Rotational Kinematics

♦Learn about : Rigid body , Translation , Rotation , Angular displacement , angular acceleration♦

Rigid body : A rigid body is a body with a definite and unchanged shape and size i.e. a body is said to be rigid if the distance between any two particles of the body remains invariant.

Motion of a rigid body

Translation: A rigid body is said to undergo translation if it moves such that it always remains parallel to itself: this means that a line connecting any two particles of the rigid body always remains parallel to itself throughout its motion.

Rotation: A rigid body is said to undergo rotation if there exists a straight line from which the distance of any particle of the rigid body remains constant throughout its motion. This straight line, whether fixed or moving is known as the axis of rotation. The rigid body is said to undergo rotation about this axis.

Angular Displacement:

Consider a rigid body undergoing rotation about an axis, perpendicular to the plane of the paper and passing through O.

Suppose 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 and ΔOA’B'(congruent)

i.e. ∠AOB and ∠A’OB’

Adding ∠AOB’ to both sides of the above equation, we get,

∠BOB’ = ∠AOA’ = θ (say)

This implies that in a given interval of time the angular displacements of all particles of the rigid body undergoing rotation are identical.

Therefore, a single variable, viz. angular displacement (θ) can be used to describe the rotational motion of the rigid body.

Angular displacement is not a vector quantity. However, for infinitesimal time intervals, the corresponding angular displacement is infinitesimal and behaves like a vector.

Instantaneous angular velocity is defined by ω= dθ/dt, the direction of ω (instantaneous) is along the axis n^

instantaneous angular accelerations are defined by :

α = dω/dt

Angular velocity is same for all the particles of a rigid body and the same is true about angular acceleration also as and the reason being equal angular displacement.

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