Conservation of Mechanical Energy

A conservative force is a force that does the same amount of work (W=Fd) on an object regardless of the path taken by the object. Examples of conservative forces are: gravity, springs, and electricity. A non-conservative force is a force that does different amounts of work depending on the path taken by the object. Friction is an example of a non-conservative force.

To make this clearer,I will use an example. If an object is moved up an inclined plane in a straight line, gravity will do some amount of work on the object, and friction will do some amount of work on the object.

However, if the object is moved back and forth as it travels up the ramp, friction acts over a longer distance and thus does more work on the object. Since the object starts and stops at the same height in both cases, gravity does the same amount of work in both cases.

Potential Energy can only be defined for conservative forces. We can have elastic potential energy or gravitational potential energy, but we can’t have frictional potential energy.

One Law that will help us solve many problems is the Law of Conservation of Mechanical Energy. Simply stated,

In the absence of friction and all other non-conservative forces, the total energy before the event is equal to the total energy after the event, and at any point during the event.

This means that as long as there are no non-conservative forces, the mechanical energy of an object will always be the same. The total mechanical energy of an object is the sum of its kinetic energy and its gravitational and elastic potential energy, or

E_TOT=KE+PE_g+PE_e

Roller Coasters provide great examples for conservation of energy problems:

Right before a roller coaster is released, it is 50m from its lowest point on its track. How fast is it going at the lowest point, ignoring friction and air resistance?

Right before the roller coaster takes off, it has no speed, so there is only potential energy. At the bottom of the roller coaster, there is no potential energy. This means that kinetic energy at the bottom is going to be equal to
the potential energy at the top.

KE=PE

PE=mgh

KE=½mv²

mgh=½mv²

Masses cancel out, so

gh=½v²

Using the numbers we have:

Another example - picture the bouncing ball to the right and try and determine all of the mechanical energy
transitions that take place. Can you tell whether or not energy is being conserved in this illustration?

And, for another great visual on conservation of energy, check out this video of renowned Physics teacher Paul G. Hewitt.

Often it is helpful to see what someone else has to say on the topic. Be sure to check out:

http://www.physicsclassroom.com/Class/energy/u5l1d.html

http://stravinsky.ucsc.edu/~josh/5A/book/work/node17.html

For Practice Problems, Try:

From The Physics Classroom:

http://www.physicsclassroom.com/Class/energy/u5l2bc.html

From the University of Oregon:

Skydiving without air resistance

Skier Going Down A Hill

Giancoli Multiple Choice Practice Questions (Select "Practice Problems) and try some. Don't worry - you won't be able to do all of them yet.