Energy

Gottfried Wilhelm von Leibniz While Newton was working on his laws of mechanics others were also working on trying to understand how and why objects move. One person engaged in this was Gottfried Leibniz,  who also invented Calculus independently of Newton.  Leibniz proposed that there was a living force (vis viva in Latin) which caused objects to move and was conserved in certain mechanical systems.  These ideas where not as successful at explaining motion as Newton’s laws, but hundreds of years later the vis viva would be recognized as kinetic energy and conservation of energy would be recognized as one of the most important principals of mechanics.  The delay was because the importance of energy was not realized until scientist started trying to explain heat and how heat and work are related.  These aspects of thermodynamics we will discuss later.  Before we discuss mechanical energy we have to first go over the mathematics of multiplying vectors.

The Dot Product

Earlier we have discussed how to add vectors, now we need to turn to multiplying vectors. There are actually two types of multiplication for vectors. The first is called the dot product (or inner product)  and this type of multiplication between two vectors results in a scalar. The other type of multiplication is called the cross product or  vector product and it results as one might guess in a vector that points perpendicular to both of the two vectors being multiplied.  For now we will only discuss the dot product as we will not make use of the cross product untill much later in the course.

Dot Product

The dot product of two vectors can be thought of as multiplying the projection of one vector in the direction of the other vector. As a formula we can write this as

\vec{A} \cdot \vec{B} = AB\cos{\theta}

where \theta is the angle between the two vectors. Notice that if the two vectors are in the same direction \theta = 0 and the vectors multiply like scalars. If the two vectors are perpendicular then \theta = 90^{\circ} and the dot product is zero. This is in fact the definition of orthogonal that the dot or inner product equals zero.
We can also define the dot product using vector components.

\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z

where A_x is the x component of A and likewise for the other components.

Work

The first type of energy we will discuss is called work. Work is defined by

W = \int \vec{F} \cdot d\vec{s}

where \vec{s} is the displacement of the object the work is being applied to.  If the force is constant then the integral just becomes W = \vec{F} \cdot \vec{s}, which will usually be the case in problems we look at. However, it is important to remember that is only true for constant force, the real definition is the integral equation above.

Kinetic Energy

Another type of energy is kinetic energy, this is the type of energy something has because of it’s motion. The kinetic energy of an object is given by

KE = \frac{1}{2}mv^2

Notice that the the kinetic energy depends on the instantaneous velocity, so an object at a given time has a certain kinetic energy. This differs from work energy which must be evaluated as the object moves between two positions.  A natural question to ask is where does the above equation come from.  Well consider a mass that only has one force acting on it so it moves with constant acceleration for a distance s.  The work done on the object is just W=Fs. From Newton’s second law the acceleration of the object is a = F/m and form our study of kinematics we know that the velocity of the object if it started at rest would be given by

v^2 = 2 \frac{F}{m} s => \frac{1}{2}mv^2 = Fs

So if the work done to the object equals its kinetic energy than the above formula must be the correct one.  It turns out that this is often the case something which we call energy conservation.

Energy Conservation

The reason energy is a useful concept is that in many situations it is conserved. That is the total energy a system has in some initial state is the same as it has at some later time. From this you can see the great advantage energy gives to solving problems over forces. With forces you have to follow what happens in your system from start to finish. With energy you can only care about two times and ignore everything that happened in between.

For energy to be useful then we need to no whether or not it is conserved.  This basically depends on the type of forces involved in the problem.  Some forces are conservative some are not.  Conservative forces are those that are path independent, that is how you go from A to B the energy use will be the same. Nonconservative forces, the path matters.  Nonconservative forces usually produce heat which is why the path you take matters.  Friction and air resistance are non conservative forces, while gravity, springs and electrical forces are conservative forces.  Any conservative force has a potential energy. This is the energy that force could give or take away from an object. If you toss a ball up in the air its kinetic energy goes away.   When it reaches its maximum height it has zero kinetic energy.  Where did the kinetic energy go?  It went into the potential for the ball to gain kinetic energy as the gravitational force pulls it down.  The relationship between force and potential energy is

U(r) = \int \vec{F}(r)\cdot d\vec{r}

This is basically the same as work, the work done by the force is the potential energy.  The difference is that work depends on the path while potential energy only depends on the end points. For conservative forces they are the same.

Potential Energy

For gravity on the surface of the Earth this gives:

U_g = mgh

For a spring this gives

U_s = k x^2

For Newton’s Law of Universal Gravity this gives

U_G = -G{{Mm}\over{R}}