# 18.2 Conservation of energy

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## Mechanical energy

Mechanical energy is the sum of the gravitational potential energy and the kinetic energy.

Mechanical energy, ${E}_{M}$ , is simply the sum of gravitational potential energy ( ${E}_{P}$ ) and the kinetic energy ( ${E}_{K}$ ). Mechanical energy is defined as:

${E}_{M}={E}_{P}+{E}_{K}$
$\begin{array}{ccc}\hfill {E}_{M}& =& {E}_{P}+{E}_{K}\hfill \\ \hfill {E}_{M}& =& mgh+\frac{1}{2}m{v}^{2}\hfill \end{array}$
You may see mechanical energy written as $U$ . We will not use this notation in this book, but you should be aware that this notation is sometimes used.

## Conservation of mechanical energy

The Law of Conservation of Energy states:

Energy cannot be created or destroyed, but is merely changed from one form into another.

Conservation of Energy

The Law of Conservation of Energy: Energy cannot be created or destroyed, but is merely changed from one form into another.

So far we have looked at two types of energy: gravitational potential energy and kinetic energy. The sum of the gravitational potential energy and kinetic energy is called the mechanical energy. In a closed system, one where there are no external forces acting, the mechanical energy will remain constant. In other words, it will not change (become more or less). This is called the Law of Conservation of Mechanical Energy and it states:

The total amount of mechanical energy in a closed system remains constant.

Conservation of Mechanical Energy

Law of Conservation of Mechanical Energy: The total amount of mechanical energy in a closed system remains constant.

This means that potential energy can become kinetic energy, or vice versa, but energy cannot 'disappear'. The mechanical energy of an object moving in the Earth's gravitational field (or accelerating as a result of gravity) is constant or conserved, unless external forces, like air resistance, acts on the object.

We can now use the conservation of mechanical energy to calculate the velocity of a body in freefall and show that the velocity is independent of mass.

Show by using the law of conservation of energy that the velocity of a body in free fall is independent of its mass.

In problems involving the use of conservation of energy, the path taken by the object can be ignored. The only important quantities are the object's velocity (which gives its kinetic energy) and height above the reference point (which gives its gravitational potential energy).
In the absence of friction, mechanical energy is conserved and
${E}_{\mathrm{M before}}={E}_{\mathrm{M after}}$

In the presence of friction, mechanical energy is not conserved. The mechanical energy lost is equal to the work done against friction.

$\Delta {E}_{M}={E}_{\mathrm{M before}}-{E}_{\mathrm{M after}}=\mathrm{work done against friction}$

In general, mechanical energy is conserved in the absence of external forces. Examples of external forces are: applied forces, frictional forces and air resistance.

In the presence of internal forces like the force due to gravity or the force in a spring, mechanical energy is conserved.

The following simulation covers the law of conservation of energy.
run demo

## Using the law of conservation of energy

Mechanical energy is conserved (in the absence of friction). Therefore we can say that the sum of the ${E}_{P}$ and the ${E}_{K}$ anywhere during the motion must be equal to the sum of the ${E}_{P}$ and the ${E}_{K}$ anywhere else in the motion.

what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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preparation of nanomaterial
how did you get the value of 2000N.What calculations are needed to arrive at it
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The fundamental frequency of a sonometer wire streached by a load of relative density 's'are n¹ and n² when the load is in air and completly immersed in water respectively then the lation n²/na is
Properties of longitudinal waves