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.

K_{i }+ U_{i }= K_{f }+ U_{f}.

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.

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 y_{f} = y_{i} + v_{yi}t - (1/2)gt^{2},
v_{y }= v_{yi }- gt. But it can also be solved using conservation of
mechanical energy.

K_{i} + U_{i} = K_{f
}+ U_{f}.

U_{f }- U_{i} = K_{i }+ K_{f}.

mg(y_{f} - y_{i}) = (1/2)m(v_{i}^{2 }- v_{f}^{2}).

v_{f}^{2
}= v_{i}^{2 }+ 2g(y_{i}-y_{f})
= 25 (m/s)^{2 }+ 2(9.8
m/s^{2})(40 m) = 809 (m/s)^{2}

v_{f} = 28.4 m/s.

A 6*10^{5} 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.

½mv_{i}^{2} = ½kx^{2}. Here x is the
distance the spring bumper is compressed, x = 0.4 m.

k = mv_{i}^{2}/x^{2} = (6*10^{5}
kg)(0.5 m/s)^{2}/(0.4 m)^{2} =
9.375*10^{5} kg m/s^{2} = 9.375*10^{5} 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

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 = ½mv
^{2}. - Gravitational potential energy: U
_{g}= mgh. - Elastic potential energy: U
_{s}= ½kx^{2}.

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 P_{1}
to position P_{2} 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.

"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.

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/s^{2})(-15 m) = -10290 J.

The resistance force F_{app} on the diver does work to convert this
mechanical energy into disordered energy

Let <F_{app}> denote the
average resistance force.

<F_{app}>*(5 m) = ∆E. <F_{app}> = (-10290 J)/(-5 m) = 2058 N, pointing
upward.

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.

F_{f }= μN = μ(3 kg)(9.8 m/s^{2})^{
}= 11.76 N.

∆E = -F_{f}d = -(11.76 N)(1.5 m) = -17.64 J.

Choose the zero of the gravitational potential energy so that E_{i
}= 0.

Then E_{f }= -17.64 J.

But E_{f
}= U_{f }+ K_{f }= -(5 kg)(9.8 m/s^{2})(1.5 m)
+ (1/2)(8 kg)v^{2}.

v^{2 }= 13.965 (m/s)^{2}, v = 3.74 m/s.