Center of Mass of any given system of
particles is considered as a point where the whole mass of all particles of system
is supposed to be concentrated for various applications of it. There are two ways
in which system of particles can be classified -
Discrete Object System (DOS)- System of
different particles distributed in a given region of space. In DOS we consider
all objects of system a point masses located at the geometric center of the
bodies.
Continuous Object System (COS)- An extended
body made up of several particles attached to each other in a continuous
fashion is taken as a Continuous Object System.
To understand the above systems please see
the below video for Introduction of Center of Mass -
Now we will understand the localization of
Center of Mass of a system but to understand the location of center of mass, we
first need to understand a physical quantity used in finding the location of
center of mass which is called - Mass Moment.
Mass
Moment is a vector quantity defined for a mass
which is calculated with respect to a fixed point or reference point. Magnitude
of mass moment is the product of mass of body and its position vector with
respect to the point about which we calculate the mass moment.
For a given system of particles Center of
Mass is defined as a point associated with the system about which sum of mass
moments of all the particles of the system is equal to zero. This is the mass
moment property of center of mass using which we can find the location of
center of mass of a given system of particle. To understand the same in detail,
please see the below video -
With the use of mass moment property of center of mass, we can also locate the center of mass of a multiple particle system in a region of space. If we are given with some particles having masses m1, m2, m3.... Located at points (x1,y1,z1), (x2,y2,z2), (x3,y3,z3).... Then we can assume that the center of mass of such a system of particles is located at position C with coordinates (xc,yc,zc) so we can calculate the position vectors of all masses with respect to the point C and make sum of all equal to zero.
This will give us the position vector of the point C which is the
center of mass of the system of particles.
To understand the logic of the above analysis properly, please see the below video -
Using the concept we studied for calculation of center of mass position vector as well as the position
coordinates, we can solve variety of problems based on different physical situations.
To understand the application of this concept, pl see the examples in below videos -
Subtractive Objects: In many situations we need to locate the center of mass of an object when from a given body a part is removed. Such systems are called subtractive objects. In such cases we first locate the center of mass of whole body which is at its geometric center if the body is symmetric in shape and the part which is removed also has its center of mass at its geometric center if it is also symmetric in shape.
Now by using mass moment property of center
of mass we can locate the center of mass of the subtractive object. See the video
below for localization of center of mass of subtractive objects -
Continuous Objects: When
we wish to calculate the center of mass of an extended body(Continuous Object
System), we consider an elemental mass in the body at a position vector defined
with respect to the coordinate system associated with the body. Now just like
the case of a multiple body system we consider this elemental mass as one
particle of the body system and use the expression of position vector of center
of mass of the multi body system.
In case of an extended body when we use the
expression of position vector for center of mass of multi body system we need
to replace the summation of all mass moments with the integration of mass
moments with respect to the origin of the coordinate system. See the video
below on localization of center of mass of a continuous extended body -
There are several continuous objects which are used frequently in problems based on different physical situations. Here we'll study about localization of these standard objects
1. Center of Mass of a Uniform Half Ring: To locate the center of mass of a half Ring, we consider
a polar element on the circumference of Ring and find the center of mass by integrating it within
limits from -90 degree to +90 degree.
2. Center of Mass of a Uniform Half Disc: To locate the center of mass of a half Disc, we consider
an elemental half ring at radius x and width dx on the surface of disc and find the center of mass
by integrating it within limits from 0 to R, the radius of disc.
3. Center of Mass of a Uniform Hollow Hemisphere: To locate the center of mass of a hollow
hemisphere, we consider an elemental ring of polar width d(theta) at an angle(theta) from the axis of
hollow hemisphere and find the center of mass by integrating it within limits from 0 to 90
degree.
4. Center of Mass of a Uniform Solid Hemisphere: To locate the center of mass of a solid
hemisphere, we consider an elemental disc at a distance x from center of hemisphere, the width
of elemental disc is taken as dx and find the center of mass by integrating it within limits from 0
to R, the radius of hemisphere.
There are several standard continuous objects and their centre of mass we studied in previous part of this article. In this article we'll study the centre of mass of Uniform Hollow and Solid Cones and applications of localization of centre of mass of different cases.
5. Centre of Mass of a Uniform Hollow Cone: To locate the centre of mass of a hollow cone, we
consider an elemental ring at a distance x from apex of the cone along the surface of cone, the
width of elemental ring is taken as dx and find the centre of mass by integrating it within limits
from 0 to the length of cone projector, the radius of hemisphere.
6. Centre of Mass of a Uniform Solid Cone: To locate the centre of mass of a solid cone, we
consider an elemental disc at a distance x from apex of the cone along the altitude of cone, the
width of elemental disc is taken as dx and find the centre of mass by integrating it within limits
from 0 to the altitude of cone.
The velocity of center of mass of a system of particles is given by the ratio of total system momentum tothe sum of all masses of particles of the system. Similarly the acceleration of center of mass of a systemof particles is given by the ratio of total external force on the system to sum of all masses of particles ofthe systemHere if on the system no external force is acting, the acceleration of center of mass will be zero hencethe velocity of center of mass will be a constant and this states that total system momentum will alsoremain constant.
This phenomenon is known as Law of Conservation of Linear Momentum for a givensystem of particles.
Law of Conservation of Linear Momentum is a very important concept used very frequently in solvingproblems related to system of particlesMotion of Center of Mass in absence of external forces : By the Law of Conservation of LinearMomentum we can state that in absence of external forces velocity of center of mass of a systemremain constant (magnitude as well as direction).
If initially in a system, center of mass is at rest then bysome process due to internal forces in the system the center of mass will not be affected and it willremain at rest or if in some case if center of mass is following a trajectory under the influence of somelaws or effects then due to internal processes, the center of mass keep on following the trajectory as itsmotion will not be affected unless there is some change in external force take place.
This is a veryimportant logic used in solving problems related to motion of center of mass as well as problems relatedto linear momentum of a system of particles.
To develop fundamental understanding and applications of Law of Conservation of Linear Momentum,see the example videos below https://youtu.be/Umhaiwyi8o0
Displacement of Center of Mass: When in a system of particles, the particles are given some
displacements then its center of mass will also get displaced, the displacement of center of mass can be
easily calculated by the same expression by which we calculate the location of center of mass.
See this
video below to understand the expression for displacement of center of mass if displacement vector of
different particles of system are given.
Velocity and Acceleration of Center of Mass: If different particles of a system are moving, the center of
mass of system may or may not move. The particles may have such motion also so that the center of
mass of system can be kept at rest. This concept will be very useful in coming time to handle variety of
problems related to center of mass of a system of particle.
See the video below to understand the
calculation of velocity and acceleration of Center of Mass of a system of particles.
To understand the calculations about motion parameters of center of mass of a system of particles,
follow the example videos given below https://youtu.be/PiTOhnyfvjE
As we know that acceleration of center of mass is due to only external forces acting on a system of particles and internal forces have no contribution in acceleration of center of mass.
Due to internal forces acting on a system of particles, individual particles of system can accelerate and
their momentum can change but overall as both action and reaction of an internal force is acting inside
the system of particle on different particles, internal forces do not effect on motion of center of mass.
To understand the concept of internal forces on a system of particles and motion of center of mass
follow the video - https://youtu.be/pwCsdFWzAYc
The basic application to understand is that center of mass motion remain unaffected by the internal
forces of system however due to internal forces and their work kinetic energy of all particles of a system
may increase but these particles move in such a way that center of mass motion remain same as before.
To grasp the application of this concept in various problems, see the example videos below -
According to Newton's Second Law, the applied force on a body is numerically equal to the rate of change of momentum of the body. Same is valid for center of mass of a system of particles as well. When a force is acting on a body or a system, it changes the momentum of system which is equal to the product of force and the time for which force is acting on the body or system.
This total change in momentum due to the applied force is called Impulse. In other words Impulse can also be given as the momentum imparted by external force/forces acting on a body or a system.
In case of a constant force applied on a body or a system, Impulse can be directly
calculated by product of force magnitude and the time for which force is applied but if the force is
varying with time then we use integration of F.dt within limits from starting to end time of the force
action.
If on a moving body one or more forces are acting for a given duration then we can write the Impulse-
Momentum equation of the body for the given duration.
To understand the applications of Impulse and
its uses in conservation of momentum, see the video - https://youtu.be/bVhMRDZQMF8
The application of Impulse is also useful in variety of problems in which external forces are present and
continuously momentum of a body or the system is changing.
In the section of collisions also impulse
play an important role in analyzing the process of collisions.
To understand some more applications of Impulse, follow the example videos below -
When two particles in motion interact only
under the internal forces of the system, and exchange momentum due to the
impulse of forces, the phenomenon is called impulse. Let's first understand the
concept of collision then we will carry on with the concept of impact, in which
bodies get in physical contact with each other.
As
we have seen in collision there is an internal which is responsible for
transfer of momentum from one body to another. As the forces of equal magnitude
act in opposite direction on the two bodies for equal duration, momentum will
be transferred from one body to another in the process.
Types
of Collisions based on material of bodies: Depending
upon different types of body materials collisions can be classified in three
categories -
a.
Elastic Collisions
b.
Partially Elastic Collisions
c.
Inelastic Collisions
For these types we need to first discuss
about the materials types as well.
The understanding of different types of
materials helps us to keep in mind that in case of elastic materials when some
work is done in deformation of material, the energy spent is stored in form of
elastic potential energy of body. In case of partially elastic material it
dissipates partially and remaining is stored which can be retrieved back. But
in case of inelastic materials whole work done in deformation is dissipated and
no energy is left in stored form.
The analysis of Collisions based on the
material of bodies will be very important in understanding the applications of
collisions in various problems.
When two bodies collide, then at the time
of impact, they deform each other. Their kinetic energy increases the elastic
potential energy of the bodies due to deformation, i.e. if bodies are elastic
in nature. In the process later, this elastic potential energy is again
released, and increases the kinetic energy of bodies. If the bodies are elastic,
then whole energy of system remains conserved and if bodies are partially
elastic or inelastic, energy dissipation takes place. The final kinetic energy
of system is less than initial kinetic energy of the system.
The amount of energy being dissipated
depends upon the type of body material used in collision. The degree of
elasticity is also measured in terms of another constant factor called Coefficient of Restitution, it defined
as the ratio of relative velocity of separation to relative velocity of
approach of the bodies undergoing collision.
For numerical analysis of collision of
bodies coefficient of restitution is a very important parameter using which we
will also understand how to evaluate the final velocities of colliding bodies
in forthcoming articles. First to understand in detail about Coefficient of
Restitution, see the video below -https://youtu.be/NMudowLlbBM
So for different types of collision we need
to remember that e=1 for elastic collisions e=0
for inelastic collisions.
When a body hits another body in such a way that their initial velocities are in the direction of the line joining their centers then after collision also their direction of motion remain same and there will not be any change in direction of motion of the bodies if bodies are spherical. Such a collision is called Head-On Collision. The line along which the normal contact force appear during the impact is same as the line joining the center of bodies and is called line of impact.
To understand the basics of Head-On Collision, see the video given below -
The type of collision depends upon the coefficient of restitution between the bodies and on this basis we can classify the Head-On Collisions in different categories like elastic collision, partial elastic collision and inelastic collisions as mentioned in above video. Now we will discuss in detail about partial elastic Head-On Collisions. See the below video for understand this -
The concept of Head-On Collisions is used
in framing variety of numerical problems. To understand the fundamental
applications of such cases, see the example videos given here -
When line of impact during collision of bodies, is different from the direction of initial motion of spherical bodies, then the direction of motion of bodies after collision changes. This is because the impulse of contact force between bodies during impact, acts in direction that is different from initial direction of motion of bodies. Such a collision is called oblique collision or multidimensional collision.
In this section we will limit ourselves to
the discussion of oblique collisions in two dimensional plane only and such
collisions are also called two dimensional collisions.
To understand oblique
collisions first we will discuss the oblique collision of a ball with a flat
surface. This will help us in understanding the detailed analysis of oblique
collisions in a plane.
Now we will analyze the two dimensional
collision of two spherical bodies in a plane. There are many numerical problems
which are framed on this concept so you need to go through this topic very
carefully and in detail.
There are many cases in which the mass of a body changes. It may increase or decrease while the body is in motion. Such cases are explained with the concept of Mass Variation. When a body is moving and its mass is changing, the momentum of body changes due to two reasons -
(1)
The presence of some external
force acting on the body.
(2)
The change in mass which may
add or extract momentum to or from the body.
To analyze such cases, we can write the equation
for momentum conservation or Impulse momentum at an intermediate time t and
t+dt. Then, integrate it for the required parameter of motion for the given
duration of motion.
To understand how to apply the concept and
how to handle such cases of mass variations, see the video below which includes
the concept analysis and solved examples for explanation -
Super Elastic Collisions: When the Coefficient
of Restitution in a given collision is more than one (e>1), the kinetic
energy of system after collision is more than that of the initial kinetic
energy of the system. In such cases, at the moment of collision some energy is
evolved. See the video below for explanation -
Rotational Motion of a body is a motion in which the body moves about a given Axis of Rotation in such a way that all its particles move in different circles, with their centers lying on the Axis of Rotation. Thus Rotational motion can be considered as an integration of several circular motions of elemental masses in the body.
Circular Motion is the motion which is
defined for a point mass or a body of which dimensions are very small compared
to the radius of the circular motion. In case of Rotational Motion, we discuss
extended bodies moving in different orientations about different possible axes
of rotations.
See the video below to understand basic
understanding of Rotational Motion -
The first thing we study in translational
motion is 'Inertia' which causes change in state of motion. Similar property used
in rotational motion is 'Moment of Inertia', which is the first and most
important parameter used in understanding and defining various other properties
of Rotational Motion. So first we need to discuss about Moment of Inertia of a
particle moving in circular motion and we will extend this knowledge in
calculation of Moment of Inertia of an extended body in rotational motion. See
the video below for Moment of Inertia -
In the analysis of Rotational Motion, moment of inertia plays a very important role. This is because all applications of rotational motions are based on inertial properties of bodies including momentum, kinetic energy, and effect of torque applied on body.
As moment of inertia of a point mass in circular motion is given by product of mass, and the square of radius of circular motion, for any extended body we can integrate the moment of inertia of an elemental mass considering within the body.
Now we will discuss the calculation of moment of inertia of some standard objects. These objects are very frequently used in different types of numerical applications and understanding the calculation process of the moment of inertia of these objects also help in handling tougher cases of the moment of inertia and analysis related to it.
See all the videos below for calculation of moment of inertia of the objects attached below:
For different types of bodies rotating about different axes of rotation, their moment of inertia is different. In calculation of moment of inertia, axes theorems play an important role. Another important term used in applications of rotational motion where moment of inertia is used is Radius of Gyration.
Radius of Gyration: It is the squared distance which when multiplied with the mass of a body gives the moment of inertia of a body about a given axis of rotation. It is also defined as a point sized body which has the same mass as that of a rotating body when it moves in a circle. If its moment of inertia is the same as that of a rotating body about the given axis of rotation, then the radius of circle is termed as radius of gyration of the rotating body about the specific axis of rotation.
See the video below to understand the Radius of Gyration -
In previous articles we discussed that axes theorems are used for calculation of moment of inertia in different cases. To understand the applications of axes theorems in different cases of moment of inertia see the example videos below -
We are all familiar with Newton's Second Law of Motion. It is the force applied on a body with acceleration of the body as F=ma (mass multiplied by acceleration), in other words, the force gives the rate at which the linear momentum of body changes.
The concept of Newton's Second Law of Motion can also be applied on a body in rotational motion. Here the law related to the torque acting on the body with the angular acceleration of the body.
In upcoming articles, we will discuss the angular momentum of a rotating body which is the angular counterpart of linear momentum. Similar to the case of translational motion, in rotational motion, applied torque on a body gives the rate of change of angular momentum of the body. This we will study in detail when we study the analysis and applications of angular momentum.
In this article, we will restrict ourselves to the understanding of Newton's Law for rotational motion and its applications.
See the video below to understand the concept and its applications
There are several physical situations on which problems on rotational motion are framed on applications of Newton's Second Law. See the example videos below to understand the applications of Newton's Second Law in different cases of Rotational Motion:
In the translational motion, we studied about the momentum of body. Momentum measures the inertia of motion in the body also the rate of change, gives the external force acting on the body. Similar to momentum, it's counterpart in rotational motion is Angular Momentum. It is equally important for analyzing the dynamics of a rotating body.
Let us first discuss the basics of angular momentum and its calculation in different cases.
We also studied about the law of conservation of Linear Momentum of a system, which states that total linear momentum of a system remains conserved if no external force is acting on the system. Similarly, we state Law of Conservation of Angular Momentum of a rotating body in which we consider, the absence of external torques on the rotating body its total angular momentum remain conserved. See the video below for understanding the Law of conservation of angular momentum.
There are many situations of rotational motion on which variety of problems are framed which involve the concept of Angular Momentum and its conservation.
To understand the applications of the concept see the example videos below.
Like impulse in translational motion, we define angular impulse in rotational motion. When an external torque is acting on a rotating body, the rate of change of angular momentum gives the magnitude of the torque acting. Using the logic, the product of torque and time for which the torque is acting on the body, gives the total change in angular momentum of the body which is called angular impulse applied on the body.
For the case if applied torque is constant, the impulse can be calculated by product of torque and the time and in case of applied torque is varying with time, we integrate the product of torque with elemental time dt during the course of motion for the time of motion.
Kinetic Energy in Rotational Motion: As a fundamental term kinetic energy is defined as half of the product of mass and square of the speed of the body in case a body is in translational motion. Rotational motion is defined for extended bodies so we can consider an elemental mass dm in the body and find its kinetic energy.
The integrating kinetic energy of this elemental mass gives us the total kinetic energy of the body in rotational motion.
