ROTATIONAL DYANAMICS

Introduction

The study of rotational motion of a rigid body is said to be rotational dynamics. A solid body is said to be rigid in which the particles are compactly arranged and their positions are not disturbed by an external force applied on it. However no real body is truly rigid. Fluid (liquid and gas) are not solid.

1.1 Equation of angular motion, Relation between linear and angular kinematics.

➽ What is angular motion of a rigid body?

➣ A rigid body is said to be in rotational or angular motion about its axis if all its particles rotate about an axis with the same angular velocity but different linear velocities.

For example motion of a wheel of a train about its axle. Kinematic equations of rotational motion can be written as,

𝛉=𝛚𝐭

𝛚=𝛚_𝟎+𝛂𝐭

𝛉=𝛚_𝟎 𝐭+𝟏/𝟐 𝛂𝐭^𝟐

𝛚^𝟐−𝛚_𝟎^𝟐=𝟐𝛂𝛉

Where,𝛉 is the angular displacement, 𝝎 is the angular velocity, 𝜶 is the angular acceleration

And t is the time interval.

➽ Find the relation between angular and linear displacement.

➣ Consider a rigid body which is rotating about an axis ‘y’ then the each particle within this body also rotates with same angular velocity but with different linear velocity.

If the particle is at a distance r from the axis of rotation and it makes angular displacement Ꝋ while moving from point A to B

and at the same time arc length(linear distance) is ‘S’ then

ꝋ=𝑺/𝒓

So, s=Ꝋ r

➽ Find the relation between linear and angular velocity.

➣ Consider a rigid body is rotating about an axis ‘y’. If the particle of a rigid body at a distance rfrom the axis of rotation is shifted from point A to B.

In this case if Ꝋ be the angular displacement ands be the arc length, then

ꝋ=𝑺/𝒓 -----------(i)

let ‘t’ be the time taken from A to B and if angularvelocity is ῳ, then

ῳ=𝜽/𝒕---------------(ii)

from (i) and (ii)

we get,

ῳ=(𝑺/𝒓)/𝒕 = 𝑺/𝒓𝒕 = 𝒗/𝒓

i.e. v= rῳ

➽ Find the relation between angular and linear acceleration.

➣ Consider a rigid body is rotated about an axis ‘y’. If the particle position within rigid body is chosen at a distance r from axis of rotation is displaced by ‘Ꝋ’ from its original position A to B making arc length ‘S’.

Then , Ꝋ=𝑺/𝒓-------------(i)

Again let, dꝊ be the small displacement within time Interval dt. So,

ῳ= 𝒅𝜽/𝒅𝒕-------------------(ii)

If angular velocity is not constant

Then angular acceleration ,

𝜶=𝒅𝝎/𝒅𝒕----------(iii)

From (i) and (ii) we get

v=r𝝎------------------(iv)

Again from linear acceleration,

a=𝒅𝒗/𝒅𝒕--------(v)

From (iv) and (v)

a=(𝒅(𝒓𝝎))/𝒅𝒕

a=(𝒓(𝒅𝝎))/𝒅𝒕

a=r𝜶

1.2 Kinetic energy of rotation of rigid body.

➽ Explain the kinetic energy of rotation of rigid body.

➣ Consider a rigid body of mass (M) consisting particles m1, m2,……..mn at a distance r1 ,r2,……….rn respectively from a axis of rotation

Yy’ and the whole body is rotating about the axis.

Though each particle within the rigid body rotating with Same angular velocity their linear velocities are different.

If v1, v2 …………vn be the velocities of Respective particle m1, m2 ……….mn . Then we can write

V1= r1 𝝎,

v2= r2 𝝎 and similarly

vn = rn 𝝎

Again rotational kinetic energy of the mass

m1 = 𝟏/𝟐m1v12 = 𝟏/𝟐m1 (r1 𝝎)2

m2 =𝟏/𝟐m2 (r2 𝝎)2 and

mn = 𝟏/𝟐mn (rn 𝝎)2

so net rotational kinetic energy of a body

K.E.r =𝟏/𝟐m1 (r1 𝝎)2 + 𝟏/𝟐m2 (r2 𝝎)2 …………….+𝟏/𝟐mn (rn 𝝎)2

K.E.r = 𝟏/𝟐[m1 (r1 )2 +m1 (r1)2 ……………….+ m1 (r1 )2 ]𝝎2

K.E.r =𝟏/𝟐 ∑▒〖(𝒎〗I ri 2 ) 𝝎2

So kinetic energy of a rotational body is (K.E.)=𝟏/𝟐 I 𝝎2

1.3 Moment of inertia, Radius of gyration.

➽ Define the term moment of inertia of a rigid body with a diagram

➣ The moment of inertia of a rigid body is defined as the sum of the product of individual mass of a particles within the rigid body and square of their perpendicular distance from the axis of rotation.

If m1, m2 ...............mn be the mass of individual particles and their perpendicular distance from the axis of rotation are r1, r2, …………rn respectively.

Then the net moment of inertia of a body about the axis of rotation can be written as,

➽ Define radius of gyration.

➣ The radius of gyration is defined as the distance from the axis of rotation to a centre of mass where all the mass of individual particles are concentrated in a given rigid body. The radius of gyration is denoted by K.