Work Energy Principle

What is Work? What is Energy?

Energy is the capacity to do work. Work is the transfer of energy into or out of a system. Now right about now, you're probably like, "Wait, a second...whoa, whoa whoa. Energy is work, and work is energy; how does that work?" This interrelation between work and energy allows the Work-Energy Principle to occur. But, more on that to come. Try to see work and energy as synonyms more than anything; doing that will help you later in the article.

Work and energy are scalar quantities; this means that work and energy have no direction associated with them: only a magnitude. However, they can have positive and negative values.

Work-Energy Principle Area of Application

Work \(W\) is done on an object when we apply a force \(F\) on it, as long as the force's component is in the direction of displacement \(d\). We can write this as:

$W=F_{||} d = Fd\cos{\theta}\mathrm{.}$

Also, recall that work is the transfer of energy into or out of a system; this means that work equals a system's change in energy. Therefore, the equation above can be appropriately rewritten as

$\Delta E =W = F_{||} d = Fd\cos{\theta}\mathrm{.}$

An outside force exerting on a system causes mechanical energy, the sum of a system's kinetic and potential energies, to be transferred into or out of the said system: so long as the force is parallel to the system's displacement. This energy-transferring process is equivalent to the work done.

Work is therefore equal to the area under a Force vs. Displacement Graph.

Fig. 2 - The work done is equal to the area under a Force vs. Displacement curve

There is no work done or \(W=0\) when:

• No displacement or force is acting on the object.
• The force is perpendicular to the motion (\(\cos{\theta} = 0\)).

Work done is negative when the force's component is in the opposite direction of the displacement. This is the case where \(\cos{\theta} = -1\) and \(\theta = 180^{\circ}\mathrm{.}\)Work has units of \(\mathrm{joules}\), \(\mathrm{J}\) or \(\mathrm{newton\,meter}\) \(\mathrm{N\,m}\). In SI base units, \(1\,\mathrm{J}=1\,\mathrm{N\,m}=1\,\mathrm{kg\,m^2/s^2}\mathrm{.}\)

What is Kinetic Energy?

Recall that energy is the ability of an object to do work.

You've likely seen kinetic energy written as

$K=\frac{1}{2}mv^2\mathrm{.}$

This is the equation for translational kinetic energy.

An object can have kinetic energy all on its own. Whereas potential energy requires a two-or-more-object system, kinetic energy only depends on an object's mass and velocity; therefore, a single-object system can have translational kinetic energy.

Explanation of the Work-Energy Principle

Any system with internal structure can have potential and kinetic energy.

For example, an object-earth system can interact with itself because the earth would exert a gravitational force on the object. Understanding the internal structure of a system is essential because it allows us to explain the law of the conservation of energy.

This energy comprises the potential and kinetic energies within the system. The law of energy conservation requires that a closed system's energy (a system that does not interact with its environment) remains constant. Therefore, a change in the system's potential energy can change its kinetic energy to keep the overall internal energy constant.

This now brings us to the Work-Energy Principle.

Work-Energy Principle Formula

Written mathematically, it looks something like this:

$W=\Delta K$

where \(W\) is the work done, and \(\Delta K\) is the change in the kinetic energy.

The Work-Energy Principle works because of energy conservation; it's just an outstanding citizen who wants to keep the balance of energy in the universe.

Proving the Work-Energy Principle

As we saw above, the Work-Energy Principle relates work to kinetic energy, but there must be some basis in math for why work and kinetic energy can be related. So, let's see if we can't prove the Work-Energy Theorem. First, we'll start with the definition of kinetic energy:

$K=\frac{1}{2}mv^2\mathrm{.}$

We will also define force \(F\) as a product of mass \(m\) and acceleration \(a\). You may recognize this as one way to write Newton's Second Law:

$F=ma\mathrm{.}$

Recall our definition of work,

$W=Fx\cos{\theta}\mathrm{,}$

where we define \(F\) as the force and \(x\) as the object's displacement due to its motion.

Let's assume that our force is completely parallel to the displacement of our object, \(\theta = 0\). Using our definition of force, we get:

$W=max\mathrm{.}$

From the kinematics for constant acceleration, we can use the equation

$v_\mathrm{f} ^2 = v_\mathrm{i} ^2 - 2ax$

Where the subscript \(\mathrm{f}\) refers to the final velocity and subscript \(\mathrm{i}\) refers to the initial velocity. Let's rearrange this to define acceleration as

$a=\frac{v_\mathrm{f} ^2 - v_\mathrm{i} ^2}{2x}\\$

and then plug this value into our definition of work and get$W = m\frac{v_\mathrm{f} ^2 - v_\mathrm{i} ^2}{2x} x$

Then, we can cancel out the \(x\)'s and distribute the \(\frac{1}{2}\\m\) to each of our \(v\)'s on the right side of the equation to get a new work equation of

\begin{align}W &= \frac{1}{2}m(v_\mathrm{f} ^2 - v_\mathrm{i} ^2)\mathrm{.} \\W &= \frac{1}{2}mv_\mathrm{f} ^2 - \frac{1}{2}mv_\mathrm{i} ^2 \\W &= K_\mathrm{f} - K_\mathrm{i} \mathrm{.} \\W &= \Delta K\end{align}

From our proof above, we see that work equals the change in the kinetic energy, proving the Work-Energy Theorem. We can find applications of the Work-Energy Principle in the examples below.

Work-Energy Principle Examples

You can't have a physics article without loads of examples, so let's dive right in.