CENTER OF MASS Revision Notes - IIT JEE/NEET Preparation | Nucleon

## CENTER OF MASS

• CENTER OF MASS
• Every physical system has associated with it a certain point whose motion characterises the motion of
the whole system. When the system moves under some external forces, then this point moves as if the
entire mass of the system is concentrated at this point and also the external force is applied at this
point for translational motion. This point is called the center of mass of the system

• CENTER OF MASS OF A SYSTEM OF 'N' DISCRETE PARTICLES
• CENTER OF MASS OF A SYSTEM OF 'N' DISCRETE PARTICLES

• POSITION OF COM OF TWO PARTICLES
• Center of mass of two particles of masses m1 and m2 separated by a distance r lies in between the
two particles. The distance of center of mass from any of the particle (r) is inversely proportional
to the mass of the particle

• CENTER OF MASS OF A CONTINUOUS MASS DISTRIBUTION
• CENTER OF MASS OF A CONTINUOUS MASS DISTRIBUTION

• CENTER OF MASS OF A UNIFORM ROD
• Suppose a rod of mass M and length L is lying along the x-axis with its one end at x = 0 and the

• CENTER OF MASS OF A SEMICIRCULAR RING
• Figure shows the object (semi circular ring). By observation we can say that the x-coordinate of
the center of mass of the ring is zero as the half ring is symmetrical about y-axis on both sides of
the origin. Only we are required to find the y-coordinate of the center of mass.

• CENTER OF MASS OF SEMICIRCULAR DISC
• Figure shows the half disc of mass M and radius R. Here, we are only required to find the ycoordinate
of the center of mass of this disc as center of mass will be located on its half vertical
diameter. Here to find ycm, we consider a small elemental ring of mass dm of radius x on the disc
(disc can be considered to be made up such thin rings of increasing radii) which will be integrated
from 0 to R. Here dm is given as

• CENTER OF MASS OF A SOLID HEMISPHERE
• The hemisphere is of mass M and radius R. To find its center of mass (only y-coordinate), we
consider an element disc of width dy, mass dm at a distance y from the center of the hemisphere.
The radius of this elemental disc will be given as

• CENTER OF MASS OF A HOLLOW HEMISPHERE
• A hollow hemisphere of mass M and radius R. Now we consider an elemental circular strip of
angular width d? at an angular distance ? from the base of the hemisphere. This strip will have an
area.

• CENTER OF MASS OF A SOLID CONE
• A solid cone has mass M, height H and base radius R. Obviously the center of mass of this cone
will lie somewhere on its axis, at a height less than H/2. To locate the center of mass we consider
an elemental disc of width dy and radius r, at a distance y from the apex of the cone. Let the mass
of this disc be dm, which can be given as

• CENTER OF MASS OF SOME COMMON SYSTEMS
• The center of mass lies closer to the heavier mass

• MOTION OF CENTER OF MASS AND CONSERVATION OF MOMENTUM
• Here numerator of the right hand side term is the total momentum of the system i.e., summation
of momentum of the individual component (particle) of the system
Hence velocity of center of mass of the system is the ratio of momentum of the system to the mass of the
system.

• Accelerat ion of center of mass of system
• Accelerat ion of center of mass of system

• Motion of COM in a moving system of particles
• Motion of COM in a moving system of particles

• COM moving with acceleration
• If an external force is present then COM continues its original
motion as if the external force is acting on it, irrespective of
internal forces.
Example:

• Momentum Conservation
• The total linear momentum of a system of particles is equal to the product of the

• IMPULSE
• Impulse applied to an object in a given time interval can also be
calculated from the area under force time (F-t) graph in the same
time interval.

• Impulsive force
• A force, of relatively higher magnitude and acting for relatively shorter time, is called impulsive force.
An impulsive force can change the momentum of a body in a finite magnitude in a very short time
interval. Impulsive force is a relative term. There is no clear boundary between an impulsive and Non-
Impulsive force

• Impulsive Tensions
• When a string jerks, equal and opposite tension act suddenly at each end. Consequently equal
and opposite impulses act on the bodies attached with the string in the direction of the string.
There are two cases to be considered.

• COLLISION OR IMPACT
• Collision is an event in which an impulsive force acts between two or more bodies for a short time,
which results in change of their velocities.

• Classification of collisions
• Classification of collisions
On the basis of line of impact

• On the basis of energy
• Elastic collision : In an elastic collision, the colliding particles regain their shape and
size completely after collision. i.e., no fraction of mechanical energy remains stored as
deformation potential energy in the bodies. Thus, kinetic energy of system after collision is
equal to kinetic energy of system before collision. Thus in addition to the linear momentum,
kinetic energy also remains conserved before and after collision

• Examples of line of impact and collisions based on line of impact
• Examples of line of impact and collisions based on line of impact

• COEFFICIENT OF RESTITUTION
• The coefficient of restitution is defined as the ratio of the impulses of reformation and
deformation of either body.

• Important Point
• A particle ‘B’ moving along the dotted line collides with a rod also in state of motion as shown in the figure.
The particle B comes in contact with point C on the rod.

• Collision in two dimension
• A pair of equal and opposite impulses act along common normal direction. Hence,
linear momentum of individual particles do change along common normal direction.
If mass of the colliding particles remain constant during collision, then we
can say that linear velocity of the individual particles change during collision in
this direction.

• VARIABLE MASS SYSTEM
• then the force exerted by this mass on the system has magnitude

• Rocket propulsion
• Initially, let us suppose that the velocity of the rocket is u.

• LINEAR MOMENTUM CONSERVATION IN PRESENCE OF EXTERNAL FORCE
• LINEAR MOMENTUM CONSERVATION IN PRESENCE OF EXTERNAL FORCE

• ARCHIMEDES PRINCIPLE
• According to this principle, when a body is immersed wholly or partially in a fluid, it loses its
weight which is equal to the weight of the fluid displaced by the body.

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