem 23
Textbook Question
Use the formula v = r ω to find the value of the missing variable.
v = 9 m per sec , r = 5 m
Verified step by step guidance1
Identify the given variables and the unknown variable. Here, the linear velocity \(v = 9\) m/s, the radius \(r = 5\) m, and the angular velocity \(\omega\) is unknown.
Recall the formula relating linear velocity, radius, and angular velocity: \(v = r \times \omega\).
Rearrange the formula to solve for the missing variable \(\omega\): \(\omega = \frac{v}{r}\).
Substitute the known values into the rearranged formula: \(\omega = \frac{9}{5}\).
Simplify the expression to find the angular velocity \(\omega\) (do not calculate the final numerical value as per instructions).
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Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Relationship Between Linear and Angular Velocity
The formula v = r ω connects linear velocity (v) and angular velocity (ω) through the radius (r). It shows that linear velocity is the product of the radius and the angular velocity, describing how fast a point on a rotating object moves along its circular path.
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Units and Dimensions in Rotational Motion
In the formula v = r ω, linear velocity (v) is measured in meters per second (m/s), radius (r) in meters (m), and angular velocity (ω) in radians per second (rad/s). Understanding these units helps ensure correct substitution and calculation of the missing variable.
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Solving for a Missing Variable in an Equation
To find the missing variable, rearrange the formula algebraically. For example, if ω is unknown, solve ω = v / r. This process requires isolating the variable on one side and substituting known values to compute the result.
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