## Conservation of energy

#### Example:

Consider an isolated, conservative system.  The system can have kinetic energy K and potential energy U.  For an isolated, conservative system, i.e. an isolated system only acted on by internal, conservative forces, the sum of the kinetic energy and potential energy is constant.

Ki + Ui = Kf + Uf.

We define

E = K + U,

where E = mechanical energy.  The mechanical energy of an isolated conservative system is conserved.  Conservation of mechanical energy is a powerful tool for solving physics problems.

#### Problem:

From a 50 m high platform a 0.1 kg stone is thrown straight upward with initial speed 5 m/s.  What is it speed 10 m above the ground?

Solution:
This problem can be solved using the kinematic equations yf = yi  + vyit - (1/2)gt2, vy = vyi - gt.  But it can also be solved using conservation of mechanical energy.
Ki + Ui = Kf + Uf.
Uf - Ui = Ki + Kf.
mg(yf - yi) = (1/2)m(vi2 - vf2).
vf2 = vi2 + 2g(yi-yf) = 25 (m/s)2 + 2(9.8 m/s2)(40 m) = 809 (m/s)2
vf = 28.4 m/s.

#### Problem:

A 6*105 kg subway train is brought to a stop from a speed of 0.5 m/s in 0.4 m by a large spring bumper at the end of its track.  What is the force constant k of the spring?

Solution:
All the initial kinetic energy of the subway car is converted into elastic potential energy.
½mvi2 = ½kx2.  Here x is the distance the spring bumper is compressed, x = 0.4 m.
k = mvi2/x2 = (6*105 kg)(0.5 m/s)2/(0.4 m)2 = 9.375*105 kg m/s2 = 9.375*105 N.

If a system of objects is isolated from its environment and does not interact with its environment at all, then certain physical quantities of the system cannot change.  They are constant or conserved.  Conservation laws are considered to be the most fundamental principles of physics.

Energy
is conserved for an isolated system.  There are, however, no truly isolated systems.  Interactions with the environment can be only approximately excluded.  So while conservation laws are very useful in analyzing many everyday situations, they strictly hold only for the universe as a whole.

Energy conservation for an isolated system is a fundamental principle of physics.  Energy for an isolated system is always conserved.  It may change forms, but the total amount of energy in an isolated system is constant.  Energy can, however, be converted from one form to another form.  Work is the conversion of one form of energy into another.  Energy comes in different forms, kinetic energy, potential energy, chemical energy, thermal energy, etc.  If an object has energy, it has the potential to do work.

There are several forms of potential energy.  Kinetic and potential energy are called mechanical energy or ordered energy.  Thermal energy is disordered energy.  Friction converts mechanical energy into disordered energy.

When no disordered energy is produced, then mechanical energy is conserved.

Kinetic energy is the energy of moving objects.
Examples are:

• the ordered kinetic energy of moving macroscopic objects
• disordered kinetic energy or thermal energy
• electromagnetic energy and radiant energy

Potential Energy is stored energy which can be converted into kinetic energy.
Examples are:

• Gravitational potential energy
• Electrostatic potential energy
• Stored Mechanical energy (springs, rubber bands, etc)
• Chemical energy (microscopic potential energy)
• Mass energy (nuclear energy)

Formulas for forms of energy we are already familiar with:

• Translational kinetic energy:  K = ½mv2.
• Gravitational potential energy:  Ug= mgh.
• Elastic potential energy:  Us= ½kx2.

When a system is acted on by an external force, then energy can be transferred into or out of the system.  An external force can do work against internal forces and change the potential energy of the system or it can be a net force changing the kinetic energy of the system.  The external force can be conservative or non-conservative.  The work done by or against a conservative force converts one form ordered energy into another form of ordered energy.  Forces that do work converting ordered energy into disordered energy are non-conservative forces.  The work done by non-conservative forces on an object as it moves from position P1 to position P2 depends on the path of the object.  Friction is an example of such a non-conservative force.  The work done by the force of friction converts ordered energy into thermal energy.

Microscopically all known forces are conservative.  Therefore, microscopically, all energy is either kinetic or potential energy.

#### Quote:

"There is a fact, or if you wish, a law, governing all natural phenomena that are known to date.  There is no known exception to this law--it is exact so far as we know.  The law is called the conservation of energy.  It states that there is a certain quantity, which we call energy, that does not change in the manifold of changes which nature undergoes.  That is a most abstract idea, because it is a mathematical principle;  it says that there is a numerical quantity which does not change when something happens.  It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number, and when we finish watching nature go through her tricks and calculate the number again, it is the same."

Richard Feynman

Macroscopically, ordered energy can easily and completely be converted into other forms.  Gravitational potential energy is an example of ordered energy.  The gravitational potential energy stored in a car on top of a hill is converted into kinetic energy and thermal energy as it rolls towards the bottom of the hill.  Hydroelectric plants convert the gravitational potential energy stored in the water in a reservoir into electric energy.  Kinetic energy also can be easily converted.  Kinetic energy is the energy an object has because it is moving.

Thermal energy cannot easily and completely converted into other forms.  Thermal energy is disordered energy.  The individual atoms and molecules that make up an object have potential and kinetic energy, but they move in a random fashion about their equilibrium positions in the object, so that the object as a whole remains at rest.  The more kinetic energy is stored in the random motion of the atoms or molecules, the higher is the temperature of the object.

#### Problem:

A 70 kg diver steps off a 10 m tower and drops straight down into the water.  If he comes to rest 5 m below the surface of the water, determine the average resistance force exerted on the diver by the water.

Solution:
The conservative system is the freely-falling diver and the water supplies the external force.
The change in the mechanical energy of the diver from when he is at rest on the tower to when he is at rest in the water is
∆E = -mg∆h = (70 kg)(9.8 m/s2)(-15 m) = -10290 J.
The resistance force Fapp on the diver does work to convert this mechanical energy into disordered energy
Let <Fapp> denote the average resistance force.
<Fapp>*(5 m) = ∆E.  <Fapp> = (-10290 J)/(-5 m) = 2058 N, pointing upward.

#### Problem:

The coefficient of friction between the 3 kg mass and the surface in the figure below is 0.4.  The system starts from rest.  What is the speed of the 5 kg mass when it has fallen 1.5 m?

Solution:
The conservative system is comprised of the masses and the frictionless table, friction provides the external force.
Ff = μN = μ(3 kg)(9.8 m/s2) = 11.76 N.
∆E = -Ffd = -(11.76 N)(1.5 m) = -17.64 J.
Choose the zero of the gravitational potential energy so that Ei = 0.
Then Ef = -17.64 J.
But Ef = Uf + Kf = -(5 kg)(9.8 m/s2)(1.5 m) + (1/2)(8 kg)v2.
v2 = 13.965 (m/s)2, v = 3.74 m/s.