The MKS unit for energy is the joule (J).

1 J = 1 N m = 1 kg m²/s²

Energy is a scalar, not a vector. It has a magnitude, but no direction.

What is energy?

Energy is the ability to do work. That's one definition, although not particularly satisfying. It's hard to define energy, but we probably know it when we see it.

Energy puts one more tool in the toolbox we use to analyze physical situations. The approach is different than the one we took with forces.

For an object moving from point A to point B, to analyze the motion using forces we need to know everything about the forces being applied to the object at all points between A and B. Using energy we can focus on what the object is doing at points A and B, and, usually, not have to worry about how it gets from one place to the other.

Every single energy problem can be handled with one master equation. One form of the equation is:

U_i + K_i + W_nc = U_f + K_f

Another form is the one the book uses:

U_i + K_i - f_kd = U_f + K_f

In these equations U represents potential energy and K is kinetic energy.

In the first equation the W_nc represents work done by non-conservative forces as the system moves from the initial state (i) to the final state (f). This is handled in a different, but equivalent, way by the book, where they are specifically writing in the energy lost to friction.

Kinetic energy is energy associated with motion. Anything that has mass and is moving has kinetic energy (K).

K = ½ mv²

If you double the mass and keep speed the same, K doubles.

If you double the speed and keep mass the same, K increases by a factor of 4!

Potential energy is energy associated with position. Usually we view potential energy as energy that is stored temporarily.

One type of potential energy is gravitational potential energy. When g is constant this is defined as:

U_g = mgy

When g is constant you are free to set the zero for potential energy. Define the zero at some convenient level, and measure all vertical distances and potential energies from there.

The critical number is not the value of the potential energy, it's the change in potential energy. ΔU is the same no matter where you define the zero.

Work links the concepts of energy and force. Work is done on an object by a force if the force, or a component of it, is along the same direction as the displacement.

W = F • d = F d cos(θ)

Work can be positive or negative, positive when the force (or some component of it) is in the same direction as the displacement and negative when these are in opposite directions. Work is zero when the force is perpendicular to the displacement, and is maximized when the force is parallel to the displacement.

For a constant force:

W = F • d = F_x Δx + F_y Δy + F_z Δz

Sometimes we won't have any forces other than those we're handling with potential energy, in which case we set W_nc = 0 in the master energy equation.

In other cases friction will be involved. Friction acts to oppose motion, and for kinetic friction the force of friction is opposite to the displacement d. In that case:

W_nc = -f_k d

One way to multiply two vectors is to take the dot product, which results in a scalar. Taking the dot product of two vectors, A and B, results in:

c = A • B = A B cos(θ)

where θ is the angle between A and B.

The dot product is ...

• zero when the vectors are perpendicular to one another
• maximum when they are parallel
• positive when the angle between the vectors is < 90
• negative when the angle between them is > 90

c = A • B = A_x B_x + A_y B_y + A_z B_z

Whenever the work done by a force can be defined in terms of potential energy the force is a conservative force. A conservative force conserves energy. The force of gravity is a good example.

To determine whether a force is conservative, consider what happens during a round trip. If the object returns to its starting point with the same kinetic energy it had when it left, and it moves under the influenec of a single force, that force is conservative.

Another way to tell that a force is conservative is that changes in kinetic energy should be path-independent. If an object moves from point A to point B and experiences a particular change in kinetic energy, all other routes from A to B will result in the same change in kinetic energy due to the conservative force.

Forces applied by springs are conservative, so we'll define a potential energy for springs. The other forces we've dealt with (friction, tension, normal) are non-conservative.

Can we ask how much work is done by a conservative force instead of dealing with potential energy? This is certainly a valid question and an equivalent approach.

The work done by a force can be related to the potential energy in this way:

W = -ΔU

Apply this to gravity:

The work done by the force of gravity when an object of mass m is raised a height Δy is:

W = -mgΔy

This applies when g is constant, at least.

The change in potential energy here is:

ΔU = +mgΔy

So, you can do one or the other. For a conservative force, consider the work done by the force or the change in potential energy. Usually we use the potential energy method.