In the chapters
explaining motion, the cause of motion have not been properly detailed. The
cause of motion is force. It is common knowledge that, a body is only at rest,
when no force is applied and a body is in motion when force is applied.
However, in both, force is acting, by keeping it at rest or setting it in
motion. And a body at rest, is said to be static or stationary. If a body is in
motion, it is said to be dynamic.
Sir Isaac Newton,
1642-1727, explained in detail, what force is by formulating three basic laws
of motion.
NEWTON’S FIRST LAW OF MOTION
The law states that,
a body will remain at rest or if it is in motion, will continue to move with
uniform speed along a straight line unless acted upon by a force.
The first law explains
the effect of force on a body in this way; if a body is at rest or in motion,
it remains in this state. This state only changes if external force is applied.
He then recognizes
that, as force is applied on the body at rest, it is reluctant to move. And as
force is applied on the body in motion to stop, it will be reluctant to
stop.
INERTIA
It is the reluctance
of a stationary body to move and the reluctance of a moving body to stop. Or,
the tendency to resists changes in the state of rest or uniform motion.
The First Law of
Motion expresses the idea of Inertia. It is otherwise known as Principle of
Inertia. That is, a body tends to keep doing what it is doing but only changes
its course when force is applied.
APPLICATION OF FIRST
LAW OF MOTION AND INERTIA
When a stationary
vehicle A takes off suddenly, probably by another vehicle B knocking its rear
end, the driver will jerk or fall back. It appears, he is reluctant to move
with the vehicle. The driver’s body is pushed forward by his seat, but his head
will remain still, in its state of rest, and it is jerked back in relation to
his body.
For this reason, neck
injuries are common in accidents where cars are hit from behind. To protect
drivers and passengers from injury, headrests are placed in cars.
Another consequence
of the first law of motion is, when a fast moving vehicle is suddenly brought
to rest by the application of the brakes, the passengers jerk or fall forward.
It appears, they are reluctant to stop as they continue to move in their
straight line of motion and unless there is little restraining force, those in
the front may hit the windscreen. Safety-belts are used to reduce this shock.
This law tells us
that, once an object is moving in a straight line with a constant velocity, it
will continue without any force applied. However, if an external force acts on
it, it will move faster or slower and probably change its direction. That is,
if air resistance and force of gravity could be eliminated, a body would go on
moving in a straight line for ever.
This is not so
because, the body would not move forever, but comes to rest after a time. It is not possible to eliminate the air
resistance, force of gravity or friction of earth, other planets and satellites.
It is only on space that these forces are absent and probably objects would
move in a straight line and constant speed.
For example, a rocket
or spacecraft is a space vehicle that travels on space into a destination,
probably Moon. It takes off from earth by firing it into space. Now, if it is made
to travel to the Moon, it would carry on in a straight line with steady speed.
On reaching the Moon, its direction and speed changes due to influence of the
Moon’s force of gravity and certainly ends its journey by landing on the Moon. The
rocket had traveled with constant speed in a straight line only on space.
On earth, when a
body or a tennis ball is thrown up into the air, its motion is opposed by air
resistance and earth’s gravity and its velocity is gradually reduced. At the
top, it is momentarily stationary. Sooner or later, it returns to the earth.
These explanations
mean that, Newton’s first law is both valid and not obeyed.
INERTIA AND MASS
Place two
rectangular blocks, one of metal and the other of wood on a smooth horizontal
table. Push them, at the same time with equal force using your hand. It is seen
that, the metal block can hardly be pushed and the wooden block can easily be
pushed. The metal block is more reluctant to move than the wooden block. Thus each
block has certain amount of inertia.
Mass is a measure of
the amount of inertia of a body.
Hence there is a
relationship between the reluctance of the blocks to move and their mass.
The metal block is more
reluctant to move than the wooden block because it is more massive or has more
mass or more inertia.
If an object changes
its direction or its velocity slightly when a big force acts on it, its inertial
mass is high. Masses are constant all over the world as there are measured
accurately by means of a chemical balance which gives a standard mass based on the International Prototype Kilogram gotten
from a particular block of metal kept in the National Bureau of Weight and
Measures in France and copies are kept in England.
WORKED EXAMPLE
1. The tendency of a body to remain at rest
when a force is applied to it is called. WAEC’03
2.
Imagine a place in the cosmos far from all
gravitational and frictional influences. Suppose that you visit that place
(just suppose) and throw a rock. The rock will
a. gradually stop.
b. continue in motion in the same direction at
constant speed.
MOMENTUM (p)
Besides inertia,
another effect which forces produce is momentum. When an object is moving, it
is said to have an amount of momentum given by its mass m and its velocity v.
The momentum of a
body is defined as the product of its mass and its velocity
Momentum = mv. ………………..i
The unit of momentum
is kgms-1.
Momentum is a vector
quantity. It has the same direction as the velocity of the body.
A runner of mass
50kg moving eastward with a velocity of 10ms-1 has a momentum of
500kgms-1 eastward.
Momentum simply
means, the quantity of motion that an object has.
Thus a bullet having
a small mass 0.01kg moving with high velocity of 1000ms-1 and a
heavy ball of mass 100kg moving with small speed of 1ms-1 has the
same momentum.
If the bullet and
the heavy ball are running at the same speed, say, 100ms-1, the
heavy ball has a greater momentum than the bullet.
Object at rest do
not have momentum, since its mass is not in motion hence momentum means ‘mass in
motion’
If the heavy ball is
at rest and the bullet is in motion, then, the heavy ball does not have
momentum while the bullet has.
The greater the
momentum of an object, the greater the force it will exert on the body it hits.
And the object is more deadly.
More powerful brakes
are required to stop a heavy lorry than a light car moving with the same speed.
NEWTON’S SECOND LAW OF MOTION
Newton’s second law
tells us what happens when an impressed or external force acts on a body at
rest or in uniform motion along a straight line.
Now, we have already
seen that an object of mass m moving
in a straight line with constant velocity, produce momentum. When a force acts
on it, it moves faster as well as change its direction and velocity.
Consequently, a momentum change occurs.
It follows that when
external force acts on a body, there is change in velocity which leads to
acceleration.
Therefore, Newton’s
second law explains the relation between force and acceleration from the change
in velocity; momentum change.
Newton’s
second law states that the rate of change of momentum of a body is directly proportional
to the applied force and takes place in the direction of the force. Or
Simply, force
is directly proportional to the rate of change of momentum produced. Or,
Force is
directly proportional to acceleration and inversely proportional to acceleration.
Force α
change in momentum
time
Suppose a force F
acts on a body of mass m moving along
a straight line with uniform velocity u
for a time t, the velocity changes
from u to v within the time interval.
Then the initial
momentum of the body is mu and its
final momentum mv. The change in
momentum is (mv – mu) or (m(v – u)), at the time
interval t.
F α mv – mu
t
F α
m(v – u)
t
but, the change in
velocity per second is the acceleration
a =
v – u (eqn of motion)
t
F α ma
i.e F = kma where k
is a constant.
The S.I. unit of force is Newton(N)
defined to make the constant k = 1.
If we take m = 1kg and a = 1ms-1
and F = 1N. Then the force of 1 newton produces an acceleration of 1ms-2
in a body of mass 1 kg. We have;
F
= ma …………………………..ii
This equation is a standard equation
of dynamics. When using the equation, the force F must be the resultant force
acting on the body.
Hence, Newton’s second law also means,
force is directly proportional to acceleration.
Also,
F = mv – mu or m(v – u) ………………iii
t
t
F = m
v ……………..………………….. iv
t
Comparison Formulae
From
F = ma. (ma)1 = (ma)2 ………..…… v
F = mv – mu .
mv
– mu
mv – mu ……………….vi
t t 1 t
F
= m v . m v m v ………………………………vii
t
t 1
t
WORKED PROBLEM
A force acts on a
body for 0.5s changing its momentum from 16.0kgms-1 to 21 kgms-1
, calculate the magnitude of the force. WAEC’03.
If a mass of 0.2kg is acted upon by a
force F which produces an acceleration a
of 4ms-2. What is the value of the force.
When taking a penalty kick, a
footballer applies a force of 30.0N for a period of 0.05s. if the mass of the
ball is 0.075kg. Calculate the speed with which the ball moves off.
A rope is being used to pull a mass of
10kg vertically upward. Determine the tension in the rope if, starting from
rest. The mass acquires a velocity of 4ms-1 in 8s. (g = 10ms-2).
IMPULSE OF A FORCE
Impulse is simply, the change in
momentum of a body.
It is an outgrowth or derivative of
the Newton’s second law.
Now,
F = mv – mu)
(from eqn. iii)
t
Multiply both
sides by t,
Then, Ft = mv – mu.
…………………… viii
The
quantity Ft (force x time) is known
as the impulse (I) of the force on
the object.
And by definition, impulse is change
in momentum.
Force
x time = change in momentum = impulse.
Ft = mv – mu =
I ……………………….ix
i.e I = Ft or mv – mu
The unit of impulse
is Ns or Kgms-1 since it is Force x time or change in momentum
Impulse is a vector quantity since
change in momentum is a vector quantity.
Impulse is mainly connected with
forces of short duration such as those arising from collisions and explosions.
WORKED EXAMPLE
1)
A force acting on a body causes a
change in the momentum of the body from 12kgms-1 to 16kgms-1
in 0.2 s. Calculate the magnitude of the impulse. WAEC’06.
2) A ball of mass 0.15kg is kicked against a
rigid vertical wall with a horizontal velocity of 50ms-1. If it
rebounced with a horizontal velocity of 30ms-1, calculate the
impulse of the ball on the wall.
DIFFERENCE BETWEEN
MASS AND WEIGHT
The tendency of a
body to remain at rest when a force is applied to it is called. WAEC’03.
A force acts on a
body for 0.5s changing its momentum from 16.0kgms-1 to 21 kgms-1
, calculate the magnitude of the force. WAEC’03.
The time rate of
change of momentum is? WAEC’01
Which of the
following statement about elastic collision is correct? WAEC’01.
Define linear
momentum.
State the law of
conservation of linear momentum.
A ball P of mass
0.25kg loses one-third of its velocity when it makes a head on collision with
an identical ball Q at rest. After the collision, Q moves off with a speed of
2ms-1 in the original direction of P. Calculate the initial velocity
of P.
State Newton’s
second law of motion.
Show that F=ma where
F is the magnitude of the force acting on a body of mass m to give it an
acceleration of magnitude a.
The engine of a
vehicle moves it forward with a force of 9600N against a resistive force of
2200N. If the mass of the vehicle is 3400kg, calculate the acceleration
produced.
An elastic collision
takes place between balls of known masses. Just before the collision, one of
the balls is moving with a known velocity while the other is stationary. Which
of the following physical quantities can be determined from the information
given. WAEC’00.
If the total force
acting on a particle is zero, the linear momentum will? JAMB’02
Jet-propelled
aircraft, Rocket propulsion, The recoil of a gun, A person walking, which of
the above is based on Newton’s third law of motion? JAMB’04.
A body of mass 12kg traveling at 4.2ms-1 collides with a second body of mass 18kg at
rest. Calculate their common velocity, if the two bodies coalesce after
collision. JAMB’08
