Rigid Body

Particle

Object without extent Point in space Solid body with small dimensions

Rigid Body

An object which does not change its shape Considered as an aggregation of particles Distance between two points is a constant Suffer negligible deformation when subjected

to external forces Motion made up of translation and rotation

Motion of a Rigid Body

Translational– Every particle has the same instantaneous velocity

Rotational– Every particle has a common axis of rotation

Centre of Mass

Centre of mass of a system of discrete particles:

rmr

m

i ii

n

ii

n

1

1

,

Centre of mass for a body of continuous distribution:

rrdm

M

M

0 ,

It is the point as if all its mass is concentrated there

Located at the point of symmetry

Conditions of Equilibrium

For particle– Resultant force = 0

For rigid body– Resultant force = 0 and– Total moments = 0

Toppling

An object will not topple over if its centre of mass lies vertically over some point within the area of the base

Figure

Stability

Stable Equilibrium– The body tends to return to its original equilibrium p

osition after being slightly displaced– Disturbance gives greater gravitational potential ene

rgy– Figure

Unstable Equilibrium

The body does not tend to return to its original position after a small displacement

Disturbance reduces the gravitational potential energy

Neutral Equilibrium

The body remains in its new position after being displaced

No change in gravitational potential energy

Rotational Motion about an Axis

The farther is the point from the axis, the greater is the speed of rotation (v r)

Angular speed, , is the same for all particles

Rotational K.E.

The term is known as the moment of inertia

2

2

1mvERot

22

2

2

1

)(2

1

mr

rm

2mrI

2

2

1 IERot

Moment of Inertia (1)

Unit: kg m2 A measure of the reluctance of the body to its

rotational motion Depends on the mass, shape and size of the

body. Depends on the choice of axis For a continuous distribution of matter:

dmrI 2

Experimental Demonstration of the Energy Stored in a Rotating Object

Moment of Inertia (2)

A body composed of discrete point masses

i

iirmI ,2

A body composed of a continuous distribution of masses

I r dmM

2

0,

Moment of Inertia (3)

A body composed of several components:– Algebraic sum of the moment of inertia of all its

components

A scalar quantity Depends on

– mass– the way the mass is distributed– the axis of rotation

Radius of Gyration

If the moment of inertia I = Mk2, where M is the total mass of the body, then k is called the radius of gyration about the axis

Moment of Inertia of Common Bodies (1)

Thin uniform rod of mass m and length l– M.I. about an axis through its c

entre perpendicular to its length

I ml112

2

– M.I. about an axis through one end perpendicular to its length

I ml13

2

Moment of Inertia of Common Bodies (2)

Uniform rectangular laminar of mass m, breadth a and length b– About an axis through its centre parallel to its bread

thI mb

112

2

– About an axis through its centre parallel to its length

I ma112

2

Moment of Inertia of Common Bodies (3)

– About an axis through its centre perpendicular to its plane

I m a b 112

2 2( )

Uniform circular ring of mass m and radius R– About an axis through its centre perpendicular to its

plane

Moment of Inertia of Common Bodies (4)

I mR 2

Uniform circular disc of mass m and radius R– About an axis through its centre perpendicular to its plan

e

Moment of Inertia of Common Bodies (5)

2

2

1mRI

– The same expression can be applied to a cylinder of mass m and radius R

Uniform solid sphere of mass m and radius R

Moment of Inertia of Common Bodies (6)

I mR25

2

Theorems on Moment of Inertia (1)

Parallel Axes Theorem

I I MdG 2

Perpendicular Axes Theorem

Theorems on Moment of Inertia (2)

I I Ix y

Torque (1)

A measure of the moment of a force acting on a rigid body

– T = F·r Also known as a couple A vector quantity: direction

given by the right hand cork-screw rule

Depends on– Magnitude of force– Axis of rotation

Work done by a torque– Constant torque: W = T

– Variable torque:

Torque (2)

W Td

0

Kinetic Energies of a rigid body (1)

Translational K.E.

KE Mvtran 12

2

Rotational K.E. – It is the sum of the k.e. of all particles comprising the body– For a particle of mass m rotating with angular velocity :

KE m rrot 122( )

If a body of mass M and moment of inertia IG a

bout the centre of mass possesses both translational and rotational k.e., then

Kinetic Energies of a rigid body (2)

12

2 mr=

2

2

1 I

22

2

1

2

1 GIMvKE

Moment of inertia of a flywheel (1)

Determination of I of a flywheel

– Mount a flywheel– Make a chalk mark– Measure the axle diameter

by using slide calipers– Hang some weights to the

axle through a cord– Wind up the weights to a

height h above the ground– Release the weights and

start a stop watch at the same time

Moment of inertia of a flywheel (2)

– Measure:– the number of revolutions n of the flywheel before the

weights reach the floor– the number of revolutions N of the flywheel after the

weights have reached the floor and before the flywheel comes to rest

Theory

Moment of inertia of a flywheel (3)

axlethe at

friction against

done

flywheel

the by gained

energykinetic

ightfalling we

theby gained

energykinetic

ightfalling wethe by

lostenergy potentialwork

nfImvmgh 22

2

1

2

1

nfImr 222

2

1

2

1

where f = work done against friction per revolution

….. (1)

– When the flywheel comes to rest: Loss in k.e. = work done against friction

Moment of inertia of a flywheel (4)

NfI 2

2

1 ….. (2)

(2) In (1)

]1[2

1

2

1 22

N

nImvmgh

)]1([2

12

2

N

n

r

Imv …. (3)

Moment of inertia of a flywheel (5)

– The hanging weights take time t to fall from rest through a vertical height h

Total vertical displacement = average vertical velocity time

tv

h

2

0

t

hv

2

Knowing v, I can be calculated from (3)

Applications of flywheels

In motor vehicle engines In toy cars

Angular momentum

The angular momentum of a particle rotating about an axis is the moment of its linear momentum about that axis.

A mr ( )2

2mr

I

Conservation of angular momentum (1)

The angular momentum about an axis of a given rotating body or system of bodies is constant, if the net torque on the object is zero

As)(

Idt

d

dt

dIT

– If T = 0, I = constant

Examples– High diver jumping from a jumping board

Conservation of angular momentum (2)

– Dancer on skates

Conservation of angular momentum (3)

Experimental verification using a bicycle wheel

– Mass dropped on to a rotating turntable

Application– Determination of the moment of inertia of a turntable

Set the turntable rotating with an angular velocity Drop a small mass to the platform, changes to a lower

value ’ If there is no frictional couple, the angular momentum is

conserved, I = I’ ’ = (I + mr2) ’

, ’ can be determined by measuring the time taken for the table to make a given number of revolutions and I can then be solved

Conservation of angular momentum (4)

Rotational motion about a fixed axis (1)

T = I d’Alembert’s Principle

– The rate of change of angular momentum of a rigid body rotating about a fixed axis equals the moment about that axis of the external forces acting on the body

)()( FpIdt

d

Rotational motion about a fixed axis (2)

Iddt

T

i.e. I = T

Compound pendulum (1)

Applying the d’Alembert’s Principle to the rigid body

Iddt

Mghs

2

2

sin .

I M k hs ( )2 2But

where k is the radius of gyration about its centre of mass G

For small oscillations

Compound pendulum (2)

M k hddt

Mgh( ) sin ,2 22

2

ddt

ghk h

2

2 2 2

SHM with period

hg

hkT

22

2

It has the same period of oscillation as the simple pendulum of length

Compound pendulum (3)

lk h

h

2 2

l is called the length of the equivalent simple pendulum

The point O, where OS passes through G and has the length of the equivalent simple pendulum, is called the centre of oscillation

S and O are conjugate to each other

The period T is a minimum when h = k (see expt. results)

Compound pendulum (4)

Torsional pendulum

I

c

where c = torsional constant

I = moment of inertia

SHM with period

c

IT 2

Rolling objects (1)

r

T

r

T

dv

2

P has two components:– v parallel to the ground– r(=v) perpendicular to the

radius OP

If P coincides with Q, the two velocity components are oppositely directed. Thus Q is instantaneously at rest

Rolling objects (2)

Hence, for pure rolling, there is no work done against friction at the point of contact

Rolling objects (3)

Kinetic energy of a rolling object

Total kinetic energy= translational K.E. + rotational K.E.

= 22

2

1

2

1 Imv

Stable Equilibrium

No toppling

Compound pendulum