DYNAMICS:

NEWTON’S LAWS
OF MOTION

Dr. M.B. Adedokun

 INTRODUCTION
In the earlier modules, motion of objects have been described
mathematically – Kinematics. In this module we want to consider what
makes objects to move the way they do.
Why does it take a long distance to stop a ship once it is motion?
Why is it harder to control a car on wet ice than dry concrete?
The answer to these and similar questions is the subject of dynamics of
motion
Dynamics – the relationship of motion to the forces that cause it
To analyze the principles of dynamics we use the kinematic quantities
displacement, velocity, acceleration along with force and mass. Isaac
Newton (1642 -1727) summarized the principles of dynamics in three
statements known as Newton’s laws of motion.
 FORCE AND MASS
  Force - what causes motion and changes in motion
  A force can be a push or pull, it produces a velocity change of the
object on which it acts.
  The unit for measuring force is newton (N)
  Forces are broadly categorized as contact forces or non- contact
force. Examples of contact forces – frictional, tension, normal and
air resistance force.
  Examples of non-contact force – electric, magnetic and gravitational
force
  Mass - a measure of of a body’s resistance to acceleration or a change
in its state of motion when a net force is applied.
 CONCEPT OF INERTIA
  INERTIA is the natural tendency for of an object to maintain a state
of rest or to keep moving in uniform motion in a straight line once
it is set in motion.
  When you try to get ketchup out of a bottle by shaking the bottle,
the ketchup in the bottle moves forward, when you jerk the bottle
backwards, the ketchup tend to still move forward due to inertia.

  If pulled quickly, a tablecloth can be removed from underneath of

dishes. The dishes have the tendency to remain still as long as the
friction from the movement of the tablecloth is not too great.

  Newton related the concept of inertia to mass.

  Mass is a measure of the inertia of a body. Massive bodies have

more inertia- more resistance to a change in motion than a less
massive object does.
NEWTON’S FIRST LAW OF MOTION
  Newton's first law states that, if a body is at rest or moving at a
constant speed in a straight line, it will remain at rest or keep
moving in a straight line at constant speed unless it is acted upon
by an unbalanced force.

  If it is at rest , it remains at rest

  If in motions, it will remain at the same velocity
  This law expresses the concept of inertia
 CONCEPT OF MOMENTUM AND IMPULSE
  MOMENTUM P of a body in motion is defined as the product of the
mass m of the object and the its velocity v.

  P = mass(kg) x velocity(m/s) ------------------1

  unit of momentum P = kgm/s(kgms-1)

  Momentum is directly proportional to the object's mass and also its

velocity. Thus, the greater an object's mass or the greater its velocity,
the greater its momentum.

  IMPULSE I of a force is defined as the product of the force and time

with which the force acts

  Impulse I = force(N) x time(s) -----------------------2

  unit of impulse is Ns.

NEWTON’S SECOND LAW OF MOTION
  Newton’s second law of motion states that the time rate change of
momentum is directly proportional to the applied resultant force and
the momentum change takes place in the direction of the force
  Resultant force FRα change of momentum/time
  Which implies that :
  FR = k(mv −mv0)/t -----------------3
  where the proportionality constant k=1
  From second law, FR= d(mv)/dt ----------------4
  For a constant mass m, equation 4 becomes
  FR = m(dv/dt) -----------------------5
  dv/dt rate of change of velocity = acceleration a
  From equation 5, FR = ma. ---------------6
NEWTON’S SECOND LAW OF MOTION…
  From equation 6,
  FR/m = a ----------7
  Equation 7, shows that by Newton's
second law, the acceleration of an
object is directly related to the net
force and inversely related to its mass.
This implies that acceleration of an
object depends on two things, force Fig. 2
and mass.
  Figure 2: if you apply more force to a
body, it accelerates at a higher rate
  Figure 3: more mass (load) requires
more force to achieve the necessary
acceleration.
Fig. 3
 NEWTON’S SECOND LAW OF MOTION…
  One newton(1N) is the amount of net force that gives an acceleration of
1m/s2 to a body of mass 1kg.
  FR = ma
  but a= v − v0/t, FR = m(v − v0/t)
  Therefore,
  FRt = mv − mv0 ------------------ 8
  Equation 8 shows that the impulse I(FRt) of a force is equal to the change
in momentum(mv − mv0).
  In vector form, ΣFi vector sum of the i-component of all the forces acting
on the body of mass m.
ΣFi = Fxi + Fyj + Fyk
where Fxi, Fyj, Fyk components of the net force each equal to the mass
multiplied by the corresponding acceleration.
 NEWTON’S THIRD LAW OF MOTION
Newton’s third law states that to every action there is an equal and opposite reaction.

If an object A exerts a force (action) on object B, then object B must exert a force(reaction) of equal
magnitude and opposite direction back on object A.

Force exerted on body A by body B FAB = − FBA Force exerted on B by body A

This law represents a certain symmetry in nature: forces always occur in pairs.
  one body cannot exert a force on another without experiencing a force itself.

 
CONSERVATION OF LINEAR CONSERVATION
  Momentum P is the product of
the mass of an object multiplied
by its velocity and is equivalent
to the force required to bring the
object to a stop in a unit length
of time (Equation 8).
  The principle of conservation of
momentum states that during
  For any array of several objects,
collisions if the net external force the total momentum is the sum
acting on a system is zero, the sum of of the individual momenta.
the momentum before collision is   If the net external force acting on
equal to the sum of the momentum. a system of bodies is zero, then
  This principle is a direct the momentum of the system
consequence of Newton’s third remains constant that is, the total
law momentum before and after is
constant.
 THE LAW OF CONSERVATION OF LINEAR MOMENTUM
  Total initial momentum pi of a system of n bodies of masses m1,m2,m3..mn
moving with initial velocities u1,u2,u3..un respectively is given as
  During collision, some of the bodies may fuse together or break apart. The
system after collision produces m number of particles with a final momentum pf
given as
  Each body of mass m , moves with velocity vi
  The law of conservation of momentum implies that total initial momentum pi =
total final momentum pf
  m1u1 + m2u2 + m3u3 … mnun+ = m1v1 + m2v2 + m3v3 … + … mmum---------- 9
  If the motion is in three dimensions, -------- 9 becomes
  m1u1x + m2u2x + m3u3x … mnxunx+ = m1xv1x + m2xv2x + m3xv3x … + … mmxumx
  m1u1y + m2u2y + m3u3y … mnuny+ = m1v1y + m2v2y + m3v3y … + … mmumy
  m1u1z + m2u2z + m3u3z … mnunz+ = m1v1z + m2v2z + m3v3z … + … mmumz
 ELASTIC COLLISION
  Collision is any strong interaction between bodies that last a relatively
short time ranging from : car accidents, neutrons hitting atomic
nuclei to a close encounter of a spacecraft with the planet Saturn.

  If the forces between colliding bodies are much larger than any
external forces, we can neglect the external forces entirely and treat
the bodies as an isolated system. For such a system, the total
momentum before and after collision are the same; conserved.

  If the forces between the bodies are so that no mechanical energy is

lost or gained in the collision, the total kinetic energy of the system is
the same before and after collision. Such a collision is called elastic
collision, both momentum and kinetic energy are conserved.
  ΣPf = ΣPi
  Σ1/2miu2i = Σ1/2miv2i
 INELASTIC COLLISION
  A collision where the total kinetic energy after collision is less
than that before collision is called inelastic collision.
  ΣPf = ΣPi
  Σ1/2miu2i ≠Σ1/2miv2i
  An inelastic collision in which the bodies stick together and move
as one body after collision is called completely inelastic collision.
 FRICTION
  Whenever two bodies interact by direct contact of
their surfaces, the interaction forces are called
contact forces; frictional forces are contact forces.
  Friction can be defined as the force that resists
motion when the surface of one object comes in
contact with the surface of another; resistance
offered by the surfaces that are in contact with
each other when they move over each other.
Friction occurs in solid, liquid and gaseous
surfaces.
  Friction is part of our day to day activities.
FRICTION IN SOME OF OUR DAILY ACTIVITIES
  We apply frictional force as we walk; friction is what holds your shoe to the
ground. The friction present on the ice is very little, this is the reason why it is
hard to walk on ice.
  When we write a frictional force is created when the tip of the pen comes in
contact with the surface of the paper.
  When the head of the matchstick is rubbed against a rough surface, heat is
generated and this heat converts red phosphorous to white phosphorous. White
phosphorous is highly inflammable and the match stick ignites., matchsticks fail
to ignite due to the presence of water. Water lowers friction.
  Friction is the force that opposes the tyre rubber from sliding on the road surface.
This friction avoids skidding of vehicles while driving .
  Friction is applied in brakes to stop a moving vehicle. kinetic energy is converted
to thermal energy by applying friction to the moving parts of a vehicle. The
friction force resists the motion and in turn, generates heat. This conversion of
energy eventually bringing the velocity to zero.
 STATIC AND KINETIC FRICTION

•  Between solid surfaces two kinds of friction occur: static and kinetic (sliding)
friction.
•  Consider a movable object resting on a table which is stationary (fig. A). If a
small amount of force Fapp is applied horizontally to the object and the
object does not move, acceleration of the object is zero, therefore the net
force on the object = 0.
•  Fn − Fg = 0 ; Fapp − Fs = 0. The applied force (Fapp) = the static frictional force
(Fs).
•  Static friction is the friction between two or more solid objects that are not
moving relative to each other.
 STATIC FRICTION
  If the applied horizontal force is increased such that Fapp is large
enough to overcome the static friction, the object then tend to
move. The value of the horizontal force when the sudden motion
occurs is the maximum force of static friction or limiting frictional force
(Fsmax) for the two surfaces in contact, when they are at relative rest.

  For all types of surfaces in contact , which are not in relative motion, is
proportional to the normal force N.

Fsmax αN, Fsmax= μs N

  where the constant μs is known as the coefficient of static friction.

  The magnitude of μs depends on the nature of the surfaces but is

independent of the area in contact, provided that the normal force
is constant.
 KINETIC FRICTION
  When Fapp the applied horizontal
force is greater than the maximum
static frictional force, the object
begins to slide over the surface.

  Once the object is sliding, there is

a force Fk acting on the object in
the opposite direction to the
direction of motion. Fk is the
kinetic frictional force with μk is independent of the
magnitude Fk = μk N. μk is the relative velocity of the
coefficient of kinetic friction. surfaces.
  Kinetic or dynamic frictional force is μk is independent of the
the opposing force between two area of the surfaces in
surfaces already in relative. To keep
two surfaces moving with respect to contact, provided that the
another Fk < Fsmax this implies that normal force FN remains
μk < μs. unchanged.
 Example
  1What is the minimum vertical force needed to move the block of wood
with mass 1kg clamped to a vertical board with a force of 10N upward
(acceleration due to gravity is 10 m/s2 and μsis 0.15)

  Solution

  Friction here is pointing downward because its fighting against the

applied upward force, before movement begins to move upward, the
resultant force equals 0

  Vertical force = F, Fs = frictional force, mg = weight of the block

  FR =F − Fs − mg = 0

  F = Fs + mg

  = Fs= μs N

  F =1.5N + 10N= 11.5N