ROTATIONAL MOTION

Introduction

In this chapter, rotational motion of a rigid body about a fixed axis of rotation is discussed. A rigid body is a system of particles in which inter particle distances do not change and the body cannot be deformed no matter how large a force is applied to it. Although a solid body is not a rigid body, it can be so considered for most of the practical applications.

Rotational Kinematics and Dynamics

In rotational motion of a body, all its particles move on circular paths having centers on a definite straight line, called the axis of rotation. Kinematics deals with motion without considering its cause, whereas dynamics deals with motion along with its cause and properties of the body.

Relations between variables of rotational and linear motion

( a ) Angular displacement:

The figure shows a rigid body rotating about a fixed axis OZ normal to the plane of the figure. P and P’ are the positions of a particle of the body at time t and t + Δt. Angle θ made by the line joining the particle to the centre of its rotation with a reference line OX shows its angular position at time t. Similarly, angle θ + Δθ is its angular position at time t + Δt. The change in angular position, of a particle is called its angular displacement. The angular displacement of the particle P is Δθ in time Δt. As the inter particle  distances do not change in a rigid body, all its  particles will have the same angular displacement in a given time. Hence, the angular displacement, Δθ, of the particle P can be considered as the angular displacement of

the rigid body. 

( b ) Angular speed and angular velocity:

The average angular speed of a particle or of the rigid body is defined as

 < ω > = time interval / angular displacement = Δ θ / Δ t

The instantaneous angular speed of a particle or of the rigid body is given by

 ω = limit       Δ θ / Δ t

        Δ t → 0

The unit of ω is radian / s or rotation / s.

The direction of angular velocity is given by the right-handed screw rule. When a right-handed screw is kept parallel to the axis of rotation and rotated in the direction of rotation of the body, the direction of advancement of screw gives the direction of angular velocity.

( c ) Scalar relation between angular velocity and linear velocity:

As shown in the figure, the particle P covers a linear distance equal to the arc length PP’ in time Δ t. Hence, average linear speed,

< v > = arc length PP' / Δ t  = r dθ / d t = r ω

Linear velocity is a vector quantity and its direction at any point on the path of motion is tangential to the path at that point. In the above equation, v, r and ω are the magnitudes of the vector quantities  v , r and ω.

( d ) Vector relation between angular velocity and linear velocity:

The position vector r w. r. t. the centre of the circular path of a particle, angular velocity ω and linear velocity V are shown in the figure.  As v is perpendicular to the plane formed by ω and r , the scalar relation v = r ω can be written in the vector form as 

v = ω × r

( e ) Angular acceleration:

The average angular acceleration in the time interval Δ t is

< α > = change in angular velocity / time interval = Δ ω / Δ t and the instantaneous angular acceleration at time t is given by 

α  =   limit        Δ ω / Δ t = d ω / d t = = rate of change of angular velocity

        Δ t → 0

α  is in the direction of Δ ω and in the case of fixed axis of rotation, both are parallel to the axis. The unit of α is rad / s 2 or rotation / s 2. 

( f ) Relation between linear acceleration and angular acceleration :

Differentiating the equation v = ω × r with respect to time gives the linear acceleration

a = d v  = d  (ω × r) = ω × dr    +  dω  × r

      dt       dt                       dt         dt

From the figure on the previous page and using right-handed screw rule it can be found that the direction of

ω × v is radial towards the centre. Hence, it is called the radial component, ar , of the linear acceleration a.

Similarly, the direction of α × r is tangential to the circular path at the position of the particle. Hence it is called the tangential component, aT , of the linear acceleration a.

Therefore

a = ar  +  aT

or a = (ar²  +  aT²)^1/2  because ar is perpendicular to aT

Even if α = 0, that is the angular velocity is constant, ar is not zero. As the angular displacement, θ, angular velocity, ω and angular acceleration, α are the same for all the particles of a rigid body, they are known as variables of the rotational kinematics.