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 (F avg ), impulse would also be the area of the rectangle whose width is ∆t and height is F avg .

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 play dough 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 m c firing a cannonball of mass m b at a speed v b . To calculate the cannon ’s recoil speed, v c , 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

an

angle θ

with the object ’s applied to an object makes

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 = 10 3 J/s × 3600 s = 3.6×10 6 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

h orse p ower,

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 because of its position energy that an object has 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 (m 1 ) and (m 2 ), separated by a distance (d), is given by: For Earth, M E = 5.98 × 10 24 kg, and R E = 6.37 above equation to calculate the value of g.

× 10 6 m. Thus, we can use the

4.2.2 Kinetic Energy: 4.2.2.1 Definition: An object ’s energy of motion, called kinetic energy, E k

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 (v i ), and final speed (v f ). 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

It cannot be negative

.

or

positive

. 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 = v i 2 / 2g or v i = √ 2gh

Example:

The maximum height reached by a ball that is thrown up with v i = (900 m 2 = 30 m/s is: h / s 2 )/ 2 ×10 m/s 2 = 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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