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Questions and Answers
Angular speed measures how fast an object moves relative to a fixed location.
False
The unit for angular speed is usually m/s.
False
The formula for angular speed ( ext{ω}) is ω = θ/t.
True
If an object completes a full rotation, its angular speed can be calculated as ω = 2π/t.
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The relationship between linear velocity ( ext{v}), angular velocity ( ext{ω}), and radius ( ext{r}) is given by the formula: ω = rac{v}{r}.
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An increase in the linear velocity of an object leads to a decrease in its angular speed when considering an increasing radius.
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Study Notes
Angular Speed
Introduction
Angular speed refers to the amount an object rotates within a certain amount of time. Unlike linear speed, which describes how fast an object moves relative to a fixed location, angular speed describes how quickly an object turns or spins around its own axis. The unit for angular speed is usually rad/s (radians per second).
Formula and Unit
The formula for angular speed ((ω)) is derived from the concept of a body moving in a circular path:
[ω = \frac{θ}{t}, \text{ where } θ = \sin^{-1}(\frac{v^2}{R^2 + v^2})]
Here (θ) represents the angle traversed by the body along a circular path, measured in radians, while (t) stands for the time taken to traverse that angle, typically measured in seconds.
For example, the angular speed of an object completing a full rotation would be:
[ω = \frac{2π}{t}, \text{ where } t \text{ is the time taken to complete the rotation}]
In general, the relationship between linear velocity ((v)), angular velocity ((ω)), and radius ((r)) can be expressed as follows:
[ω = \frac{v}{r}]
This implies that an increase in the linear velocity of an object leads to an increase in its angular speed when considered in relation to an increasing radius.
Solved Examples
Problem 1:
If the Earth takes 365 days to complete a revolution around the sun, we can calculate its angular speed using the formula:
[ω = \frac{θ}{t} = \frac{2π}{365} \times \frac{1}{24} \times \frac{1}{60} \times \frac{1}{60} = 1.9923 \times 10^{-7} \text{ rad/s}]
Problem 2:
If a wagon wheel has a radius of 1 meter and a linear speed of 5 meters per second, its angular speed can be calculated as:
[ω = \frac{v}{r} = 5 \text{ rad/s}]
These solved examples demonstrate how to calculate the angular speed of objects based on their linear speed and time taken for rotation.
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Description
Learn about angular speed, which measures how quickly an object rotates around its axis. Explore the formula for angular speed using radians per second as the unit of measurement. Practice solving problems involving linear velocity, radius, and time to determine angular speed.
