Conservation of Conservation of Mechanical EnergyMechanical Energy

IntroductionIntroduction

““The laws of conservation are the The laws of conservation are the cornerstone of physics.cornerstone of physics.””

DefinitionDefinition

When a physical quantity is conserved, the When a physical quantity is conserved, the numeric value of the quantity remains the numeric value of the quantity remains the same throughout the physical process.same throughout the physical process.

Although the form of the quantity may Although the form of the quantity may change, the final and initial value is change, the final and initial value is consistent.consistent.

ExampleExample

LetLet’’s Explains Explain

The kinetic energy of an object falling The kinetic energy of an object falling solely under the influence of gravity is solely under the influence of gravity is constantly changing.constantly changing.

During this time the gravitational potential During this time the gravitational potential energy is also changing.energy is also changing.

Individually?Individually?

These quantities are not conserved These quantities are not conserved individually; however, as a system, they individually; however, as a system, they are.are.

Recall thatRecall that

A system is defined as a definite quantity A system is defined as a definite quantity of matter enclosed by boundaries.of matter enclosed by boundaries.

In general the amount of energy remains In general the amount of energy remains constant when no mechanical work is constant when no mechanical work is done on or by the system, and no energy done on or by the system, and no energy is transmitted to or from the system.is transmitted to or from the system.

ExampleExample

The Law of Conservation of EnergyThe Law of Conservation of Energy

The total energy of an isolated system The total energy of an isolated system is always conserved!is always conserved!

ConversionsConversions

Within an isolated system, energy may be Within an isolated system, energy may be converted from one form to another.converted from one form to another.

However, the total amount of all forms of However, the total amount of all forms of energy is not going to change!energy is not going to change!

Random ThoughtRandom Thought

Did you know that total energy can neither Did you know that total energy can neither be created nor destroyed?be created nor destroyed?

This means that energy as a whole, This means that energy as a whole, (taking the entire universe as our system) (taking the entire universe as our system) is conserved and constantly being is conserved and constantly being changed from one form to another.changed from one form to another.

Conservation of EnergyConservation of Energy

We can say that because:We can say that because:W = W = ΔΔKE + KE + ΔΔPE PE

and the net work done on the system is to be 0and the net work done on the system is to be 0

Then, Then,

ΔΔKE = - KE = - ΔΔPE PE

We can expand…We can expand…

We can expand this equation and say that:We can expand this equation and say that:

KEKEii + PE + PEii = KE = KEff + PE + PEff

According to this equation, the sum of the According to this equation, the sum of the kinetic and potential energy remains the kinetic and potential energy remains the same before and after.same before and after.

ExampleExample

Example continuedExample continued

With the absence of non-conservative With the absence of non-conservative forces such as air resistance and friction, forces such as air resistance and friction, the trading of energy is exactly even.the trading of energy is exactly even.

Fortunately for the skydivers, this is not Fortunately for the skydivers, this is not the case. They have their parachutes the case. They have their parachutes which create resistive forces that slow which create resistive forces that slow them down.them down.

Allowing for a nice, Allowing for a nice, smooth landing!smooth landing!

So, the rule isSo, the rule is

In any isolated system of objects In any isolated system of objects interacting only through conservative interacting only through conservative forces, the total mechanical energy is: forces, the total mechanical energy is:

E = KE + PEE = KE + PE

ExampleExample

A diver of mass m drops from a board 10.0 A diver of mass m drops from a board 10.0 meters above the waters surface. (a) Use meters above the waters surface. (a) Use conservation of mechanical energy to find conservation of mechanical energy to find the divers speed 5 meters above the the divers speed 5 meters above the surface. (b) his speed when he surface. (b) his speed when he

hits the water. hits the water.

(Neglect air resistance)(Neglect air resistance)

ExampleExample

Suppose the same diver vaults off the Suppose the same diver vaults off the springboard, leaving it with an initial speed springboard, leaving it with an initial speed of 3.50 m/s upwards. Use the law of of 3.50 m/s upwards. Use the law of conservation of energy to find his speed conservation of energy to find his speed when he strikes the water.when he strikes the water.

ExampleExample

A waterslide is 21.9 A waterslide is 21.9 meters high. With meters high. With what speed will a 60 what speed will a 60 kg woman be kg woman be travelling when she travelling when she reaches the bottom?reaches the bottom?