Centre of Mass

Centre of Mass

1. Centre of mass of a body or a system of particles is the point at which the whole mass of the system or body is supposed to be concentrated and moves as if

the whole external force is applied at that point.

2. The motion of centre of mass of a body represents the motion of the whole body.

3. Position of centre of mass in Vector notation can be denoted as follows:

If r₁, r₂, r₃. . . are the position vectors of particles of masses m₁, m₂, m₃...... then the position vector of their centre of mass is



Velocity of centre of mass

If ....... are the velocities of particles of masses

m₁, m₂, m₃......., m_n , theN velocity of their centre of mass.



i.e total momentum of the system is the product mass of the whole system and the velocity of the centre of mass.



LAW OF CONSERVATION OF LINEAR MOMENTUM :

In the absence of net external force, the total linear momentum of a system remains constant. i.e. = constant

Effect of internal forces- The linear momentum of particles remains constant under the influence of internal forces .

The linear momentum is conserved in all types of collisions (elastic and inelastic)

or



where u₁and u₂and v₁and v₂are the velocities of two particles with masses m₁and m₂ before and after the collision.

In the absence of external forces, the linear momenta of individual particles can change but the total linear momentum of the whole system remains constant.

The law of conservation of linear momentum is based on the Newton's laws of motion. This is the fundamental law of nature and there is no exception to it.

If two particles of masses m₁and m₂are moving with velocities 1and 2at right angles to each other, then the velocity of their centre of mass is given by



Acceleration of centre of mass.

If , ........ are the accelerations of particles of masses m₁, m₂, m₃........mnthen the accelaration of their centre of mass is



In the absence of external forces,

a) the centre of mass of a system is at rest if the centre of mass is initially at rest.

b) if the centre of mass of a system is moving with constant velocity, it continues to move with the same velocity.

For a ring the centre of mass is its centre where there is no mass.

For a triangular plane lamina, the centre of mass is the point of intersection of the medians of the triangle.

The centre of mass of a uniform square plate lies at the intersection of the diagonals.
Out of a uniform circular disc of radius R, if a circular sheet of r is removed; the Centre of mass of remaining part shits by a distance . d is the distance of
the centre of the smaller part from the original disc.
Out of a uniform solid sphere of radius R, if a sphere of radius r is removed, the centre of mass of the remaining part, shifts by . d is the distance of the
smaller sphere from the centre of the original sphere.

Note:- When shell in flight explodes, The acceleration of centre of mass before and immediately after explosion is a_cm= g downward.



Collisions

The event or the process in which two bodies, either coming in contact with each other or due to mutual interaction at a distance apart, affect each others

motion (velocity, momentum, energy or the direction of motion) is defined as a collision between those two bodies.

The particles come closer before collision and after collision they either stick together or move away from each other.

The law of conservation of linear momentum is necessarily conserved in all types of collisions whereas the law of conservation of mechanical energy is not


Types of collisions :

(i) Elastic collision or perfect elastic collision

(ii) Semi elastic collision

(iii) Perfectly inelastic collision or plastic collision

One dimensional collision: The collision, in which the particles move along the same straight line before and after the collision, is defined as one dimensional

collision.

According to the law of conservation of kinetic energy



According to the law of conservation of momentum


Newton's law of elastic collision - The relative velocity of two particles before collision is equal to the negative of relative

velocity after collision i.e (v₁- v₂) = -(u₁- u₂)


Important formulae and features for one dimensional elastic collision.

The velocity of first body after collision



The velocity of second body after collision



m₁= m₂then v₁= 0 and v₂= u₁. Under this condition the first particle come to rest and the second particle moves with the velocity of first particle before

collision. In this state there occurs maximum transfer of energy.

Exchange of energy is maximum when m₁= m₂. This fact is utilised in atomic reactor in slowing down the neutrons. To slow down the neutrons these are made

to collide with nuclei of almost similar mass. For this hydrogen nuclei are most appropriate.


Target Particle at rest : If m₂ is at rest, before collision





If m₂ is at rest and kinetic energy of m1before collision with m₂ is E. The kinetic energy of m₁and m₂ after collision is



and



One dimensional inelastic collision:

The collision, in which the kinetic energy of the system decreases as a result of collision, is defined as inelastic collision

According to law of conservation of energy



Q = other forms of energy like heat energy, sound energy etc. Q0

According to Newton's law of inelastic collision (v₁- v₂) = -e(u₁- u₂)

e = Coefficient of restitution


Coefficient of Restitution (e) -





or -

e is dimensionless and carries no limit.

Limits of e 0 < e < 1

For plastic bodies and for perfectly inelastic collision e = 0. e = 1 for perfect elastic collision.



SEMI - ELASTIC COLLISIONS

1) The velocity of first body after collision



and velocity of second body



2) Loss of energy in inelastic collision



If a body falls from a height h and strikes the ground level with velocity and rebounds with velocity v upto a height h₁then the coefficient of restitution is

given by

If the body rebounds again and again to heights h₁, h₂, h₃.... then



The total distance covered by the body for infinite number of collisions

Time taken by the body in falling through height h is

Thus the total time taken by the body in coming to rest is

For a semielastic collision 0 < e < 1



Perfectly inelastic collision:

The collision, in which the two particles stick together after the collision, is defined as the perfectly inelastic collision.

For a perfectly inelastic collision e = 0

According to law of conservation of momentum





Explosions

In an explosion, linear momentum is conserved, but kinetic energy is not conserved. The kinetic energy of the system after explosion increases. The internal

energy of the system is used for the above purpose.

If a stationary shell breaks into two fragments, they will move in opposite directions, with velocities in the inverse ratio of their masses.



Note:- When a stationary shell explodes, its total momentum is zero, before or after explosion.



Recoil of Gun

If a stationary gun fires a bullet horizontally, the total momentum of the gun + bullet is zero before and after firing.

If M and are the mass of the gun and velocity of recoil of the gun, m and is the mass of the bullet and velocity of the bullet, then

M + m = 0 i.e. |MV| = |mv|

or magnitude of momentum of the gun is equal to magnitude of momentum of the bullet.

The bullet has greater Kinetic energy than the gun.