Rigid Body - GEOCITIES.ws
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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:
n
r
m i ri
i
1
n
mi
i
1
,
Centre of mass for a body of continuous
distribution:
r
M
0
rdm
M
,
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
position after being slightly displaced
Disturbance gives greater gravitational potential
energy
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.
ERot
1 2
mv
2
1
m ( r ) 2
2
1 2
mr 2
2
E Rot
1 2
I
2
The term I mr 2 is known as the moment of
inertia
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:
I r 2 dm
Experimental Demonstration of the
Energy Stored in a Rotating Object
Moment of Inertia (2)
A body composed of discrete point masses
I mi ri ,
2
i
A body composed of a continuous distribution
of masses
M
I r 2dm ,
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
centre perpendicular to its
length
1
I ml 2
12
–
M.I. about an axis through one end perpendicular to
its length
1 2
I ml
3
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
breadth
1
I mb 2
12
–
About an axis through its centre parallel to its
length
1
I ma 2
12
Moment of Inertia of Common
Bodies (3)
–
About an axis through its centre perpendicular to its
plane
I
1
m (a 2 b 2 )
12
Moment of Inertia of Common
Bodies (4)
Uniform circular ring of mass m and radius R
–
About an axis through its centre perpendicular to its
plane
I mR2
Moment of Inertia of Common
Bodies (5)
Uniform circular disc of mass m and radius R
–
About an axis through its centre perpendicular to its
plane
1
I mR 2
2
–
The same expression can be applied
to a cylinder of mass m and radius R
Moment of Inertia of Common
Bodies (6)
Uniform solid sphere of mass m and radius R
2
I mR 2
5
Theorems on Moment of Inertia (1)
Parallel Axes Theorem
I I G Md
2
Theorems on Moment of Inertia (2)
Perpendicular Axes Theorem
I Ix Iy
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 corkscrew rule
Depends on
–
–
Magnitude of force
Axis of rotation
Torque (2)
Work done by a torque
–
–
Constant torque: W = T
Variable torque:
W
Td
0
Kinetic Energies of a rigid body (1)
Translational K.E.
KE tran
1
Mv 2
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 rot
1
m ( r )2
2
Kinetic Energies of a rigid body (2)
=
1
mr 2
2
1 2
I
2
If a body of mass M and moment of inertia IG
about the centre of mass possesses both
translational and rotational k.e., then
KE
1
1
Mv 2 I G 2
2
2
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
Moment of inertia of a flywheel (3)
Theory
kinetic energy kinetic energy
work done
potential energy lost
gained by the gained by the against friction
by the falling weight
falling weight
flywheel
at the axle
1 2 1 2
mgh mv I nf
2
2
1 2 2 1 2
mr I nf
2
2
….. (1)
where f = work done against friction per revolution
Moment of inertia of a flywheel (4)
–
When the flywheel comes to rest:
Loss in k.e. = work done against friction
1 2
I Nf ….. (2)
2
1 2 1 2
n
(2) In (1) mgh mv I [1 ]
2
2
N
1 2
I
n
v [m 2 (1 )] …. (3)
2
r
N
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
v0
h
t
2
2h
v
t
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 )
mr 2
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
d
d
As
T I
( I )
dt
dt
–
If T = 0, I = constant
Conservation of angular
momentum (2)
Examples
–
High diver jumping from a jumping board
Conservation of angular
momentum (3)
–
Dancer on skates
–
Mass dropped on to a
rotating turntable
Experimental verification using a bicycle wheel
Conservation of angular
momentum (4)
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
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
d
( I ) ( Fp)
dt
Rotational motion about a fixed
axis (2)
d
I
T
dt
i.e. I = T
Compound pendulum (1)
Applying the d’Alembert’s
Principle to the rigid body
d 2
I s 2 Mgh sin .
dt
But
I s M (k 2 h2 )
where k is the radius of gyration
about its centre of mass G
Compound pendulum (2)
2
d
2
2
M ( k h ) 2 Mgh sin ,
dt
For small oscillations
d 2
gh
2
2
2
dt
k h
SHM with period
k 2 h2
T 2
hg
Compound pendulum (3)
It has the same period of oscillation as the
simple pendulum of length
k 2 h2
l
h
l is called the length of the equivalent simple
pendulum
Compound pendulum (4)
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)
Torsional pendulum
c
I
where c = torsional constant
I = moment of inertia
SHM with period
I
T 2
c
Rolling objects (1)
d 2r
v
r
T
T
Rolling objects (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 (3)
Hence, for pure rolling, there is no work done
against friction at the point of contact
Kinetic energy of a rolling object
Total kinetic energy
= translational K.E. + rotational K.E.
1 2 1 2
=
mv I
2
2
Stable Equilibrium
No toppling
Compound pendulum