1

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.

The reference frame in which the first law holds is called
an

inertial frame.

0

0









a

then

,

F

If

r

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

) .

B

A

r

F

F

F











2

3.2. Newton’s Second Law

The relation between the net force F

r

applied on an

object, its mass m and the resulting acceleration a is given
by Newton’s second law

(3.1)

In the case when mass m varies, the more general expression
for the force is used

(3.1a)

The net force acting on a particle is equal to the
product of the particle mass and its acceleration
(for constant mass).

const

m

for

a

m

F

r









v

m

p

where

dt

p

d

F

r













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.

3

Newton’s second law, cont.

Eq. (3.1a) transforms into (3.1) for a constant mass m

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

.

Example: forces acting on a body on the ramp

weight

Q = mg

reaction (normal) force N
frictional force F ≤ μ N

equation of motion:

a

m

dt

v

d

m

dt

dm

v

dt

v

d

m

dt

)

v

m

(

d

F

r























 



























t

,

dt

r

d

,

r

F

dt

t

r

d

m

2

2

)

t

(

r



a

m

F

Q

N















4

3.3. Newton’s Third Law

This can be written as the vector relation

(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.

When two bodies interact by exerting
forces on each other, the forces are equal
in magnitude and opposite in direction.









21

12

F

F

5

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

R

z

– Earth’s radius

T = 24 hrs

The circular motion around the Sun is a couse of another
acceleration

2

2

2

2

2

1

4

3

4

s

cm

,

R

T

R

R

v

a

z

z

z

















2

13

2

7

2

2

6

0

10

5

1

10

3

4

s

cm

,

cm

,

s

a













6

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 results in
acceleration

(3.3)

In the noninertial frame moving with acceleration
vs. the inertial frame this accelaration is equal

Hence

Introducing above into (3.3) one gets in the inertial frame

(3.4)









i

a

m

F

i

a



F



0

a

a

a

i











0

a

a

a

i











0

a













 









0

a

a

m

F

7

Inertial forces, cont.

Eq. (3.4) can be transformed as follows

(3.5)

where

is

the inertial force.

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 .

The centripetal acceleration of the reference frame
is equal , where ω – angular velocity,
ρ – radius of the circle.
In this case in the rotating frame where the particle
is at rest one obtains

,

where the centrifugal force is given by .

0

a

m

F

a

m











0

F

F

a

m











0

0

a

m

F









0

F











2

0





a

0

0







F

R

Q

















2

0

0

m

a

m

F







8

4. Galileo’s Transformation

We select two inertial reference frames S and S’ where
S’moves in respect to
S with a constant velocity v

0

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

From the above equation it folows

that:

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.













i

t

v

r

r

0

































i

t

v

z

k

y

j

x

i

z

k

y

j

x

i

0































t

t

z

z

y

y

t

v

x

x

0































t

t

z

z

y

y

t

v

x

x

0

9

If position vectors and are functions of time, then making
use of TG and
diferentiating vs. time one obtains

(4.2)

or

It can be then concluded that observers in different reference
frames register different velocities. The velociy has no absolute
meaning.



r





r



































dt

z

d

dt

dz

dt

y

d

dt

dy

v

dt

x

d

dt

dx

0

























0

0

v

v

v

v

v

v



0

v

Transformation of velocity

Taking the time derivative of Eq.(4.2), one obtains

Because

is constant, the last term in above equation

is zero and one gets

Observers on different frames register the same

acceleration,

in other words acceleration is invariant vs. GT.

Transformation of acceleration

dt

v

d

dt

'

v

d

dt

v

d











0

0

v

















a

a

dt

'

v

d

dt

v

d

10

The law of momentum conservation

in particular applies for

collisions.
For the S frame one can write for two colliding particles with
velocities and

(4.3)

Making use of GT transformation for velocity

one obtains the expession valid for reference frame S’

or

(4.4)

The right side of Eq.(4.4) is constant ( ), 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.

The law of momentum conservation vs. GT











const

v

m

v

m

2

2

1

1

























0

2

2

0

1

1

v

v

v

v

v

v

1

v



2

v

























const

v

m

v

m

v

m

v

m

0

2

2

2

0

1

1

1

























0

2

1

2

2

1

1

v

m

m

const

v

m

v

m

const

v 



0