Angular Quantities
Angular Quantities
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Describe and analyse rotational motion using angular quantities.
Because the radian is based on the pure idea of "the radius being laid along the circumference", it often gives simple and natural results when used in mathematics and some physics.
What is a Radian?
What is a Radian?
The angle made when we take the radius and wrap it round the circle.
π radians = 180°
π radians = 180°
So 1 radian =180°/π
So 1 radian =180°/π
= 57.2958...° (approximately)
= 57.2958...° (approximately)
Angular Displacement
Angular Displacement
The first variable to be considered is the Rotational equivalent of Displacement, the Angular Displacement (θ), which is defined as the change in Angular Position between two points and can be found by;
d = r θ
d = r θ
Where ;
Where ;
θ = The Angular Displacement ( in Radians )
θ = The Angular Displacement ( in Radians )
d = The Arc Length (m)
d = The Arc Length (m)
r = The Radius of the Circle (m)
r = The Radius of the Circle (m)
Angular Velocity
Angular Velocity
An object moving in a circular path will have an Angular Velocity (ω), which is defined as the rate of change of Angular Displacement and can be found by :
ω = (change in θ)/(change in time)
ω = (change in θ)/(change in time)
ω= ∆θ/ ∆t
ω= ∆θ/ ∆t
Where :
Where :
ω = Angular velocity ( radians s-1 )
ω = Angular velocity ( radians s-1 )
θ = Angular Displacement ( radians )
θ = Angular Displacement ( radians )
t = time ( s )
t = time ( s )
In order to convert between Linear ( Tangential ) Velocity and Angular Velocity for an object, the following formula can be used;
In order to convert between Linear ( Tangential ) Velocity and Angular Velocity for an object, the following formula can be used;
Since linear velocity is constant v= d/t and since d= rθ
v=(rθ) /t = r (θ/t)
and θ/t = ω
So;
v = r ω
v = r ω
Two other useful methods of describing Angular Velocity :
Time Period (T) =Time taken for 1 complete revolution ( s )
Revolutions Per Second (frequency) = ( 1 / T )
The Time Period is related to the Angular Velocity by;
Angular Acceleration
Angular Acceleration
An Object which changes is Angular Velocity will have an Angular Acceleration (𝛼), which is defined as the rate of change of Angular Velocity and can be found by :
Where :
α = Angular Acceleration ( radians s-2 )
ω = Angular Velocity ( radians s-1 )
θ = Angular Displacement ( radians )
t = time ( s )
To convert between linear acceleration and angular acceleration we use the following;
So,
a = r𝛼
a = r𝛼
Extra resources:
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