Newton’s Second Law of Motion : Concept of Momentum

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Statement:

  • The rate of change of momentum of a body is directly proportional to the impressed (applied) force and takes place in the direction of force.
  • To apply Newton’s second law of motion, the body should be acted upon by unbalanced force. Thus to apply this law we should consider a net force acting on the body.
  • Force is a vector quantity.  The direction of the net force is in the same direction as that of the acceleration. Thus, if the direction of the acceleration is known, then the direction of the net force is also known.


Concept of Momentum:

  • Catching a ball which is moving with a high speed is more difficult than catching a ball which is moving with lesser velocity. It is easier to catch a ball with less mass than to catch a ball with more mass.
  • The physical quantity which depends on the velocity of body and its mass and which determines how much force is required to bring them to rest is known as momentum.
  • The product of mass and velocity of a body is called momentum of body.  It is denoted by p. Thus magnitude of momentum is given by p = mv.
  • Since mass is scalar and velocity is a vector, momentum is a vector quantity whose direction is the same as that of velocity.
  • The S.I. unit of momentum is kilogram metre per second (kg m s-1). The c.g.s. unit is gram centimetre per second (g cm s-1).  Dimensions of momentum are  [M1L1T-1]
  • If body is moving along a straight line path then it is said to possess linear momentum. If body is rotating then it is said to possess angular momentum.

Explanation :

  • Let us assume a body of mass m moving with velocity u. Let a constant force F acts on the body for t seconds and changes its velocity from u to v.

Change of Momentum

  • Where, k is constant and unit of force is defined such that value of k is 1.
  • Therefore in scalar form mathematical expression of Newton’s second law of motion is given by  F  =  ma. This equation is also called the fundamental equation of classical mechanics.

Note:

  • When mass of a body is constant, then its acceleration is directly proportional to the force acting on it.When force acting on a body is constant, then its acceleration is inversely proportional to the mass of the body.
  • When  a rocket is launched upward in a space, its acceleration  increases. As the fuel in rocket is burned and exhausted to propel the rocket, the mass of the rocket decreases continuously. But the propulsion force is almost constant. Hence acceleration increases.


Newton’s First Law as Special Case of Newton’s Second Law:

  • Mathematical expression of Newton’s second law of motion is given by F  =  ma.
  • Let F = 0. Substituting this value in above equation value of ‘a’ the acceleration is found to be zero. Thus there is no change in the magnitude and direction of velocity, when there is no force.  It means body at rest is remaining at rest and the body in uniform motion in straight line is remaining in the uniform motion in the same straight line.  Thus Newton’s first law of motion can be considered as special case of the second law of motion.

Units of Force:

Definition of Unit Force:

  • Force = Mass   x  Acceleration . Hence   Unit force = Unit mass   x  Unit acceleration.
  • Unit force is that force which produces unit acceleration in a body of unit mass.
  • Absolute units of force are newton and dyne.
  • Gravitational units of force are kgf and gf
  • Dimensions of force are [M1L1T-2]

Definition of 1 newton (1N):

  • S.I. unit of force is newton (N).
  • 1 newton  = 1 kg   x  1 ms-2
  • One newton  is that force which produces an  acceleration of 11 ms-2  in a body of mass 1 kg.

Definition of 1 dyne:

  • c.g.s.  unit of force is dyne
  • 1 dyne  = 1 gm   x 1 cm s-2  
  • One dyne  is that force which produces an  acceleration of 1 cm s-2 in a body of mass 1 gram.

Definition of 1 kilogram force (1 kgf):

  • It is a force which is required to produce an acceleration of 9.8 m s-2 in a body of mass 1 kg.
  • 1 kgf = 1 kg x 9.8 m s-2 , Thus 1 kgf = 9.8 N.

Definition of 1 gram force (1 gf):

  • It is a force which is required to produce an acceleration of 980 cm s-2  in a body of mass 1 g.
  • 1 gf = 1 g X 980 cm s-2 , Thus 1 gf = 980 dyne

Relation between newton and dyne:

  • 1 newton  = 1 kg   x  1 m s-2 1 , thus newton  = 1000 g   x  100 cm s-2
  • Thus 1 newton  = 100000 g cm s-2 i.e 1 newton  = 105 dyne


Impulse of a Force:

  • When large force acts on a body for short interval of time then the body is said to be subjected to impulse.
  • Impulse is denoted by letter J.
  • e.g. Hammer struck on nail, ball struck by bat.

Change of Momentum 02

  • Thus  impulse of force = change in the momentum of the body.
  • The S.I. unit of impulse is newton-second (Ns). In c.g.s. system its unit is dyne second.
  • Its dimensions are[M1L1T-1]. These dimensions are same as that of momentum.
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