Angular displacement:
Take a rigid body undergoing rotation related to an axis, perpendicular to the plane of the paper and passing through O. Consider that A and B are any two particles of the rigid body at the position 1 while A¢ and B¢ are their subsequent locations when the body is at the position 2.
Since the body undergoes rotation,
OA = OA′ and OB = OB′
Further AB = A′B′, since the body is rigid.
ΔOAB ≡ ΔOA′B′(congruent)
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![1406_Angular displacement,Velocity and Accelerations.png](https://www.expertsmind.com/../CMSImages/1406_Angular displacement,Velocity and Accelerations.png)
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i.e. ∠AOB = ∠A′OB′
Adding ∠AOB′ to both sides of the above relation, we get
∠BOB′ = ∠AOA′ = θ (say)
This shows that in a given interval of time the angular displacements of all particles of the rigid objects undergoing rotation are identical.
Therefore, a single variable, viz. angular displacement (θ) may be used to define the rotational motion of the rigid object.
Angular velocity:
The power of change of angular displacement with respect to time is called as angular velocity.
Average angular velocity: The power of change of angular displacement with related to time is called as angular velocity.
To describe average angular velocity it is necessary to specify the interval in which we are talking for average. ![1820_Angular displacement,Velocity and Accelerations1.png](https://www.expertsmind.com/../CMSImages/1820_Angular displacement,Velocity and Accelerations1.png)
Instantaneous angular velocity: Concurrent velocity seems angular velocity at a particular instant. It is mathematically shown as ,
![1042_Angular displacement,Velocity and Accelerations2.png](https://www.expertsmind.com/../CMSImages/1042_Angular displacement,Velocity and Accelerations2.png)
Angular accelerations (ω):
The rate of change of angular velocity with related to time is called as angular acceleration.
Average angular acceleration: It is important to denote the time interval to in which we are talking about average.
In a given time interval t1 to t2 the average angular acceleration is described as,
![629_Angular displacement,Velocity and Accelerations3.png](https://www.expertsmind.com/../CMSImages/629_Angular displacement,Velocity and Accelerations3.png)
Instantaneous angular acceleration: Concurrent angular acceleration means angular acceleration at a particular dot instant at t = t1 mathematically it is shown as,
![572_Angular displacement,Velocity and Accelerations4.png](https://www.expertsmind.com/CMSImages/572_Angular%20displacement,Velocity%20and%20Accelerations4.png)
Direction of angular acceleration: If value of ω increasing then direction of a will be equal as direction of ω and vice-versa.
Equation of angular motion:
ω(t) = ωo + at
θ(t) = θo + ωot + 1/2αt2
ω2(t) = ω02 +2αθ
Here ωo = magnitude of the initial angular velocity
ω(t) = value of the angular velocity after time t.
θo = Initial angular position.
θ(t) = Angular position after time t.
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