Edexcel IAL - Mechanics 3- 4.1 Angular Speed- Study notes - New syllabus
Edexcel IAL – Mechanics 3- 4.1 Angular Speed -Study notes- New syllabus
Edexcel IAL – Mechanics 3- 4.1 Angular Speed -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 4.1 Angular Speed
Angular Speed
When a particle moves in a circular path, its position can be described using an angular displacement rather than a linear displacement. The rate at which this angular displacement changes with time is called the angular speed.
Definition of Angular Speed
Angular speed, denoted by \( \omega \), is defined as the rate of change of angular displacement \( \theta \) with respect to time:
\( \omega = \dfrac{d\theta}{dt} \)
The SI unit of angular speed is radians per second (rad s\(^{-1}\)).
Relationship Between Angular Speed and Linear Speed
If a particle moves in a circle of radius \( r \) with linear speed \( v \), then
\( v = r\omega \)
This relation allows conversion between linear and angular quantities.
Angular Speed and Period
The period \( T \) is the time taken for one complete revolution. Since one revolution corresponds to an angular displacement of \( 2\pi \) radians,
\( \omega = \dfrac{2\pi}{T} \)
A larger angular speed corresponds to a shorter period.
Uniform Circular Motion
In uniform circular motion, the angular speed \( \omega \) is constant. Although the speed remains constant, the velocity is continually changing direction.
Example :
A particle moves in a circle of radius \( 0.4 \) m with constant speed \( 2 \,\text{m s}^{-1} \). Find its angular speed.
▶️ Answer/Explanation
Using the relation
\( \omega = \dfrac{v}{r} \)
\( \omega = \dfrac{2}{0.4} = 5 \)
Conclusion: The angular speed is \( 5 \,\text{rad s}^{-1} \).
Example :
A particle completes one revolution every \( 0.5 \) s. Find its angular speed.
▶️ Answer/Explanation
The period is
\( T = 0.5 \)
Using
\( \omega = \dfrac{2\pi}{T} \)
\( \omega = \dfrac{2\pi}{0.5} = 4\pi \)
Conclusion: The angular speed is \( 4\pi \,\text{rad s}^{-1} \).
Example :
A particle moves in a horizontal circle with angular speed \( 6 \,\text{rad s}^{-1} \). Find its linear speed if the radius of the circle is \( 0.25 \) m.
▶️ Answer/Explanation
Using the relation
\( v = r\omega \)
\( v = 0.25 \times 6 = 1.5 \)
Conclusion: The linear speed is \( 1.5 \,\text{m s}^{-1} \).