### Classical mechanics

# 3. Energy, work and power

The idea of 'energy' is a cornerstone of physics. It is essential for understanding so much of the world around us and to describe so many phenomena that to say physics is the study of matter and energy is an understatement.

Energy is everywhere, so much so that it has become an abused term these days. The same is true, perhaps even more so, of work and power. People refer to everything from the electric current in their homes to their weekend gardening chores as forms of energy, work or power; and they are right to some extent, but not entirely.

The ideas discussed in this article are elementary but extremely important because they will find use elsewhere, in more complex discussions, such as Lagrangian mechanics and Hamiltonian mechanics or even in nuclear physics or, most famously, in quantum and condensed matter physics.

# Energy and work

Defining energy is harder than it might appear. The joke is that energy, work and power are best defined in terms of each other cyclically. It seems like a safe thing to do (it is) but it is hardly a satisfying approach.

In physics, energy is best defined mathematically rather than verbally. Also, there is no universal equation governing energy but the many forms of energy have their own descriptions. Generally speaking

Energy is a property that must be transmitted

toa system to perform work on it orbya system for it to perform some work.

This is where we encounter some hazy descriptions. What exactly is work? For now, take work to mean exactly what you normally do: gardening, jogging, reading a newspaper, working on your computer, teaching, cycling etc. To do all these, a body has to lose *something*, and, if given the same thing, you will be able to do some more of these activities. That *thing* is energy.

## Kinetic energy

In all these examples, specifically, our conversations revolve around a form of energy known as kinetic energy. We will better define this in a moment, but first let us see what other type of energies exist: turning on a light bulb is said to involve electric energy; heating a tough of water involves heat or thermal energy; mixing two reactive liquids can involve chemical energy; when the earth pulls you down we say there exists gravitational energy; when a system opposes this gravitational energy it must contain within itself an appropriate amount of energy that it can potentially use elsewhere, which we simply call its potential energy. Likewise there are several different forms of energy that exist mutually or exclusively as the situation demands.

Regardless of what energy we refer to, we always measure energy in joule. However, electron volt and watt too are units that can be used to directly or indirectly imply the involvement of energy, but, at the end of the day, they can all be written in terms of a joule.

The energy possessed by an object due to its motion is called kinetic energy.

It is defined mathematically, for an object of mass $m$ moving with velocity $v$, as

$$\begin{equation}E_k = {1 \over 2} m v^2 \label{eq:ke}\end{equation}$$

Energy is a scalar since, for a vector $\mathbf{v}$, the squared value $\mathbf{v\cdot v} = |\mathbf{v}|^2 = v^2$ is a scalar.

## Work

We shall now make a minor detour from energy to discuss what it means to do work. In physics,

Work is said to be done when a force displaces a mass.

As a result, when either the applied force, the displacement or both are zero, no work is done. Spinning your arms, therefore, means no work is done. Sliding a feather away from you, though, does count as work.

Mathematically,

$$W = \mathbf{F} \cdot \boldsymbol{\Delta} \mathbf{s}$$

which means if a force and displacement exist perpendicular to each other (not too common a case) no work is done. More practically, it means that the more the difference is between the directions of the force and displacement, between zero and perpendicularity, the less is the work done.

Think of work and energy as two ways of looking at the same thing. Indeed this is what the **work-energy theorem** tells us:

Work is given by the change in the kinetic energy of a system.

That is,

$$W = \Delta E_k$$

It is possible to extend this definition to define *positive* and *negative* work, which is generally (in physics) defined as work done *on* and *by* the system respectively. Therefore, positive work sees an increase in energy and negative work sees a decrease in the same.

We can, in fact, use the above two equations to arrive at eq. \eqref{eq:ke}.

$$\begin{align*}W &= \mathbf{F} \cdot \boldsymbol{\Delta} \mathbf{s} \\[.5em]\Delta E_k &= {\textrm{d}\mathbf{p} \over \textrm{d} t} \cdot \Delta v \textrm{d}t \\[.5em]\textrm{or, }\;\textrm{d} E_k &= m ( \textrm{d} \mathbf{v} ) \cdot \mathbf{v}\end{align*}$$

where $\mathbf{p} = m\mathbf{v}$ is the momentum of the body. Force and momentum are closely related to each other as we discussed in chapter 2.

Now observe that

$$\textrm{d} v^2 = \textrm{d} ( \mathbf{v\cdot v} ) = \mathbf{v} \cdot ( \textrm{d}\mathbf{v} ) + ( \textrm{d}\mathbf{v} ) \cdot \mathbf{v} = 2 \, \mathbf{v} ( \textrm{d} \mathbf{v} )$$

which gives us

$$\begin{align*}\d E_k = {1\over 2} m \d v^2 \\[.5em] \therefore E_k = {1 \over 2} m v^2\tag{\ref{eq:ke}}\end{align*}$$

just as we said earlier. However, note that the energy expended by a rotating body is different from one exhibiting linear (or approximately linear) motion, but we will put that aside as discussion for another day.

## Potential energy

The other important type of mechanical energy we are interested in is potential energy: the energy potentially contained in a body which is at rest or, more scientifically,

Potential energy is the energy contained in a body placed inside a field.

The field we are all in, the gravitational field of the earth, is what gives us all our most familiar flavour of potential energy: **gravitational potential energy** or **GPE** for short. Here we will use the standard symbol for potential energy $E_p$, or we will favour $V$ where such usage is unambiguous.

The potential energy of a body, like $E_k$, depends on its mass. However, since the body is explicitly *not* in motion, the $E_p$ does *not* depend on $\mathbf{v}$. Instead, it depends on the field that the body is placed in, which, in case of $E_p$, means the energy must somehow depend on $\mathbf{g}$, the acceleration due to the Earth's gravity.

To see how let us turn to the idea of work again and recall that, in this case, the force acting on the body is simply its weight $m\mathbf{g}$, which leaves us with

$$\begin{align*}W &= \mathbf{W} \cdot \boldsymbol{\Delta} \mathbf{s} \\[.5em]\textrm{d} E_p &= m \mathbf{g} \cdot \d\mathbf{s} \\[.5em]\end{align*}$$

This equation contains a rather interesting term: $\d\mathbf{s}$. If an object is forbidden from moving, as far as potential energy is concerned, just what is the displacement term doing here? It turns out that this is not so much the displacement caused by the force but, rather, the displacement a force could *potentially* cause if the body moved.

Ask yourself, 'if the force in question causes the body to move, by how much would it displace the body?' This is our $\d\mathbf{s}$ term. In case of attractive forces, think of this as the separation between two bodies. A magnet can potentially move an iron nail over a distance between itself and the nail and no more or less. An object kept at a height $h$ above the earth, say, on a table, can *potentially* be dropped through a displacement of $h$ by the earth right up to the floor. Therefore we write the potential energy as

$$\begin{equation}E_p = m \mathbf{g} h \label{eq:pe}\end{equation}$$

where $m$ is the mass of the body, $\mathbf{g}$ is the acceleration it experiences due to the earth but just as well due to different factors depending on the system it is in, and $h$ is the separation between two bodies in case of attractive force fields. In fact it is in the presence of attractive forces that we will see this equation being used repeatedly although there is no valid reason to treat this as a given.

# Mechanical power

A term often coupled with work and energy is the term **power**. Like the other two, power is also frequently misused in everyday life. It usually refers to either electric power (although not strictly as a product of current and potential difference) or physical strength.

Power is the rate of doing work.

Mathematically,

$$\begin{equation}P = {\textrm{d} W \over \textrm{d} t} \label{eq:p}\end{equation}$$

where $P$ is the power associated with the system. The logic is fairly simple: the faster one works, the more energy they expend in a given time, the more power they have. In mechanics, though, power is of much less importance than energy; in current electricity or electronics, however, power can be an important characteristic describing the capability of a device.

# Units of measurement

Energy, as is evident from eq. \eqref{eq:ke} has the dimension $[M] [L]^2 [T]^{-2}$. Naturally, the same is true of eq. \eqref{eq:pe} too. The unit of energy is a **joule**, named after the Englishman James Prescott Joule who contributed greatly towards early studies of energy and even defined it. A joule is also the unit of work.

One joule is the work done by a force of one newton displacing some point on a mass over a distance of one metre.

A joule is, therefore, equivalent to a newton-metre. There are other forms of energy, with other descriptive formulae (e.g. in terms of relativity); there are also alternate definitions of power, as said above, and of the joule (e.g. in terms of electricity as the heat given out when an ampere of current passes through an ohm of resistance for a second). All of these, however, are best discussed in context and it is only basic mechanical energy, work and power that we will concern ourselves with at this point.

Understanding these will help us understand other ideas of classical mechanics. Start with chapter 1 or chapter 5 for a better idea of how energy comes into play when we try to understand various systems around us.