Motion Of A Particle

 

Principles in  the formation of the Equation of Motion of a Particle :-  

For Motion in Straight Line ;

Let the mass of the body (particle) be    m    and let the distance of the particle measured from a suitable origin be   x   at the time  t   .

Then acceleration is      (d/dt)(dx/dt)

therefore by Second Law of Motion

m[(d/dt)(dx/dt)] = Forces in the direction of    x   increasing

The forces usually are ;-

1)---   g    vertically downwards in a gravitational field like earth .
2)---  tension in a string
3)---  reactions or stress at point where the point may be in contact with other particles .
4)---  forces of attraction 
5)---  forces of resistance to motion by any means may be by atmosphere , friction, winds or by any other means .

In  taking the forces we must fix the sign properly if the forces acts along the line .

If a force  F   acts at an angle    A     to the straight line , then the resolved part of the force along the line is    FcosA  . If   g  acts on on a particle and if the particle is moving along the horizontal line , then   g    , being vertical , has no resolved part horizontally  and   g   does not effects the Motion .

Also note that velocity,    v=dx/dt     , and acceleration

     (d/dt)(dx/dt) = dv/dt = (dx/dt)(dv/dx) = v(dv/dx)  

For Motions in two dimension;

Let two perpendicular axes are   X      and    Y     at the plane of motion . The forces are resolved in these two directions and respectively

   m(d/dt)(dx/dt)       and       m(d/dt)(dy/dt)