# Concept of Dynamics

About the Author : Mickheal Navani is a Young physicists, and holds a BSc in Physics from Green land University India. He writes for majority of science blogs. Follow him @ [email protected]

Preamble
Apart from  kinematics, dynamics
deals with motion and the force that causes motion.  Wikipedia  defines dynamics as a branch of applied mathematics (specifically classical mechanics) concerned with the study of forces and torques and their effect on motion, as opposed to kinematics, which studies the motion of objects without reference to its causes.  Thus,
dynamics is defined as the aspect of mechanics under which are we study about
the system of forces acting on a body that is in motion.
Sir Isaac Newton had at a time carried out an elaborate work on motion
and the force that caused motion.  Its
findings were summarized in what has now come to stay as the celebrated
Newton’s law of uniform motion.  Thus,
having a clear understanding of force and motion requires an adequate
understanding of these laws.

Inertia and the Newton First Law of
Uniform Motion
The first law has much to do about a quantity known as inertia.  When a body is at rest, it has its tendency
to continue to remain in the state of rest.
This tendency is inertia.  In the
same way, object in motion also has its tendency to continue its motion on a
straight line.  This is also inertia.
Using the above, the inertia of a body hence is defined as the tendency
of a body at rest to continue to be in union on a straight line.  The mass of a body is the measure of its
inertia ie the greater the mass of a body, the greater the force required to
change its states of rest or uniform motion.
The property of matter is called the inertia mass.
Newton first law of uniform motion is the summary of the above
explanation.  The law states as follows
every object will continue in its present state of rest or uniform motion in a straight line , unless it is been acted upon by an
external force.  The above provided the
force that  causes motion, just like the concept of heat and temperature
Newton Second law of Uniform Motion
While the first law admits that
force causes motion, the second law sets out to describe the force that causes
motion.  In doing this , a quantity known
as momentum is used.  The mass of a body
and its velocity ie
p= mass
x velocity
where p is the momentum
Note
however that momentum can change since the velocity that is used in defining
momentum can change as well.  Thus there
are
(I)
Initial momentum  (p)
(II)
Final momentum (p)
Where p1 = mu and P2
= mv, u and v are the initial and final velocity respectively.
If m is the mass in kilogram (kg) and
u and v are the velocities in m/s then momentum will be kgm/s. Note that the
fact that momentum and change in momentum is used in describing forces that cause motion.
This as well has been  explained by the Newton’s law of uniform motion which states as
follow:
F
α êp
t
Since êp    p2
– p1
(change
of momentum)
F α    p2 – p1 t
F α
mu – mv t
F α
m(v-u)  (since  v-u  =
acceleration) T
:.  F
α  ma
F =kma (k = I ie unit constant)
Thus, the above implies that
F = ma
Hence, by definition, the
force that caused the motion of an object is defined as the product of the
mass of the body and its acceleration.
Impulse and the momentum conservation principle
The impulse is a force that acts over a little
period of time ie
I  = Ft
I is the impulse, t is the
time and f is the force.  Using the
second law of uniform motion ie
F
êp
T
Which implies that
Ft
êp
The above consequently
implies that the impulse I also defined as the change in momentum (êp)  of a of a body.   Recall that
êp  =  p2 – p1
Which implies that
êp  =  mv
–  mu
êp  =  m(v-u)
êp  =  mêv
Again, the above implies that
impulse also define as the product of the mass of a body and its change in
velocity.  The unit of impulse is Ns
(Newton second)
Note  that
Ft  =  êp
Ft  =  p2-
p1
The above implies that
momentum can be conserved.   For this to
be realistic, the force f should be zero
(O).  This implies the momentum
conservation principle which states as follows:
If the total force acting on a system is
zero, then the vector sum of the momentum is constant.  Momentum is conserved in the following
systems
(i)
During collision
(ii)
During explosion
Collision
Collision is one of the system in which momentum is
constant.  This simply implies that the
sum of momentum before collision is the same as the sum of momentum after
collision.
Collision can however be
elastic, if not, it is inelastic.  For
elastic collision, apart from momentum being conserved kinetic energy is also
conserved.  For illustration, the
following diagram illustrate two bodies which are in collision.
m1u1                                              m2u2  Before collision
m1v1                                                m2v2  After collision
Then for conservation of momentum we write: m1u1
+  m2u2             m1v1 +  m2v2      ……………………….
(i)
and  for that of kinetic energy we
also write ½ m1u1
+  m2u2             m1v1 +  m2v2     ……………………..(ii)
Note that (i)  and (ii) are
simultaneously obtained only if the collision is elastic.
Example of elastic collision is the collision among the molecules of
ideal gases.
Collision is inelastic if kinetic energy is not conserved. For instance,
a hall dropped into a muddy ground, its collision is not elastic ie inelastic
Explosion
Apart from collision (elastic and inelastic), momentum is also conserved
during explosion.  For instance, when a
bullet is being fire from a gun.  During
explosion, though momentum is conserved but just like in inelastic collision,
kinetic energy is not conserved.
Kinetic energy is used in differentiating between inelastic collision and
explosion, though in the two, kinetic energy is not conserved.  During collision, the sum of the kinetic energy
after collision is less than sum of its kinetic energy before collision while
sum of the kinetic energy before explosion, is less than the sum of kinetic
energy after collision.
Example
A force of 15N is used to push a door for 5 seconds.  What is the impulse of the door?
Solution
t  –  ?
p  = 15N
t  =   5s
I
=  F x t
t
=  15  x 5
=  75N secs
Example
Calculate the momentum of a
body of mass 4 kg moving with a velocity 50ms-1
m =  4kg
v  =  50ms
p  =   m  x v
=   4 x 50
– 200kgm-1
Example
When taking a penalty kick, a
footballer applies a force of 30,0N for a period of 0,05s.  I f the mass of the ball is 0.075kg,
calculate the speed with which the ball moves off.
Solution
f   =  30N
t   =   0.05s
u   =  0
v   =  0.075kg
m
= 0.075kg
From  Newton’s
second law of motion
Ft   = m(v-u)
30
x 0.05 = 0.075(v – 0)
V  =  30  x 0,05
=20m/s
0.075
Example
A 1kg cement is pushed along
brick of a building by a force of 50N   A
fictional force of 15N opposes the motion.
Calculate the acceleration given to the cement.
Solution
Force  F
=  50N   – 15N
= 35N
Since F
=  ma
35  =
I  x a
A
35     =
35 ms-1

Application of Newton’s and conservation of Momentum
Laws
1.
Recoil of a Gun
Before a
gun is fired, the initial momentum of the gun and bullet is equal to zero. When
the gun is fired, the bullet moves forward with a certain velocity say vr.Therefore
its momentum is equal to m1v1, where m2 is the
mass of the bullet.  Suppose the mass of
the gum is m and also moving with velocity v2, from the law of
conservation of linear momentum.
m1v1  +   m2v2
m1v1  =   m2v2
Which
shows that their moment is in opposite direction ie the momentum is on opposite
direction ie the momentum of the gun opposes that of the bullet.  This is why the gun jerks backward or
recoils.  It should also be noted that
the mass of the gun is heavier than that of the bullet ie (m1>>m2)
which then makes both the kinetic energy and the velocity of the bullet higher,
Then that of the gun.
2.
Jet and Rocket Propulsion
Engines that are in Jet and rockets has a
combustion chamber in which the fuels (jet fuel in the case of jets) are
converted into gases to be burnt to provide the energy needed by the jets of
flight.  As those gases are expelled
downward at very high speeds from the aircraft, an equal but opposite momentum is
given to the aircraft to enable them to fly.
Summary
1.
Momentum (p) of a body is defined as the
product of its mass and its velocity (p=mv).
2.
Impulse is the product of the force and the
time during which the force acts (I = ft)
3.
Newton’s first law of motion states that a
body will continue to be in a state of rest or of uniform motion in a straight
line unless acted upon by an external force,
It is also called the law of inertia.
Newton’s second law of motion states
that the rate of change of momentum is proportioned to the force and takes
place in the direction of the force.
Newton’s third law states that action
and reaction are equal and opposite.
4.
Principle
of conservation of linear momentum stares that in a system of colluding
objects, the total momentum is always conserved provided there is not external
force acting on the system.
5.
There are two types of collision
(i)
Elastic
collision
(ii)
Inelastic collision
6.
Newton
law can be seen in
(i)
Recoil
of gun
(ii)
Jet and rocket propulsion
(iii)
Why walking is possible
7.
Mass is the quantity of matter in a body.   The
unit of mass in kg and that fo weight is the Newton (N)
References
1.https://en.wikipedia.org/wiki/Isaac_Newton
2.Sciencedirect: fundamental concept of aerodynamics
3. New school physics by John Nelson (Revised 2015)