potential energy and mechanical energy
conservation
speech transcript
Liceo Scientifico Isaac newton – Roma
potential energy and
mechanical energy conservation
in accordo con il
Ministero dell’Istruzione, Università, Ricerca
e sulla base delle
Politiche Linguistiche della Commissione Europea
percorso formativo a carattere
tematico-linguistico-didattico-metodologico
scuola secondaria di secondo grado
professor
Serenella Iacino
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potential energy and mechanical energy
conservation
Potential Energy and Mechanical Energy Conservation
In this video the basic points of mechanical energy conservation will be illustrated.
A stationary object, under specific conditions, has energy and has the ability to do
work.
For example, a book on a shelf has energy, which will be shown in the form of
kinetic energy if it falls down.
Forms of energy, stored in an object, are called Potential Energies of which we
have two types defined by the force to which they are linked:
1. Gravitational potential energy that is the work done by the gravity on an object;
2. Elastic potential energy that is the energy of a stretched or compressed spring
and it represents the work done by an elastic force to return the spring to its
initial position.
3. An object of mass m, at a height h above the ground, has gravitational potential
energy equal to the work which would be done by gravity to move the object
from the height h to the ground.
In general, the work done by gravity to move an object of mass m, from a height
hA above the ground, to a height hB above the ground, is equal to the difference
between Gravitational Potential Energy in the position A and B.
A man dives into the sea from the cliffs of Acapulco, from a heigth (h) of 40 m; his
potential energy dicreases by 25000 j.
As you can see graphically. How heavy is the man? This is the weight.
We can observe that the gravitational potential energy depends only on the height
h and doesn’ t depend on the chosen path to go from the initial position to the final
position.
For example, we can calculate the work done by gravity to move an object from the
point A to the point B through two different routes.
If AB measures h and the plane is inclined at an angle teta , the work done by
gravity through vertical route AB is…
The work done by gravity through the route ACB is the sum of the work through AC
and the work through CB.
So, the two pieces of work are equal and the work done by gravity through the
closed route ( ACBA ) is zero: in fact the work done by gravity through the route AC
is equal to mgh, through the route CB is equal to zero and through the route BA is
equal to ( - mgh ).
Thus we have the concept of the conservative force: a conservative force does work
which is independent of the route taken and only depends on the initial position A
and final position B.
Therefore gravity is a conservative force.
An example of closed route is given by a Roller Coaster: in this ride, the wagon
goes up and down and then returns to its initial position.
Ignoring friction, the force of gravity and the normal are the only forces acting upon
the wagon.
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potential energy and mechanical energy
conservation
But the normal N does work equal to zero because it’s perpendicular to the
displacement s, so only gravity does work.
When the wagon goes up , the angle is obtuse and the work is negative and the
kinetic energy decreases:
When the wagon goes down, the angle between the gravity and the displacement is
acute so the work is positive and the kinetic energy increases.
If the wagon arrives at a position A, which is height h above the ground, with
velocity VA, its potential energy is mgh, and from the work – energy theorem, we
have:….
So, if the wagon arrives at a position B , which is also h above the ground, with
velocity VB, its potential energy is again mgh, and from the work – energy
theorem, we have this formula.
Therefore we observe that vb=va ; when the wagon returns to the starting point, it
will have the same initial velocity, and so the variation of kinetic energy is equal to
zero.
From which the work of gravity, through a closed route, is equal to zero.
We know that the elastic potential energy of a compressed spring is this one,
which represents the work done by the elastic force to pull the spring back to its
original length.
We can observe that the work depends only on the compression x and so on the
initial and final positions of the spring, therefore the elastic force is a
conservative force.
However not all forces are conservative.
A force is called “Non conservative” if its work depends on the chosen route,
between the initial position and the final position.
Friction is an example of a non conservative force; in fact it always opposes
motion and does negative work that is not stored as potential energy but is
transformed into sound or thermal energy.
Therefore, the work done by friction through a closed route is not zero; for example
we can calculate this work through a square route of side s, as we can see in the
picture.
In Physics the Mechanical Energy of an object is the sum of its potential and
kinetic energies:
It is conserved only in the systems where only conservative forces are involved, in
fact if in a system we don’ t have conservative forces, the mechanical energy can
change as in the case of the friction force in which the mechanical work is
transformed into sound or thermal energy.
To demonstrate this, we must remember the work – energy theorem for which we
have:
The work done by a conservative force is equal to the difference in potential energy
between the initial and the final position:
Equating the two expressions, we have…
So we can conclude the initial and final energies are equal .
If there is no friction, a ROLLER COASTER is a demonstration of Energy
Conservation.
When the wagon goes up , the velocity decreases and the kinetic energy is
transformed into gravitational potential energy.
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potential energy and mechanical energy
conservation
At the highest point of the track, the wagon has the highest gravitational potential
energy.
When the wagon goes down, its velocity increases and the potential energy is
transformed into kinetic energy, so the total mechanical energy remains constant.
In the same way ignoring friction, mechanical energy is conserved also when an
object is approaching a spring at high velocity.
When the object compresses the spring, its kinetic energy decreases and is
transformed into elastic potential energy.
When the motion is reversed, the potential energy decreases while the kinetic
energy increases and when the object leaves the spring , the kinetic energy returns
to its initial value.
Let’ s make an example with the ball in a pinball machine.
To fire the ball of mass m, suppose we compresse the spring, having a constant
equal to K, by length x. Ignoring friction, we want to know what is the launch
velocity of the ball. We observe that gravity and the normal act upon the system
but don’ t do work because they are perpendicular to the displacement ; so the
elastic potential energy is transformed into kinetic energy of the spring.
From energy conservation, we have this equation from which the velocity is this
one.
Let’ s give another example.
In a water park, two children enter the swimming – pool using two different slides,
both without friction; Two slides have the same height h, the first one with a
regular slope, the second one very steep at first, then becomes horizontal.
If v1 is the velocity at the end of the first slide, and v2 is the velocity at the
end of the second slide, according to energy conservation , we have……
So as you can see v1 is equal to v2.
In reality, conservative and non conservative forces act upon an object, so the total
work is the sum of the work done by the conservative forces and the work done by
the non conservative forces.
Now, from the work – energy theorem , the total work is equal to the change in
kinetic energy from which , equateing the two expressions, we have this …………..
But the work done by a conservative force is equal to the difference in potential
energy.
So, substituting, we have this formula.
This relationship demonstrates that the work done by a non conservative force is
equal to the change of mechanical energy. So, the law of energy conservation
is no longer valid.
Materiale sviluppato da eniscuola nell’ambito del protocollo d’intesa con il MIUR
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