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Linear Momentum


(1) Introduction


  • We have already studied about the newton's laws of motion and about their application
  • It becomes difficult to use Newton's law of motion as it is while studying complex problems like collision of two objects,motion of the molecules of the gas,rocket propulsion system etc
  • Thus a further study of newton's law is required to find some theorem or principles which are direct consequences of Newton's law
  • We have already studied one such principle which is principle of conservation of energy.Here in this chapter we will define momentum and learn about the principle of conservation of momentum .
  • Thus we begin this chapter with the concept of impluse and momentum which like work and energy are developed from Newton's law of motion

(2) Impulse and momentum



  • To explain the terms impulse and momentum consider a particle of mass m is moving along x-axis under the action of constant force F as shown below in the figure


    Impulse  and momentum
  • If at time t=0 ,velocity of the particle is v0 then at any time t velocity of particle is given by the equation
    v = v0 + at
    where a = F/m
    can be determined from the newton's second law of motion .Putting value of acceleration in above equation\
    we get
    mv = mv0 + Ft
    or
    Ft = mv - mv0                   -(1)
  • right side of the equation Ft, is the product of force and the time during which the force acts and is known as the impluse
    Thus
    Impulse= Ft
  • If a constant force acts on a body during a time from t1 and t2,then impulse of the force is
    I = F(t2-t1)                  -(2)
    Thus impulse recieved during an impact is defined as the product of the force and time interval during which it acts
  • Again consider left hand side of the equation (1) which is the difference of the product of mass and velocity of the particle at two different times t=0 and t=t
  • This product of mass and velocity is known as linear momentum and is represented by the symbol p. Mathematically
    p = mv                  --(3)
  • physically equation (1) states that the impulse of force from time t=0 to t=t is equal to the change in linear momentum during

  • If at time t1 velocity of the particle is v1 and at time t2 velocity of the particle is v2,then
    F(t2-t1)=mv2-mv1                  -(4)
  • so far we have considered the case of the particle moving in a straight line i.e along x-axis and quantities involved F,v, and a were all scalars
  • If we call these quantities as components of the vectors F,v and a along x-axis and generalize the definations of momentum and impulse so that the motion now is not constrained along one -direction ,Thus we got
    Impulse=I=F(t2-t1)                  -(5)
    Linear momentum=p=mv                  -(6)
    where
    I=Ixi+Iyj+Izk
    F=Fxi+Fyj+Fzk
    p=pxi+pyj+pzk
    v=vxi+vyj+vzk
    are expressed in terms of their components along x,y and z axis and also in terms of unit vectors
  • On generalizing equation (4) using respective vectors quantities we get the equation
    F(t2-t1) =mv2-mv1                  -(7)
  • So far while discussing Impulse and momentum we have considered force acting on particle is constant in direction and maagnitude
  • In general ,the magnitude of the force may vary with time or both the direction and magnitude may vary with time
  • Consider a particle of mass m moving in a three-dimensional space and is acted upon by the varying resultant force F. Now from newtons second law of motion we know that
    F=m(dv/dt)
    or Fdt=mdv
  • If at time t1 velocity of the particle is v1 and at time t2 velocity of the particle is v2,then from above equation we have


  • Integral on the left hand side of the equation (8) is the impulse of the force F in the time interval (t2-t1) and is a vector quantity,Thus

    Above integral can be calculated easily if the Force F is some known function of time t i.e.,
    F=F(t)

  • Integral on the right side is when evaluated gives the product of the mass of the particle and change in the velocity of the partcile

  • using equation (9) and (10) to rewrite the equation (8) we get

  • Equivalent equations of equation (11) for particle moving in space are

  • Thus we conclude that impulse of force F during the time interval t2-t1 is equal to the change in the linear momentum of the body on which its acts

  • SI units of impulse is Ns or Kgms-1



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