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Preparatory Program in Basic
Science(PYSC001)
PART (I)
PYSICS(2)
Coordinator:
Prof.Dr.Hassan A.Mohammed
UNIT II :
MOMENTUM & ENERGY
3 - MOMENTUM & IMPULSE
3.1 Momentum :Impetus(im + petus):in motion
Momentum is a vector quantity that reflects an object’s
ability to do work or cause damage. This ability is directly
proportional to both the object’s mass and velocity.
Therefore, momentum is defined as:
Momentum = mass × velocity,
or,
p=m×v
The SI unit of momentum is: kg.m/s.
Any object at rest has zero momentum.
On the other hand, a moving object with either a large
mass or large velocity has a large momentum.
Examples: a supertanker (large mass), and a bullet
(large velocity).
3.2 Impulse:
3.2.1 Definition
A force can be applied to an object only for
a limited duration (∆t). The longer that ∆t
is, the greater that the effect will be on the
object (causing a change in its velocity and
momentum).
Therefore, we introduce impulse as the product of the applied force
and the time interval during which the force acts:
Impulse = force on object × time interval; or
The adjacent graph shows the force’s behavior.
From the above definition of impulse, impulse in
this graph should be the area under the force
curve. If we can determine the average impact
force (Favg), impulse would also be the area of the
rectangle whose width is ∆t and height is Favg.
3.2.2 Relationship of Impulse to Momentum:
This means that the impulse on an object equals the
change in its momentum.
3.2.3 Important notes :
1. Whenever we exert a force on an object, we also exert an impulse
on it.
2. When the force is not constant, we take its average to calculate the
impulse.
3. We may say that an object has momentum, but may not say that it
has impulse.
4. We may say that an impulse on an object causes a change in its
momentum. Alternatively, we may say that a change in an object’s
momentum causes an impulse.
5. Bouncing results in a change of direction of v and p
producing larger impulse and force
a) If we drop two balls of equal mass, one made of playdough and the other of highly elastic rubber, the first ball
stops upon hitting the ground, producing an impulse
∆p = mv – 0 = mv. The second ball reverses direction,
producing an impulse ∆p = mv – (– mv) = 2mv.
b) Following the same principle,
a karate expert breaks a stack
of bricks by bouncing his hand off it.
3.2.4 Car-Crash Example
The figure shows a car whose driver lost control of the
brakes while going at a constant speed, v.
To stop the car, the driver must choose between
three options:
(a) Driving into a haystack,
(b) Driving into a brick wall, or
(c) Bouncing off a concrete wall.
All options will cause the same change in momentum:
I = ∆p = mv - 0 = mv.
However, as is indicated in the figure, there are important
differences between these options, as follows:
a) The haystack option extends the impact time, which
greatly decreases the impact force and reduces harm and
damage.
b) The brick-wall option shortens the impact time, which
results in a very high impact force that would cause great
harm and damage.
c) The concrete-wall option is similar to the previous one,
but with the added effect of bouncing. This doubles the
change in momentum and, consequently, the impact force.
3.3 Conservation of Momentum
3.3.1 Derivation
From Newton’s 3rd law, we learned that the interaction forces between
two objects (A and B) are given
By:
we can conclude from this that:
The impulse from A on B
cause a change in the momentum of
object),B
and vice versa. Thus, we have:
Hence, the momentum of the A-B system is the same
before and after the interaction.
This is the law of “conservation of momentum :
Momentum is never gained or lost in an interaction.
3.3.2 Collisions
3.3.2.1 Definition
Collision is a special kind of interaction in which the
interacting objects come in contact with each other so as
to exchange momentum and energy.
A collision is distinguished by an impact that separates
between what happens before and after the collision
3.3.2.2 Types
There are two types of collisions:
a. Elastic Collisions in which the
colliding objects are not permanently
deformed and do not generate heat.
Example:collision between the
billiard balls in the figure.
b. Inelastic Collisions in which the
colliding objects become distorted and
generate heat. This is usually
associated with tangling, sticking, or
coupling between the colliding
objects. Example: collision between
the two cars in the figure.
3.3.2.3 Discussion
1. In a collision between two objects, momentum is exchanged. This
exchange depends on the details of the collision, such as the velocity of
the colliding objects, and whether the collision is elastic or not.
2. As was discussed earlier, momentum is conserved in a collision,
which means that: before after.
This is true for both elastic and inelastic collisions.
3. At impact, the impulses of the two colliding objects are equal and
Opposite:
which means that the momentum gained by one object equals the momentum
lost by the other.
3.3.3 Simple Examples
3.3.3.1 Zero Initial Speeds
From momentum conservation, the total final
momentum must be zero. This can only happen if the
two objects move away along the same line(linear
motion). Thus, we only need to consider scalar
momenta and speeds. We have:
As a specific example, consider a cannon
of mass mc firing a cannonball of mass mb at
a speed vb. To calculate the cannon’s recoil
speed, vc, we substitute in the above
equation:
3.3.3.2 Zero Final Speeds
From momentum conservation,
the total initial momentum must be zero.
We have:
3.3.3.3 Worked Exercise
A 2-ton car going 40
km/h is hit at the rear
by a 5-ton truck going
50 km/h.
4- ENERGY, POWER & SIMPLE MACHINES:
4.1 Work, Energy, Power
4.1.1 Work:
4.1.1.1 Definition
Work is the exertion of force through a distance.
Work is a scalar quantity that is directly proportional
to the applied force and to the distance the object
moves because of the force.
Thus, we say:
Work = force × distance, or:
The SI unit for work is the joule, defined as:
[J ≡N.m].
4.1.1.2 General Equation for Work
The above equation is only true for linear motion: when
the applied force and the object’s displacement are along
the same line (θ = 0).
If, on the other hand, the force applied to an object makes
an angle θ with the object’s displacement, a more general
equation for work is:
4.1.2 Energy
4.1.2.1 Definition
Energy is the capacity of an object to do work.
Example:
Muscles (energy source) enable creatures to move (work).
On the other hand, doing work often results in stored energy.
Example:
Digesting food (work) produces (energy) for the body. Thus,
we say that work and energy are interchangeable.
Since energy and work are interchangeable, we use for
energy measurements the same unit as for work, the
joule[J].
4.1.2.2 Different Forms of Energy:
Energy takes many forms: mechanical, chemical, electric,
nuclear, etc. It is stored in plants, foods, batteries, and fuels.
Specific examples:
1. Waves: All waves carry energy. Light and sound are
two examples. Both light and sound waves carry energy
that depends on the wave’s frequency and intensity.
2. Heat: Heat usually results from burning a substance. The
amount of heat generated depends on the temperature,
type, and amount of the substance.
4.1.2.3 Observing and Using Energy:
Examples:
1. A man can do work by exerting a force through a
distance; but this requires food, so he converts the
energy in food into work.
2. We burn coal to generate heat that can be converted
into electricity and then into many modern forms of work:
4.1.2.4 Conservation of Energy (and matter)
A fundamental law of nature (as decreed by Allah) is:
Energy is conserved; it neither
increases nor decreases.
Thus, energy can only convert from one form to another.
For a closed system, this law can be summarized as:
4.1.2.5 Common Energy Units:
1.A calorie is defined as the energy needed to raise the
temperature of one gram of water by 1 ºC at normal
temperature (18 ºC) and pressure (1 atmosphere).
A calorie is related to the joule as:
[1 cal = 4.184 J ≈4.2 J].
Although the calorie is not an SI unit, the SI permits using
it in heat applications.
2. In food products, the energy available in a food item upon
digestion is commonly expressed is the kilocalorie
(sometimes written as Calorie, with a capital C), where
[1 Cal = 1kcal = 1000 cal].
3. A common unit of energy is the BTU (British thermal unit).
1 BTU is the energy needed to raise the temperature of 1 lb
of water by 1 ºF (Fahrenheit).
A BTU relates to the joule as: [1 BTU = 1.054 kJ].
4. A common unit of energy is the kilowatt-hour (kWh), which
is the standard unit of electricity consumption. It relates to
the joule as:
( 1watt = 1J/s)
1 kWh = 103 J/s × 3600 s = 3.6×106 J or 3.6 MJ.
Larger businesses and institutions sometimes use the
megawatt-hour [MWh]. The energy outputs of large power
plants over long periods of time, or the energy consumption
of nations, can be expressed in gigawatt-hours [GWh].
4.1.3 Power:
4.1.3.1 Definition:
is the rate of change of work or energy. In its
simplest form, it is defined as:
Like work, power is a scalar quantity. The SI unit for
power is the Watt, defined as
4.1.3.2 Common Power Units:
1. A unit of power, commonly used in regard to motors, is
the horsepower, which is defined as:
1 hp = 746 W ≈ 0.75 kW.
2. Another power unit, commonly used to describe an air
conditioner’s cooling capacity, is the British Thermal Unit
per hour, [BTU/h], where:
1 BTU/h = 0.293 W.
4.2 Mechanical Energy:
Mechanical energy is the energy that arises from an object’s
position or velocity, and is the sum of potential and kinetic
energies.
4.2.1 Potential Energy
4.2.1.1 Definition:
Potential energy Ep : is the energy that an object has
because of its position under the influence of a certain
force
The work done on an object to change its position must equal the
change in potential energy (by the law of conservation of energy).
Therefore:
4.2.1.2 Forms of Potential
Energy
Potential energy can be mechanical or
non-mechanical.
Examples of mechanical potential
energy:
1. Water trapped behind a dam has
gravitational potential energy because of
its height above the base of the dam.
We can use this energy to run a
hydroelectric station
2. A drawn bow has elastic potential
energy stored in the bow and string.
3. A compressed or expanded spring has
elastic potential energy.
Examples of non-mechanical potential energy:
1. An electric circuit has electric potential energy stored in the
battery or voltage source.
2. At the molecular level, chemical potential energy is stored in the
relative position of atoms in molecules
3. There is potential energy in food, arising from molecular binding
4. Potential energy is stored in fuels, such as coal and natural gas.
5. Nuclear potential energy is stored in the atom’s nucleus.
4.2.1.3 Gravitational Potential Energy
To raise an object of mass (m) to height
(h) requires
work = force × distance = m·g·h.
Once at that height, the object will
possess a potential energy equal to the
work done to place it at that location:
4.2.1.4 Important Notes
1. Potential energy is relative, which means that its value depends on
the point of reference.
Example: If we raise a 1-kg book 1 m above a table that is 1 m above
ground, the books potential energy relative to the table is 10 J, and is
20 J relative to the ground.
2. Potential energy can be positive or negative.
Example:
Consider the book in the previous example,
and assume that the ground-ceiling (+J)
distance is 3 m. This means that
when we place the book 1 m above the
table’s surface, its potential energy relative to
the ceiling is -10 J. Thus, instead of doing
work on the book to bring it down a distance of 1 m, the book will do
the work (losing potential energy).
3. An object’s potential energy
at a certain location is the same
regardless of how the object
reaches that location. In the
adjacent sketch, a man of mass
(m) climbs to certain height (h)
using three different routes.
Regardless of the route,
his potential energy at h will be the same in all three cases.
4. A more general form of gravitational potential energy between two
objects of masses (m1) and (m2), separated by a distance (d),
is given by:
For Earth, ME = 5.98 × 1024 kg, and RE = 6.37 × 106 m. Thus, we can use the
above equation to calculate the value of g.
4.2.2 Kinetic Energy:
4.2.2.1 Definition:
An object’s energy of motion, called kinetic energy,
Ek depends on the object’s mass and speed.
Your moving car has kinetic energy that keeps it going.
Blowing wind has kinetic energy that can be converted
into electric energy by means of windmills.
Work is needed to change an object’s speed, and (by
the law of conservation of energy ) this work must
equal the kinetic energy gained
by the object:
4.2.2.2 Derivation:
An object of mass m is subjected to a force (F) over a
distance d during a time interval (t). It has acceleration
(a), initial speed (vi), and final speed (vf).
The work done on it to reach this speed must equal the
kinetic energy it gained.
4.2.2.3 Important Notes
1. Kinetic energy must always be either zero or positive.
It cannot be negative.
2. Work can convert to potential energy which is then
converted to kinetic energy, or vice versa.
3. Kinetic energy is directly proportional to mass. If a
truck goes as fast as a car of half its mass, the truck’s
kinetic energy is double the car’s.
4. Kinetic energy is proportional to the square of the
speed, which means that if you double your car’s
speed, you quadruple its kinetic energy. Since energy is
the capacity to do work, for either good or bad,
uncontrolled kinetic energy can be very dangerous.
Furthermore, doubling the speed substantially increases
fuel consumption
4.2.3 Example of Mechanical Energy
4.2.3.1 Pile Driver
If we let an object (like the pile driver
fall, it starts gaining speed and losing
height. This means that it gains kinetic
energy and loses potential energy. By
the conservation law, the potential
energy it loses should equal the kinetic
energy it gains. Thus, if it falls a
distance (h) and gains speed (v), we
have:
4.2.3.2 Throwing a
Ball
If we throw a ball up with a
speed vi, its speed becomes
zero at the highest point it
reaches, h.
Using the same equations as
above, we find that:
h = vi2 / 2g or vi =√ 2gh
Example: The maximum
height reached by a ball that is
thrown up with vi = 30 m/s is: h
= (900 m2 / s2)/ 2 ×10 m/s2
= 45 m.
4.2.3.3 Bicycle up a Hill
Assume that the bicycle in the
figure is traveling over
frictionless ground. At the
bottom of the hill, all of its
mechanical energy is kinetic.
As it starts rising up the hill, it
loses kinetic energy and gains
potential energy (with the total
energy remaining constant).
Its potential energy is
maximum at the peak, after
which the bicycle starts
regaining kinetic energy and
losing potential energy.
4.2.3.4 Pendulum
An example similar to the previous
one is the pendulum. Assume that
we have a frictionless pendulum,
with bob of mass (m) and string of
length (l).
At point (3), the pendulum’s total
energy is all potential, whereas it is
all kinetic at point (1). At any
intermediate point (such as point
(2)), the energy is a mixture of
kinetic and potential energies.
From the values in the figure, we
have:
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