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Center of mass is a point of a body or group of bodies that represents the motion of a body when a external force is applied on the body or system. In this complete video lesson series we have discussed the need for the concept of center of mass. We have derived mathematical equation for the center of mass of a system using the concept that the algebraic sum of moments about center of mass is zero.
We have also derived equations for velocity and acceleration of the center of mass and proved that there is no effect of internal force on the motion of center of mass.
We have further discussed how center of mass is impacted when a body is added or when a mass is removed from a given system.
Here are the list of posts under the concept of center of mass.
We know that center of mass of shifts a new and heavy position of the system when a certain mass is removed. Basing on the concept, we would like to solve two problems. We would like to measure the shift in the center of mass of disc and sphere when a certain portion of the mass is removed from one side of the body. If the mass of the system is removed from the center of mass of the system, center of mass of the system remains the same.
For a two dimensional body like disc, mass is directly proportional to the area of the disc. It is simply because mass is the product of volume and density of the disc. Volume can be further can be written product of area of the disc and thickness of the disc. As thickness is same, mass of the disc is directly proportional to the area of the body. Area is further directly proportional to the square of the radius of the disc.
We can use the mathematical derivation that we have derived in the previous post to measure the shift in the center of mass of the system as shown in the video below.
Shift in the center of mass of sphere when a portion of the sphere is removed
In the same way, we can also measure the shift in the center of mass of a sphere when a portion of mass is removed from it from one side. The only difference in this case when compared with the disc, is sphere is a three dimensional body. Hence mass of the body is directly proportional to the volume of the sphere. It means, mass of the sphere is directly proportional to the cube of the radius of the cube. The problem can be further solved as shown in the video below.
If a certain mass from the system is removed, center of mass obviously changes from its existing position. When some mass of the system is removed, new center of mass shifts towards the heavier part of the system.
We know that center of mass of a system is a point of a system that represents the actual motion of a system when some external force is applied on the system.For a geometrically symmetric body, center of mass is at its geometrical center and it even coincides with center of gravity.
We can use the basic principle of center of mass in solving this conceptual problem to derive a mathematical equation. We know that the algebraic sum of moments about center of mass of the system is always equal to zero. It means the moments of the system about one direction is equal in magnitude and opposite in direction to the moments of the same system in the opposite direction.
Let the original center of mass of the system is at O. Let a small mass is removed from the system whose center of mass is at one point and the remaining portion of the system has center of mass at some other point. We can equate the moment of the system that has two masses. They are removed mass and remaining mass.
By simplifying the above equation, as shown in the below video lesson, we can the location of new center of mass of the system. We can find the shift in the center of mass as the product of mass removed and the distance between original center of mass and removed center of mass divided by the mass of the remaining portion.
Adding a new mass to the existing system automatically alter the center of mass of the existing system. As extra mass is added to the system, that extra mass also shows its influence on the motion of the system and hence new center of mass will be generated to the system. It actually shifts towards the heavier part of the system.
Center of mass is a point of a system which represents the actual motion of the system. For any system, it is some where in the system and it can have coordinates along three axes of the Cartesian coordinate system.
Here in this post we would like to solve a small problem in the given video lecture below to demonstrate the impact of adding a new mass to the existing system.
In the given problem, three particles of different masses has center of mass some where. Now as per the given problem, a new mass is added to the system, its center of mass is shifted to orizion of the system. So we can equate each coordinate of center of mass to zero.
By simplifying the given mathematical equation and using the value from the data of the first condition given in the problem, we can find the position of the fourth particle added to the system as shown below.
Three identical spheres are arranged in such a way that each one is in contact with the other two. We would like to identify the center of mass of the system. We shall take some point as the reference point with respect to which, we identify the center of mass position. Let the center of the first sphere is that reference point.
Center of mass is a point which represents the motion of the system when some external force is applied on it. We have all ready explained how to find the coordinates of center of mass of a system in the previous posts. Here we are going to use that concept to solve this problem.
Let all spheres are having identical radius and their mass is not given in the problem. We can express the mass as the product of volume and density. As the spheres are made up of same materiel, their densities are same and volume is directly proportional to the cube of radii of the sphere.
To identify the center of mass of the system, first we need to identify the center of mass of each sphere and that shall be taken as the reference point of each sphere. We can use simple mathematical understanding as shown in the video lesson presented below, to identify the location of center of mass of each sphere.
The first sphere is at the orizion and the second sphere is on the x- axis. Its center of mass is at a distance double the radius of the sphere as shown. Similarly, we can identify the location of center of mass of third sphere.
By subsisting the values in the formula of center of mass, we can measure the center of mass of the system as shown below.
Center of mass of T Shaped body
Let us consider a T shaped body having equal dimensions of rods connected. We would like to find the center of mass of that system. As the rod is one dimensional body, mass of the body is directly proportional to its length. We need to fix a reference point from which we can measure the center of mass of the system. Let the junction of the T shape is the reference point. It is the center of horizontal rod and its center of mass lies there itself.
The vertical rod has mass uniformly distributed over the entire length and hence its mass is at its geometrical center of the rod. Thus we can identify the location of center of mass of each rod as shown in the video below. In the place of mass, we can substitute the length of the rod itself. By substituting the values in the mathematical formula of center of mass, we can the center of mass of the system as shown in the below video.
Center of mass is a point of a system which represent the motion of a system. We can find the center of mass of any system using mathematical equations we have derived.In the previous posts we had discussed how to find the center of the system using its mathematical derivation.
Let us consider a system which has three identical particles having equal masses placed at the three corners of a equilateral triangle of known side. We would like to measure the center of mass of this system. Let each particle has different masses of 1,2 and 3 kilogram at three corners of the triangle and the side of the triangle is only one meter.
Let one kilogram and 2 kilogram are on the horizontal axis and third body of mass three kilogram is at the third corner of triangle as shown in the video presented.
To find the center of mass of any system, we shall consider a reference point and let one kilogram in this problem is that reference point. We are going to find out center of mass of this system using that reference point. We shall treat that point as horizon and shall measure the position of all particles with reference to that point.
As shown below, using simple mathematical techniques like Pythagoras rule, we have determined the coordinates of each particle of the system as shown.
As we know the mass of all particles and their coordinates, we can measure the center of mass of this system as shown below. This system has particles in a plane and hence center of mass will have both components of x- axis and y-axis.
Motion of center of mass is effected because of external force applied and internal forces cannot effect the motion of center of mass. Center of mass is a point of a system which represents the motion of the system when some force is applied on it. To prove the influence of the motion we can consider the derivation we have made for the acceleration of the center of mass as shown in the earlier post. By differentiating the acceleration of center of mass, we can get the equation for the for force acting on the system.
It is mathematically proved in the presented video, it is proved that the force on the center of mass of the system is the product of mass of the system and the acceleration of center of mass. The force is actually sum of internal as well as external forces. But we know that internal forces can not influence the motion of center of mass. It is simple like internal collisions inside a gas cylinder can not move it physically from one place to other and we need to apply external force to move it from one place to other.
So in the place of forces, we can simply write the external forces acting on the system. Thus external force is the product of mass of the system and the acceleration of center of mass.
If the external force is equal to zero, then the product of mass of the system and its acceleration of center of mass is equal to zero. Product of two quantities will be zero when at least on of them is equal to zero. Mass is a fundamental physical quantity and it can not be zero. There is no particle in the nature that has zero mass. Hence the acceleration of center of mass of the system is equal to zero.
So it is clearly exhibited that only external forces can influence the motion of center of mass but not internal forces. It is very clear that internal forces cannot show any influence on the motion of the center of mass.
Let us consider a small example like a bomb thrown from the ground at an angle to the ground from the horizontal. In the middle of the path, let it got exploded into so many pieces. This explosion is happened because of internal forces but not due to any external internal forces. Different particles of the bomb may travel in different directions but the center of mass particle of the bomb continue its parabolic path.
Motion of center of mass can be understood in terms of velocity and acceleration of center of mass. We have derived equations for coordinates of center of mass in the post earlier.Position of center of mass is expression in terms of displacement and differentiating it once with respect to time gives velocity of center of mass.
As the mass of the particle is constant, it is not going to vary with time during differentiating and hence it can be written out side the mathematical differentiation.Thus,we need to differentiate displacement of each particle with time. The rate of change of displacement is velocity. Thus we get velocity of each particle. Hence, we can get the velocity of centre of mass of the system.
In the place of product of mass and velocity,we can substitute the momentum of the each particle. Thus, by adding the momentum of each particle, we can write the momentum of the system. Adding all the particles masses can give the total mass of the system. The product of mass of the system with velocity of center of mass of the system gives the momentum of center of mass.
By differentiating the velocity of center of mass equation with time, we can get the acceleration of center of mass of the system. Rate of change of velocity of each particle with time give acceleration of each particle. Thus we can write the equation for acceleration of center of mass of the system as shown in the video lesson below.
We can write the product of mass and acceleration of the system as the force acting on the system.
Centre of mass is a point of a system and its coordinates can be expressed in the vector format. Center of mass is a point of system which can be used to represent the motion of the system.We have learned in the previous post that, we can derive mathematical derivation of centre of mass location and it is some where in between the system of two bodies. It can be understood that the center of mass point is towards the heavy mass location of the system.
For a one dimensional body mass is along only one dimension like length and the center of mass has only one coordinate, may be along X-axis or Y-axis.
If the body is in a plane, its mass is spread over two dimensions and hence its center of mass also has two coordinates.To express the center of mass, we need at least two coordinates. If the system is in the three dimensions, then the center of mass of this system also has three coordinates.
Center of mass of a system can be identified even for a system with large number of particles just by extending the mathematical format.
We can express coordinates of center of mass in the mathematical vector format also. By adding unit vectors along each direction of the coordinate system, we can assign a vector format as shown in the video below.
To measure the magnitude of center of mass, we can find the vector magnitude of center of mass system as shown.
Center of mass is a point of a body or a system which represents the actual motion of a body or a system. There need not any mass at the center of mass point physically. For a body, it can be out side the body and it can be inside the system. If different point of a body are having complicated and different kinds of motion, the real motion of the body is exhibited by center of mass.
The algebraic sum of moments about the center of mass of a system is equal to zero. Different particles of the body exhibits and applies moment on the center of mass particles in different directions but their algebraic sum is equal to zero.
Moment is defined as the product of mass and distance of the point about the center of mass.
We can derive mathematical expression for the center of mass of a system using the concept that the algebric sum of moments is equal to zero.
The location of center of mass of a system depends on the reference point from which we are measuring the center of mass of the system.
Let us consider a system of two particles having different masses. From a reference point these two particles are at two different distances. Let us assume that the cenere of mass of the system is some where between them and we can equate the moments about that point with all other points.
By simplifying the mathematical equations as shown in the video, we can get the mathematical expression as shown below.
When ever the reference position changes the value of cenrte of mass appears like changes with respect to that point.
Centre of mass is a point of a body which represents the
actual motion of the body. In mechanics we are always interested in studying
the motion of the body. To move a body from one position to another position,
we shall apply sufficient external force. The body will be able to move only
when the applied forces sufficient to overcome the inertia of the body. Once if
the body starts moving, we would like to study the motion of the body like what
is the path of the body.
If all particles of the body are having a similar kind of
displacement and the path, that kind of the motion is called as translatory motion.
In translatory motion any point of the body represents the actual motion of the
body. But practically all the particles of the body every time won’t represent
the actual motion of the body. In that case, to understand the translatory motion
we shall consider a point which actually represents the motion of the body. The
point of the body which represents the actual motion of a body is called
Centre of mass. If we want to replace the body and put a point we shall put the
point and the location of the Centre of mass.
It may happen sometimes that at the point of Centre of Mars,
there won’t be mass at all. For example in the case of the motion of the ring on
a floor, we can study the motion of the ring taking the Centre of mass at the
centre of the ring and actually at the centre of the ring there is no content
and there is no mass. Centre of mass doesn’t mean there shall be mass and it is
just a point which represents the actual motion of the body, Having a mass at
the Centre of mass is not mandatory.
Taking any point of the body to study the motion of the body
every time will not give you the correct way of understanding the motion of the
body.
If we consider a point of the surface of a tire of a vehicle
who is moving to study the motion of the tire, we get the motion of that point
is a complicated one as shown in the video. But actually the tire is moving in
a more organized way and to study the motion of the tire we shall consider a
point called Centre of mass. For any regular and organized bodies, Centre of
mass is at the geometrical centre of the system. Different particles of body
may be representing different kinds of motion, but the actual motion of the
body is represented by the Centre of mass.
Motion of a single body are a system of the particles which
are together treated like a system, can be studied basing on the concept of
Centre of mass.
Let us consider the sun and the earth system. We know that
the earth is revolving around the Sun in a specified orbit. We also know that
it is because of the gravitational force between the sun and the earth, the
Earth is revolving around the Sun in a specified orbit. We also know that
during the process earth is also having a spin motion. Now we would like to
study the motion of the earth around the Sun. Taking any point on the surface
of the earth and if we identify the path taken by it around the Sun, it will be
as shown in the given video. It can be noticed that the party is quite
complicated and we don’t express the path of the earth in that way. It is
simply because the point that we have taken the earth is not the Centre of mass
of the earth. If we consider the point of Centre of mass of the earth and study
the motion of the earth around the Sun, we can see a clear elliptical path.
Therefore Centre of mass is a point of your body or a system
which represents the actual motion of the body. There may be mass are there may
not be mass at the Centre of mass point and if it is representing the motion of
the body are system, it shall be treated like a Centre of mass.
For all uniform and geometrically organized
shape bodies Centre of mass coincides with the geometrical Centre and the
Centre of gravity of the body. Centre of gravity is a point of your body or
system which represents the place where the acceleration due to gravity is
acting on the body.