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Center of mass

Module by: Sunil Kumar Singh. E-mail the author

Summary: There is a characteristic geometric point of the three dimensional body in motion, which behaves yet as a particle.

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Our study of motion has been limited up to this point. We referred particle, object and body in one and same way. We considered that actual three dimensional rigid body moved such that all constituent particles had same motion i.e. same trajectory, velocity and acceleration.

An actual body, however, can move differently to this simplified paradigm. Consider a ball rolling down an incline plane or consider a stick thrown in air. Different parts of the body have different motions. While translating in the air, the stick rotates about a moving axis. In the nutshell, this means that these bodies do not behave like a particle as assumed earlier.

Figure 1: The stick does not behave like a particle.
Stick thrown in air
 Stick thrown in air  (cm1a.gif)

Apparently, it would be quite complicated to describe motions of parts or particles, having different motions, in an integrated manner. There is but one surprising simplifying characteristic of these motions. There is a characteristic geometric point of the three dimensional body in motion, which behaves yet as a particle. This point is known as "center of mass" in short "COM". It has following two characterizing aspects :

  • COM appears to carry the whole mass of the body.
  • All external forces appear to apply at COM.

Figure 2
Rolling ball on an incline and Stick thrown in air
(a) The force appears to operate on the COM is “mgsinθ”. (b) The force appears to operate on the COM is “mg”.
Figure 2(a) (cm2a.gif)Figure 2(b) (cm3a.gif)

Significantly, the center of ball, which is COM of rolling ball, follows a straight linear path; whereas the COM of the stick follows a parabolic path as shown in the figure. Secondly, the forces appear to operate on the COMs in two cases are “mgsinθ” and “mg” as if they were indeed particle like objects. This concept of COM, therefore, relieves us of the complexities that otherwise we would have faced describing motions of rigid bodies.

We must here note the word "appear" in describing characteristics of COM. In reality, COM is a geometric point. It need not be even the part of the body. Think of a hollow sphere, whose center is its COM. Is there any material there? Also, consider the motion of the stick again. Each particle constituting the stick is pulled by gravitational force. What it means that external forces, in actuality, may not act on COM alone as they appear in equivalent term.

Further the concept of COM is equally valid to a system of particles, which may not be in contact like in the case of three balls placed at the vertices of a triangle. This follows from the fact that a rigid body, after all, is an arrangement of particles, where inter-particle distances are extremely small.

In words, we can, therefore, define COM as :

Definition 1: Center of mass (COM)
COM of a system of particles is a geometric point, which assumes all the mass and external force(s) during motion.

Mathematical expression of COM for a system of particles

Let us now look at a baseball bat closely. It has a peculiar shape. Where should the COM lie? By experience, we can say that it should lie on the heavier side. This perception comes from the realization that heavier part has most of the mass. It gives us the clue about COM that it lies on the heavier side of an irregularly shaped body. Now, let us consider the motion of spherical sphere of uniform density. Here, mass is evenly distributed. Where should the COM lie? Obviously, COM coincides with the center of spherical ball.

It emerges from the discussion that COM is a statement of spatial arrangement of mass i.e. distribution of mass within the system. The position of COM is given a mathematical formulation which involves distribution of mass in a volumetric space, assuming as if all mass is concentrated there :

Figure 3: COM assumes as if mass of all particles is concentrated there.
COM for a system of particles
 COM for a system of particles  (cm4.gif)

Total mass x Position of COM = Mass of individual particle x Position of particle Total mass x Position of COM = Mass of individual particle x Position of particle

Position of COM = Mass of individual particle x Position of particle Total mass Position of COM = Mass of individual particle x Position of particle Total mass

This relation is read as "position of COM is the mass weighted average of the positions of particles".

Further, COM is a geometric point in three-dimensional volume and it involves distribution of mass with respect to some reference system. This expression of the position of COM is description of a position in three dimensional volume. Now, substituting with appropriate symbols, we have :

r COM = m i r i M r COM = m i r i M

It is important to note here that this formulation is not arbitrary. We shall find subsequently that this mathematical expression of COM, as a matter of fact, leads us to formulation of Newton's second law for a system of particles, while keeping its form intact.

We can express the position of COM in scalar component form as :

x COM = m i x i M y COM = m i y i M z COM = m i z i M x COM = m i x i M y COM = m i y i M z COM = m i z i M

We know that there are only two directions involved with each of the axes. As such, we assign positive value for distances in reference direction and negative in opposite direction.

Special cases

(a) Particle system along a straight line :

COM of particle system along a straight line is given by the scalar expression in one dimension:

Figure 4: Scalar COM expression in one dimension
COM for a system of particles along a straight line
 COM for a system of particles along a straight line  (cm5.gif)

x COM = m i x i M x COM = m i x i M

Evidently, COM of particles lie on the line containing particles.

(b) Two particles system

The expression of COM reduces for two particles as :

x COM = m 1 x 1 + m 2 x 2 m 1 + m 2 x COM = m 1 x 1 + m 2 x 2 m 1 + m 2

As only one coordinate is involved, it is imperative that COM lies on the line joining two particles. There are two additional simplified cases for two particles system as :

(i) Origin of coordinate coincides with position of one of the particles :

The expression of COM is simplified when position of one of the particles is the origin. For two particles system, x 1 = 0 , x 2 = x , x 1 = 0 , x 2 = x ,

x COM = m 2 x m 1 + m 2 x COM = m 2 x m 1 + m 2

where "x" is the linear distance between two particles.

(ii) The masses of the particles are equal :

In this case, m 1 = m 2 = m m 1 = m 2 = m (say)

x COM = x 1 + x 2 2 x COM = x 1 + x 2 2

When origin of coordinate coincides with position of one of the particles, x 1 = 0 , x 2 = x , x 1 = 0 , x 2 = x , :

x COM = x 2 x COM = x 2

This result is on expected line. Center of mass (COM) of two particles of equal masses is midway between the particles.

Example 1

Problem : Two particles of 2 kg and 4 kg are placed at (1,4) and (3,-2) in x and y plane. If the coordinates are in meters, then find the position of COM.

Solution : Using expression in scalar component form :

Figure 5: Positions of the particles are shown on "xy" coordinate system.
Two particles system
 Two particles system  (cm6.gif)

x COM = m 1 x 1 + m 2 x 2 m 1 + m 2 x COM = m 1 x 1 + m 2 x 2 m 1 + m 2

x COM = 2 x 1 + 4 x 3 6 = 7 3 m x COM = 2 x 1 + 4 x 3 6 = 7 3 m

and

y COM = m 1 y 1 + m 2 y 2 m 1 + m 2 y COM = m 1 y 1 + m 2 y 2 m 1 + m 2

y COM = 2 x 4 + 4 x ( - 2 ) 6 = 0 y COM = 2 x 4 + 4 x ( - 2 ) 6 = 0

Thus, position of COM lies on x - axis at x = 7/3 m.

COM of rigid body

Rigid body is composed of very large numbers of particles. Mass of rigid body is distributed closely. Thus, the distribution of mass can be treated continuous. The mathematical expression for rigid body, therefore, is modified involving integration. Integrating for a differential mass element, ⅆm, we have :

x COM = 1 M x m y COM = 1 M y m z COM = 1 M z m x COM = 1 M x m y COM = 1 M y m z COM = 1 M z m

Evaluation of above integrals is simplified, if the density of the rigid body is uniform. In that case,

ρ = M V = m V m M = V V ρ = M V = m V m M = V V

Substituting,

x COM = 1 V x V y COM = 1 V y V z COM = 1 V z V x COM = 1 V x V y COM = 1 V y V z COM = 1 V z V

Symmetry and COM of rigid body

Evaluation of the integrals for determining COM is very difficult for irregular shaped bodies. On the other hand, symmetry plays important role in determining COM of a regularly shaped rigid body. There are certain simplifying facts about symmetry and COM :

  1. If symmetry is about a point, then COM lies on that point. For example, COM of a spherical ball of uniform density is its center.
  2. If symmetry is about a line, then COM lies on that line. For example, COM of a cone of uniform density lies on cone axis.
  3. If symmetry is about a plane, then COM lies on that plane. For example, COM of a baseball bat lies on the central plane.

The test of symmetry about a straight line or a plane is that the body on one side is replicated on the other side.

Example 2

Problem : A small circular portion of radius r/4 is taken out from a circular disc of uniform thickness having radius "r" and mass "m" as shown in the figure. Determine the COM of the remaining portion of the uniform disc as determined from the center of the disc.

Figure 6: A small circular portion of radius r/4 is taken out.
COM of remaining portion of circular disc
 COM of remaining portion of circular disc  (cm7.gif)

Solution : If we start from integral to determine COM of the remaining disc portion, then it would be a really complex proposition. Here, we shall make use of the connection between symmetry and COM. We note that the COMs of smaller and bigger discs are their respective centers. How can we use this fact to find center of mass of the remaining portion ?

The main idea here is that we can treat regular bodies with known COM as particles, which are separated by a known distance. Then, we shall employ the expression of COM for two particles to determine the COM of the remaining portion. We must realize that when a portion is removed from the bigger disc on the right side, the COM of the remaining portion shifts towards left side (heavier side).

The test of symmetry about a straight line or a plane is that the body on one side is replicated on the other side. We see here that the remaining portion of the disc is not symmetric about y-axis, but is symmetric about x-axis. It means that the COM of the remaining portion lies on the x-axis on the left side of the center of the bigger of the disc. It also means that we can employ the expression of COM for one dimension.

While employing expression of COM, we use a trick. The bigger disc (with known COM) is equivalent to two particles system comprising of (i) remaining portion (with unknown COM) and (ii) smaller disc (known COM). Now, x-component of the COM of larger disc is :

Figure 7: A small circular portion of radius r/4 is taken out.
COM of remaining portion of circular disc
 COM of remaining portion of circular disc  (cm8.gif)

x COM = m 1 x 1 + m 2 x 2 m 1 + m 2 x COM = m 1 x 1 + m 2 x 2 m 1 + m 2

But, x-component of the COM of large disc coincides with origin of the coordinate system. Further, let us denote the remaining disc by subscript "r" and the smaller circular disk by subscript "s".

0 = m r x r + m s x s m r + m s 0 = m r x r + m s x s m r + m s

m r x r + m s x s = 0 x r = - m s x s m r m r x r + m s x s = 0 x r = - m s x s m r

As evident from figure, x s = r 2 x s = r 2 . We, now, need to find mass of smaller disc, m s m s , and mass of remaining portion, m r m r , using the fact that the density is uniform. The density of the material is :

ρ = m V = m π r 2 t ρ = m V = m π r 2 t

where "t" is thickness of disc.

m s = V ρ = ( π r 2 4 ) t x m π r 2 t = m 4 m r = m - m s = m - m 4 = 3 m 4 m s = V ρ = ( π r 2 4 ) t x m π r 2 t = m 4 m r = m - m s = m - m 4 = 3 m 4

Putting these values in the expression of the position of COM, we have :

x r = - m s x s m r = - ( m 4 ) ( r 2 ) 3 m 4 = - r 6 x r = - m s x s m r = - ( m 4 ) ( r 2 ) 3 m 4 = - r 6

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