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3. Force and motion
3.1. Newton’s First Law
The first scientist who discovered that moving with constant velocity does not
require a force was Isaac Newton (observing the frictionless motion of the
Moon and the planets).
This is determined by the law
If no net (resultant) force acts on a body, the body’s velocity
cannot change; that is, the body cannot accelerate.


If Fr  0, then a  0
The reference frame in which the first law holds is called an inertial frame.
If several forces act on a body,
we determine the net force as a vector
sum of all forces (the net force of two
forces is shown in the figure) .



Fr  FA  FB
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3.2. Newton’s Second Law
The relation between the net force Fr applied on an object, its mass m and
the resulting acceleration a is given by Newton’s second law


Fr  m a for m  const
(3.1)
The net force acting on a particle is equal to the product
of the particle mass and its acceleration (for constant
mass).
In the case when mass m varies, the more general expression for the force is
used




dp
(3.1a)
Fr 
where p  m v
dt
The net force acting on a particle is equal to the time
rate of change of the momentum
The linear momentum (simply momentum) is a vector quantity which is changed
only by the external net force.
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Newton’s second law, cont.
Eq. (3.1a) transforms into (3.1) for a constant mass m




d(m v)
dv  dm
dv
Fr 
m
v
m
 ma
dt
dt
dt
dt

Newton’s second law can be considered as a definition of force acting on a
particle. In many cases we know the force from experience and need to know
the path of a particle. In this case one solves the so called equation of motion
enabling to find

r (t ) .




d 2 r t     d r 
m
 F r ,
,t
dt 2
 dt 


Example: forces acting on a body on the ramp
weight Q = mg
reaction (normal) force N
frictional force , in general defined as F ≤ μ N,
where μ - coefficient of friction
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Forces acting on a body on the ramp, cont.
Equation of motion (II Newton’s law):
  

N  Q  F  ma


Sum of forces N and Q on the left side of equation is a vector, which
magnitude is equal Q sin 
and then the above equation can be written in a scalar form as
Q sin   F  ma
When frictional force F has its maximum value one obtains
mg sin   mg cos   ma
and finally
a  g(sin    cos  )
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3.3. Newton’s Third Law
When two bodies interact by exerting forces on each
other, the forces are equal in magnitude and opposite
in direction.


FAB - force on A from (or due to) B, FBA - force on B from A.
This can be written as the vector relation


FAB   FBA
(3.2)
Eq. (3.2) holds when both forces are measured at the same time.
In the atomic scale the third law is not always obeyed.
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3.4. Inertial and noninertial reference frames
The reference frame is “inertial” if Newton’s three laws of motion hold.
In contrast, reference frames in which Newton’s laws are not obeyed are
labeled “noninertial.”
The frame which rests (or moves with constant velocity) in respect to the
distant „stable” stars is inertial.
The Earth in many practical cases can be considered as inertial.
We should remember however, that the Earth rotates around its axis which
gives a small acceleration. On the equator one gets
v2
4 2
cm
2
a1 
  R z  2 R z  3,4 2
Rz
T
s
Rz – Earth’s radius
T = 24 hrs
The circular motion around the Sun is a couse of another acceleration
a2 
4 2
3  10 s 
7
2
 1,5  1013 cm  0,6
cm
s2
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3.5. Inertial forces
In order to use Newton’s laws in noninertial frames, one introduces
apparent forces called inertial forces.

In the inertial frame the applied force F results in

acceleration ai


F  m  ai
(3.3)
In the noninertial frame moving with acceleration

a0 vs. the inertial frame this accelaration is equal
  
a  ai  a0
Hence
  
ai  a  a0
Introducing above into (3.3) one gets in the inertial frame
  
F  m a  a0 



(3.4)
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Inertial forces, cont.
Eq. (3.4) can be transformed as follows
 

ma  F  ma0
  
ma  F  F0


where F0  m a0 is the inertial force.
(3.5)
According to (3.5) the sum of real and apparent forces is employed
to write the second Newton’s law in the noninertial reference frame.
Example of an inertial force
In the rotating reference frame one introduces

the apparent force called centrifugal force F0 .
The centripetal acceleration of the reference frame


is equal a0   2  , where ω – angular velocity,
ρ – radius of the circle.
In this case in the rotating
 frame
 where the particle
is at rest one obtains Q  R  F0  0 ,



where the centrifugal force is given by F0  ma0  m 2  .
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4. Galileo’s Transformation
We select two inertial reference frames S and S’ where S’moves in respect to
S with a constant velocity v0 along the x –axis.
Assumptions (following from eperiments):
• t = t’
• measurements of length in both frames give
the same results (i=i’, j=j’, k=k’)
If for t=t’=0 the origins O and O’ coincide, then
according to the assumptions one obtains



or
r  r  v t i
0







i x  j y  k z  i x   j y   k z  v 0t i
From the above equation it folows that:
x  x   v 0t
y  y 


z  z 
t  t 
x   x  v 0t
y   y


z   z
t   t
Galileo’s
reverse
transformation (4.1)
transformation
(GT)
GT is a base of the classical relativity principle: fundamenal laws of physics
are the same in two reference frames for which Galileo’s transformation holds.
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Transformation of velocity


If position vectors r and r  are functions of time, then making use of GT and
diferentiating vs. time one obtains
dx dx 


 v0 
dt
dt

dy dy 



dt
dt


dz dz
or


dt
dt







v  v  v 0
(4.2)
v  v  v0
It can be then concluded that observers in different reference frames register
different velocities. The velociy has no absolute meaning.

v0
Transformation of acceleration
Taking the time derivative of Eq.(4.2), one obtains



d v d v' d v 0


dt
dt
dt

v
Because 0 is constant, the last term in above equation is zero and one gets


d v d v'

dt
dt


a  a
Observers on different frames register the same acceleration,
in other words acceleration is invariant vs. GT.
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The law of momentum conservation vs. GT
The law of momentum conservation in particular applies for collisions.


For the S frame one can write for two colliding particles with velocities v1 and v 2



m1 v1  m2 v 2  const
(4.3)
Making use of GT transformation for velocity




v1  v1  v 0

 
v2  v2  v0

one obtains the expession valid for reference frame S’






m1 v1  m1 v 0  m2 v 2  m2 v 0  const



or




m1 v1  m2 v 2  const  m1  m2 v 0
(4.4)
The right side of Eq.(4.4) is constant ( v 0  const ), hence the law of momentum
conservation is also valid in the moving frame S’.
Conclusion: The law of momentum conservation is invariant in all inertial frames
moving at constant velocities relatively to each other.

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