CLASSICAL MECHANICS
BLOCK โ4 : MECHANICS OF SYSTEM
OF PARTICLES
PRESENTED BY: DR. RAJESH MATHPAL
ACADEMIC CONSULTANT
SCHOOL OF SCIENCES
U.O.U. TEENPANI, HALDWANI
UTTRAKHAND
MOB:9758417736,7983713112
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Classical Mechanics: Mechanics of System of Particles - Prof. Arya, Cheat Sheet of Classical Mechanics
An in-depth exploration of the fundamental concepts in classical mechanics, focusing on the mechanics of a system of particles. It covers key topics such as degrees of freedom, constraints, generalized coordinates, d'alembert's principle, and lagrange's equation of motion. The document aims to equip the reader with a comprehensive understanding of the principles governing the motion of a system of particles, enabling them to solve complex problems in this domain. The detailed explanations, mathematical derivations, and illustrative examples make this document a valuable resource for students and researchers interested in the field of classical mechanics.
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CLASSICAL MECHANICS
BLOCK โ 4 : MECHANICS OF SYSTEM
OF PARTICLES
PRESENTED BY: DR. RAJESH MATHPAL
ACADEMIC CONSULTANT
SCHOOL OF SCIENCES
U.O.U. TEENPANI, HALDWANI
UTTRAKHAND
MOB:9758417736,
Structure
๏ด 4 .1 Introduction
๏ด 4 .2 Objectives
๏ด 4 .3 System of Particles
๏ด 4 .4 Degree of Freedom
๏ด 4 .5 Constraints
4.5.1 Holonomic Constraints and Nonholonomic Constraints
4.5.2 Scleronomic Constraint and Rheonomic Constraints
๏ด 4 .6 Forces of Constraint
4.1 INTRODUCTION
๏ด We have study that the applications of Newtonโs laws of motion require the
specification of all the forces acting on the body at all instants of time.
๏ด But in practical situations, when the constraint forces are present, the
applications of the Newtonian approach may be a difficult task.
๏ด You will see that the greatest drawback with the Newtonian procedure is
that the mechanical problems are always tried to resolve geometrically
rather than analytically.
๏ด When there is constrained motion, the determination of all the insignificant
reaction forces is a great bother in the Newtonian approach.
๏ด In order to resolve these problems, methods have been developed by
DโAlembert, Lagrange and others.
๏ด The techniques of Lagrange use the generalized coordinates which will be
discussed and used in this unit.
๏ด You will see that in the Lagrangian formulation, the generalized coordinates
used are position and velocity, resulting in the second order linear
differential equations
4 .3 SYSTEM OF PARTICLES
๏ด We know that a system of particles means a group of particles inter-related.
The equations for a system of particles can be readily used to develop
those for a rigid body. An object of ordinary size known as a โmacroscopicโ
system โ contains a huge number of atoms or molecules. A very important
concept introduced with a system of particles is the center of mass.
๏ด Let us consider a system consisting of N point particles, each labeled by a
value of the index i which runs from 1 to N. Each particle has its own mass
m
i
and (at a particular time) is located at its particular place r
i
. The center
of mass (CM) of the system is defined by the following position vector-
Center of mass ๐ ๐ถ๐
= ( 1 / M )ฯ
๐
๐
๐
where M is the total mass of all the particles.
๏ด This vector locates a point in space which may or may not be the position
of any of the particles. It is the mass-weighted average position of the
particles, being nearer to the more massive particles. Let us consider a
simple example to understand this. Let us consider a system of only two
particles, of masses m and 2 m, separated by a distance l. Let us choose
the coordinate system so that the less massive particle is at the origin and
the other is at x= l, as shown in Figure ( 1 ). Then we have m 1
= m, m 2
= 2 m,
x 1
= 0 , x 2
= l. (The y and z coordinates are zero of course.) By the definition
x
CM
= ( 1 / 3 m) (mร 0 + 2 mรl)= ( 2 / 3 )l
๏ด The total force on the ith particle consists of the (net) external force on it, plus the net
force due to the interactions with other particles in the system, which we call the internal
forces. We can write this out as follows-
๐น ๐
= ๐น ๐
๐๐ฅ๐ก
- ฯ ๐
ิฆ ๐น ๐๐
.....( 4 )
where
ิฆ ๐น ๐๐
denotes the force exerted on the ith particle by the jth particle.
๏ด To get the total force on the whole system, we simply add up all these forces-
๐น ๐ก๐๐ก
= ฯ ๐
ิฆ ๐น ๐
๐๐ฅ๐ก
- ฯ ๐
ฯ ๐โ 1
ิฆ ๐น ๐๐
.....( 5 )
๏ด In the double sum on the right the terms cancel in pairs by Newton's 3 rd law
(for example, ๐น 12
- ๐น 21
= 0 ). The double sum thus gives zero, therefore-
๐น ๐ก๐๐ก
= ฯ ๐
ิฆ ๐น ๐
๐๐ฅ๐ก
.....( 6 )
๏ด The total force on the system is the sum of only the external forces. Since there are very many
internal forces, the fact that they give no net contribution to the total force is why it is possible for
many applications to treat an object of macroscopic size as a single particle. The internal forces
do play important roles in determining some aspects of the system, such as its energy.
๏ด For each particle individually, we have Newton's 2 nd law-
๐น ๐
=
๐๐ ๐
๐๐ก
.....( 7 )
๏ด Adding these for all the particles, and using the above result for ๐ ๐ก๐๐ก
, we find two forms of the 2 nd
law as it applies to systems of particles-
Newtonโs 2 nd law for systems-
๐น ๐ก๐๐ก
๐๐ฅ๐ก
=
๐๐ ๐ก๐๐ก
๐๐ก
= M ๐ ๐ถ๐
.....( 8 )
Here ๐ ๐ถ๐
=
๐๐ฃ ๐ถ๐
๐๐ก
is the acceleration of the CM. We see that the total external force produces
an acceleration of the center of mass, as though all the particles were located there.
๏ด The total kinetic energy of all the particles is
1
2
ฯ
๐
๐
๐
2
1
2
ฯ
๐
๐
๐
๐ถ๐
2
1
2
ฯ
๐
๐
๐
2
๐ถ๐
2
๐
๐ถ๐
1
2
๐ถ๐
2
1
2
ฯ
๐
๐
๐
2
๐ถ๐
. ฯ
๐
๐
๐
๏ด But the value of last term is zero, therefore we have-
1
2
ฯ
๐
๐
๐
2
1
2
๐ถ๐
2
1
2
ฯ
๐
๐
๐
2
๏ด We can interpret the terms on the right as-
- The first term is what the kinetic energy would be if all the particles really were
at the CM and moving with its speed. We often call this the kinetic energy of the
CM motion.
- The second term is the total kinetic energy as it would be measured by an
observer in the CM reference frame. We call this the kinetic energy relative to the
CM, or sometimes the internal kinetic energy.
๏ด The total kinetic energy is the sum of these two terms-
Kinetic energy of a system, K =
1
2
๐๐ฃ ๐ถ๐
2
- ๐พ ๐๐๐. ๐ก๐ ๐ถ๐
๏ด This breakup of the kinetic energy into that of the CM plus that relative to the
CM is an example of the general property. We will see that this property also
holds for angular momentum. It holds for linear momentum too, but the second
part, the total linear momentum relative to the CM, is always zero.
๏ด One use of this relation is to define the average force that acts during a
specified time interval. Let the force act for time ฮt, producing a net
change โ ๐ิฆ in the total momentum. The average force is given by-
Average force
๐๐ฃ
โ ๐ิฆ ๐ก๐๐ก
โ๐ก
๏ด This is useful in cases where the force is an unknown function of time and
we would like to describe its average effect over some specific time
interval without having to investigate the detailed behaviour.
4 .4 DEGREE OF FREEDOM
๏ด โThe minimum number of independent variables or coordinates required to
specify ( or define) the position of a dynamical system, consisting of one or
more particles, is called the number of degrees of freedom of the system.โ
๏ด Let us consider the example of the motion of a particle, moving freely in
space. This motion can be described by a set of three coordinates (x, y, z)
and hence the number of degrees of freedom possessed by the particle is
three.
๏ด Similarly, a system of two particles moving freely in space needs two sets of
three coordinates (x 1
, y 1
, z 1
) and (x 2
, y 2
, z 2
) i.e. six coordinates to specify its
position. Thus, the system has six degrees of freedom. If a system consists of
N particles, moving freely in space, we require 3 N independent coordinates
to describe its position. Hence, the number of degrees of freedom of the
system is 3 N.
4 .5 CONSTRAINTS
๏ด Generally, the motion of a particle or system of particles is restricted by one
or more conditions. The restrictions on the motion of a system are called
constraints and the motion is said to be constrained motion.
๏ด A constrained motion cannot proceed arbitrarily in any manner. For
example, a particle motion is restricted to occur only along some specified
path, or on a surface ( plane or curved) arbitrarily oriented in space. The
motion along a specified path is the simplest example of a constrained
motion.
4. 5. 1 Holonomic Constraints and
Nonholonomic Constraints
๏ด The constraints that can be expressed in the form f(x
1
, y
1
, z
1
: x
2
, y
2
, z
2
; x
n
, y
n
z n
; t) = 0 , where time t may occur in case of constraints which may vary with
time, are called holonomic and the constraints not expressible in this way
are termed as non-holonomic.
๏ด The motion of the particle placed on the surface of sphere under the
action of gravitational force is bound by non-holonomic constraint ( r
2
- a
2
) โฅ