Transcript Rigid Body - GEOCITIES.ws

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)
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
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

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
v0
h
t
2
2h
v
t
Knowing v, I can be calculated from (3)
Applications of flywheels
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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)

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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)
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Application
–
Determination of the moment of inertia of a turntable
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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)
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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 2r
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