Conservation of energy

There are three quantities in physics that are always conserved:

• momentum,
• angular momentum and
• energy

The law of conservation of energy states that

• The total energy of the universe is constant.
• Therefore, if energy in one system appears to disappear, it can't have been lost completely. It musthave been transformed to some other form.
• Energy is neither created nor destroyed. It just changes form.

There are only two forms of mechanical energy:

• kinetic and
• potential.

In this section, we practice with problems in which energy transforms between them.

For ideal mechanical systems, the total energy E is seen as

E = KE + PE or

In conservation of energy problems, Work plays an important role :

• to transfer energy from one place to another.
• ie, an object lifted has work Fd done on it in raising it a given height.
• At this height, the object now has a value of PE equal to Fd, the work done on it.
• We know that in doing work in the vertical dimension, the force used is just the weight of the object being moved, and that weight is equal to mg.
• so.. W = Fd = mgd. This is the same as the PE formula.. PE = mgh !

In conservation of mechanical energy problems, we idealize our systems such that they are isolated from outside sources of energy transfer, and make the assumption that

• when there is a change in PE, it can be accounted for, or manifest as a change in KE, and we say that the total energy E remains constant.
• Note that this is much like the conservation of momentum law we studied in a previous chapter. In that case we stated that in a closed, isolated system, that momentum remained constant, and that

In energy problems we use the same principle:

This says that the total energy E of a closed, isolated system remains constant, and so does the sum of KE + PE before, during and after any event that results in a transformation of the mechanical energy.

A useful expansion of this formula can also be written as

We can use this relationship to solve for such values as the velocity of an object after it has gained kinetic energy and lost potential.

YOU TRY: Do the practice problems 9-11 on page 261-2. For number 9, you should get about 3.1 x 10³ J for the kinetic energy before going up the hill, and a value of 3.7 m as the increase in vertical heigth as the kinetic energy drains away and is transformed into PE. If you can get these results for number 9, you should have no difficulty with number 10 and 11

lab: finding the changes in PE and KE in a real situation (not closed and isolated).. Dynamics Cart on a Ramp

View some exemplary write ups for the Dynamics cart lab. 1 2

Homework: Set 4: problems 57-66