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Centre of  Mass and momentum.

posted Nov 5, 2017, 10:54 PM by Upali Salpadoru

Centre of Mass.




You may have learned the ‘Centre of mass’ as the balancing point of a body. But this is meaningless when it comes to a group of objects. Imagine two objects in space like the Earth and the moon.Where is the centre of mass?

The figure shows an astronaut tied to a space vehicle with a cord of negligible mass.


Where is the centre of mass of man and vehicle?


Fig.1. Two objects in space.

We can illustrate this picture as a simplified diagram.


If the two objects are connected by a stiff rod, as shown by the red line, we can imagine what would happen if we exert a force at the points shown by the white arrows A,B, C and D.

Starting from D for every force there will be some rotation except for force at one point.



And that point will be the Centre of Mass.

According to the mass ratio this has to be at A.

.


Centre of Mass

The point in a body or system of bodies at which the whole mass may be considered as concentrated. Definition - Merrium Webster dictionary


Mass and Momentum.

Mass in the absence of a force has no motion. Then the momentum is zero. When a force is applied it gains velocity It accelerates.  F=ma        that is    a = F/m .


The vigour of motion depends on the mass of the object and the velocity it has gained.

This is the vector quantity we describe as the….. MOMENT.

Moment =  Mass x Velocity. …….    P = mv

P    is the symbol for momentum.


Impulse.

Force multiplied by time gives a quality known as impulse.

F x t = Impulse…    Impulse =  Change in Momentum……….Δ p = F Δt

Vector quantities are underlined.  t is for time.




Centre of Mass and momentum.


When two masses A and B collide and move in separate ways , the centre of mass will move in the same direction.

Velocity of centre of mass is constant. (Vcm



Vcm   =  Total momentum / Total mass



Example 1.

2 kg. mass A  of velocity 0.5 ms-1 is colliding with B having a mass of 3 kg at 0. 4 ms-1 .

  1. Total momentum before collision.   

  2. Velocity of centre of mass.

  3. If A and B gets linked together, what will be it’s velocity


Solution.


a)Momentum of A ….= + 2x 0.5…= + 1.0 kgms-1

  Momentum of B ….=  - 3  x 0.4  =-  1.2 kgms-1

  Total momentum before = - 0.2  kgms-1


b) Vcm   = - 0.2 /mA = mB

         =  - 0.2/ 5……….- 0.04 ms-1


c) As the velocity of centre of mass does  not change the linked object will move at the same velocity as above . kis =  0.04 ms-1←   m


Example 2.


When a rock having a mass of 71 kg. was blasted it fragmented into three pieces.

Particle

Mass.

Velocity.

Momentum

1,

2

3

10 kg

36 kg.

25 kg.


10 ms-1  to East.

15 ms-1  to West.


360

375



Answer the following.

  1. Total momentum  before blasting.

  2. Draw a vector diagram for 2 and 3 after the blast.

  3. The resultant R.



  1. Momentum of piece no. 1.

  2. Velocity of no.1.


Answers

  1. P = mv  …...62 x0 = 0.

  1. Figure in column one.


  1. R2 = p1 2 + p2 2.

                  R = 3602  + 3752

                     = 129600 + 140625

                      =   270225

                 R   = 520

      Tan  Θ = 375 / 360  = 1.04

             Θ = 46°  

      R = 520 kg.ms-1  46°  south of west.

4.  Momentum of piece 1 has to be equal and opposite to R

    That is = 520 kg.ms2 ° west of North.


5.  Mass of no 1 piece= 62-36 - 25

                                  =



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