## Work and energy

6-2-98

Energy gives us one more tool to use to approach physical situations. When we were analyzing situations in terms of forces and accelerations, we would usually freeze the action at a particular instant in time, draw a free-body diagram, set up force equations, figure out accelerations, etc. With energy the approach is usually a little different; often you can look at the starting conditions (initial speed and height, for instance) and the final conditions (final speed and height, perhaps), and not have to worry about what happens in between. The initial and final information often tells you all you need to know.

### Work and energy

Whenever a force is applied to an object, causing the object to move, work is being done by the force. If a force, F, is applied but the object doesn't move, no work is done; if a force is applied and the object moves a distance s in a direction other than the direction of the force, less work is done than if the object moves a distance s in the direction of the applied force.

The physics definition of "work" is:

The unit of work is the unit of energy, the joule (J). 1 J = 1 N m.

Work can be either positive or negative: if the force has a component in the same direction as the displacement of the object experiencing the force, the force is doing positive work, but if the force has a component in the direction opposite to the displacement, the force does negative work. If you pick a book off the floor and put it on a table, for example, you're doing positive work on the book, because you supplied an upward force and the book went up. If you pick the book up and place it gently back on the floor again, though, you're doing negative work, because the book is going down but you're exerting an upward force, acting against gravity.

### Kinetic energy

An object has kinetic energy if it has mass and if it is moving. It is energy associated with a moving object, in other words. For an object traveling at a speed v and with a mass m, the kinetic energy is given by:

### The work-energy theorem

There is a strong connection between work and energy, in a sense that when there is a net force doing work on an object, the object's kinetic energy will change by an amount equal to the work done:

Note that the work in this equation is the work done by the net force, rather than the work done by an individual force.

### Gravitational potential energy

Let's say you're dropping a ball from a certain height, and you'd like to know how fast it's traveling the instant it hits the ground. You could apply the projectile motion equations, or you could think of the situation in terms of energy (actually, one of the projectile motion equations is really an energy equation in disguise).

If you drop an object, it falls down, picking up speed along the way. This means there must be a net force on the object, doing work. This force is the force of gravity, with a magnitude equal to mg, the weight of the object. The work done by the force of gravity is the force multiplied by the distance, so if the object drops a distance h, gravity does work on the object equal to the force multiplied by the height lost, which is:

work done by gravity = W = mgh (h = height lost by the object)

An alternate way of looking at this is to call this the gravitational potential energy. An object with potential energy has the potential to do work. In the case of gravitational potential energy, the object has the potential to do work because of where it is, at a certain height above the ground, or at least above something.

### Conservation of energy

We'll take all of the different kinds of energy we know about, and even all the other ones we don't, and relate them through one of the fundamental laws of the universe.

The law of conservation of energy states that energy can not be created or destroyed, it can merely be changed from one form of energy to another. Energy often ends up as heat, which is thermal energy (kinetic energy, really) of atoms and molecules. Kinetic friction, for example, generally turns energy into heat, and although we associate kinetic friction with energy loss, it really is just a way of transforming kinetic energy into thermal energy.

The law of conservation of energy applies always, everywhere, in any situation. There is another conservation idea associated with energy which does not apply as generally, and is therefore called a principle rather than a law. This is the principle of the conservation of mechanical energy.

### The conservation of mechanical energy

Mechanical energy is the sum of the potential and kinetic energies in a system. The principle of the conservation of mechanical energy states that the total mechanical energy in a system (i.e., the sum of the potential plus kinetic energies) remains constant as long as the only forces acting are conservative forces. We could use a circular definition and say that a conservative force as a force which doesn't change the total mechanical energy, which is true, but might shed much light on what it means.

A good way to think of conservative forces is to consider what happens on a round trip. If the kinetic energy is the same after a round trip, the force is a conservative force, or at least is acting as a conservative force. Consider gravity; you throw a ball straight up, and it leaves your hand with a certain amount of kinetic energy. At the top of its path, it has no kinetic energy, but it has a potential energy equal to the kinetic energy it had when it left your hand. When you catch it again it will have the same kinetic energy as it had when it left your hand. All along the path, the sum of the kinetic and potential energy is a constant, and the kinetic energy at the end, when the ball is back at its starting point, is the same as the kinetic energy at the start, so gravity is a conservative force.

Kinetic friction, on the other hand, is a non-conservative force, because it acts to reduce the mechanical energy in a system. Note that non-conservative forces do not always reduce the mechanical energy; a non-conservative force changes the mechanical energy, so a force that increases the total mechanical energy, like the force provided by a motor or engine, is also a non-conservative force.