Work is a scalar quantity that can be either positive or negative, depending on the force doing the work.  If the force is negative, the work, therefore, will be negative too, due to the equation W = Fd. Whether or not if the force is negative or positive depends on whether the force has a component that points in the same (positive), or opposite (negative) direction of the displacement.  Since the equation is Fcos(theta), and cos(90) = 0, then that shows that any force perpendicular to the motion will cause no work.  Similarly, if theta = 0, cos(0) = 1, meaning that all of the work is used, because it is pointing parallel to the displacement. Without movement, work done by a force acting on an object is zero. When an object moves down, work is positive. When an object moves up, work is negative.

When work is done on an object, kinetic energy (KE) changes.  The equation for Kinetic Energy is KE = 1/2mv^2.  The work-energy theorem states that the work (W) done by the net external force equals the difference between the objects final kinetic energy and initial kinetic energy.

W = KEfinal - KEinitial

If the net force does positive work, the kinetic energy increases. If the net force does negative work, the kinetic energy decreases. If the work is zero, the kinetic energy remains the same. The work-energy theorem does not apply to the work done by an individual force, unless it is the only one present. It deals with NET force.  This is because in the previous equation, a higher final value will reult in a positive number (and increase) and a higher intial value will result in a a negative number (decrease).

According to the work-energy theorem, a moving object has kinetic energy, because work was done to accelerate the object from rest to a speed. Conversely, an object with kinetic energy can perform work, if it is allowed to push or pull on another object.

This diagram illustrates a satellite moving about the Earth in a circular orbit and in an elliptical orbit. The only external force that acts on the satellite is the gravitational force.

In a circular orbit, the gravitational force is always perpendicular to the displacement of the satellite, and does no work.

In an elliptical orbit, there can be a component of the force along the displacement, and work can be done.

The work done by the force of gravity can be either positive or negative. Only the difference in vertical height should be considered when calculating the work done by gravity.

PE = mgh

Vertical distance doesnt need to be measured from Earths surface, only from the objects initial height to its final height, as long as you stay constant.

Gravitational potential energy (PE) is the energy that an object has by virtue of its position. This concept is associated only with conservative forces. An objects potential energy is the difference between two potential energies. This allows zero level heights to be taken anywhere, as long as both heights are measured from the same zero level. The gravitational potential energy depends on the object, the Earth, and the height.   Also, remember that for an object gain energy from it's height, it must have put energy into its getting there.

If Mr. Crymes lifts a bowling ball over his head, the ball gains potential energy, yet he had to expend energy to get it up to that height.

Also, if an object moves straight up, the equation transforms to set PE equal to negative KE at the bottom.  This is because at the bottom, PE = 0 and KE = Max, and at the top, PE = Max and KE = 0.

In the diagram, gravity exerts a force on the basketball. Work is done by the gravitational force as the basketball falls from the initial height to the final height.

When an object is moved from one place to another, the work done by the gravitational force does not depend on the choice of path. The work done by gravity depends only on the initial and final heights, and not on the path between them. A conservative force is one that in moving an object between two points, does the same work, independent of the path taken between the points. Also, a force is conservative when it does no net work on an object moving around a closed path, starting and finishing at the same point. Some conservative forces include: gravitational force, elastic spring force, and electric force. Within these forces, the same amount of positive and negative work is done.

A non-conservative force is one that in moving an object between two points, the work depends on the path of motion between the points. Some non-conservative forces include: static and kinetic frictional forces, air resistance, tension, normal force, and propulsion force of a rocket. Within these forces, more positive work, or negative work may be done, either opposing, or propelling the object. This causes greater amounts of work to be done over the objects paths, so that the work depends on the choice of path. The concept of potential energy is not defined for a non-conservative force.

In normal situations, conservative forces (like gravity), and non-conservative forces (such as air resistance) act on an object at the same time. According to the work-energy theorem, the work done by the net external force equals the change in the objects kinetic energy.

W = Wnc + Wc

When calculating non-conservative forces, the non-conservative force plus the conservative force will equal the change in kinetic energy (see the math notes for the equations).  When dealing solely on the non-conservative force, it will ewual the change in kinetic energy plus the change in potential energy.

The principle of conservation of mechanical energy states that the total mechanical energy of an object remains constant as the object moves, provided that the net work done by external non-conservative forces is zero. The total mechanical energy (TME) remains constant all along the path between the initial and final points. A quantity that remains constant throughout motion is said to be "conserved."

E = KE + PE

While energy is constant, however, KE and PE may be transformed into each other. Kinetic energy of motion is converted into potential energy of position, for example when a moving object coasts up a hill. Potential energy of position is converted to kinetic energy of motion when an object is allowed to fall. The principle of conservation of mechanical energy can be applied even when forces act perpendicular to the path of a moving object. Tension is always perpendicular to the circular path of motion.

And because E can be subtituted for KE and PE values, the non-conservative force equation can be rewritten to look like this:

Wnc = Ef - Ei.  And when Wnc = 0, you can manipulate that equation to get Ef = Ei, showing that conservative forces retain all of their energy.

Most moving objects experience non-conservative forces (friction, air resistance). Non-conservative forces can do either positive or negative work. Work is positive when it has a component in the direction of the displacement, and speeds up the object. Work is negative when it has a component opposite to the displacement and slows the object down.  And when a non-conservative force acts upon a system, the total energy is less at the end than it was at the top.

Power incorporates both the concepts of work and time. Power is work done over time.

P = W / t

Average power is the rate at which the net force does work. Out of two cars, one may go faster more quickly because it has a more powerful engine. Power can also be defined as the rate at which energy is changing. This is because the work-energy theorem relates work to the change in an objects energy.

To find the average power of a system, you take the change in energy over the time.  So:

P = E / t, where P is the average power and E is the change in energy.

When you manipulate the power equation to involve distance, you will end up with:

P = Fv  (the process is in the notes)

Often, situations arise where the force is not constant, but changes with the displacement of an object..