Newton's 2nd Law Of Motion

Newton's 2nd Law Of Motion

1.    The rate of change of momentum of a body is directly proportional to the external force and the change in momentum takes place in the direction of the force.
2.    Newton’s second law of motion leads to a formula useful for measuring force.\(\overline{F}=m\overline{a}\)  .
3.    Force is a vector.  It is always in the direction of change in momentum. Force is also always in the direction of acceleration.
4.    SI unit of force is newton (N).  If a force acting on a mass of 1 kg produces in it an acceleration of 1 m s^-2 in its direction, it is called a newton. 
5.    CGS unit of force is dyne.  If a force acting on a mass of 1 gm produces in it an acceleration of 1 cm s-2 in its direction, it is called a dyne.
6.    One newton = 10⁵ dyne.
7.    Gravitational units of force:  kilogram weight (kg.wt) and gram weight (gm.wt) are called the gravitational units of force. 1 kg.wt or kg. f = 9.8 N,  
       1 gm.wt or gm.f = 980 dyne.
8.    To calculate a force ‘F’, there are several useful variants of the formula \(\overline{F}=m\overline{a}\).
9.     \( F = \frac{{mv - mu}} {t} \)   ,    10. \( F = m\frac{{dv}} {{dt}} \)s    ,       11.\( F = v\frac{{dm}} {{dt}} \)   
Derivation of F = ma : Consider a body of mass ‘m’ moving with initial velocity u. Let a force F acts on the body for time ‘t’ so that the velocity of the body after time ‘t’ is v.

    
Initial momentum of the body (P_i) = m u
    Final momentum of the body (P_f) = m v
    Now, change in momentum of the body = Pf – Pi = mv – mu = m (v – u)
    Time taken for this change in momentum = (t – 0) = t
    \( \therefore \) Rate of change of momentum = \( \frac{{change\,\,of\,\,momentum}} {{time\,\,taken}} \) =\( \frac{{m\,\,(v\, - \,u)}} {t} \) = m a  \( \left( {\because a = \frac{{(v\, - \,u)}} {t}} \right) \) 
F=\( \frac{{d\vec p}} {{dt}} \)= Rate of change of momentum
F=ma