Page 38 - 4167

not depend on action of other forces. It is principle of forces
superposition(principle of independent action of forces).Then the total
acceleration is given by:

    n 
a (3.8)
 aa 1  a 2  ...  a n   i
   i 1
The law F  m a  i , is fulfilled for each force and we can sub-
i
stitute 
 F
a  i m
i
 1 n  n  
a  F  F  m a 
i
i
m  i 1 i1 (3.9)
The resultant force is the force that causes the same action as
all forces applied to an object together:
 n 
F   F i
 i 1 (3.10)

So, instantaneous acceleration is the second derivative of vector
 2 
displacement with respect to time a  d r 2 .The differential
dt
equation of motion is given by
 d 2 
r
F  m
dt 2 (3.11)
and is called the main equation of motion. In terms of components:

d 2 x
F  m  a  m
x
x
dt 2
d 2 y
F  m  a  m
y y 2
dt
2
d z
F  m a  m
z
z
dt 2 (3.12)
The basic equation is called so because if forces that act on an
object and its mass are given it is possible to determine all characteristics

(parameters) of motion.
 
Note, that Newton's second law in the form F  m a  is valid
only for objects with constant mass. In general case the concept of

impulse is introduced:

37