Work and Energy

In this post, we will explore where the formula for work comes from.

Work

Exactly what is work? First, let us think about the characteristics of work in relation to force and displacement. It's pretty obvious that the net force acting on an object and the displacement will be pointing to the same direction. Also, the more we displace an object, there is more work done, and the more we put force on an object, there is more work done. We immediately see a plausible definition of work

Suppose a constant force $\vec F$ acts on an object and displaces by $\vec s$ in the same direction as the $\vec F$. Then the work done is $$W = Fs.$$

Now, we see a problem when we try to apply this definition on each individual force that makes up the net force: the direction of the force and displacement may not necessarily be the same. We know, however, that a force that is orthogonal to the displacement hasn't done any work. Also, the total work must be the sum of all the work done by individual forces. Thus, if we use the direction of displacement as the $x$-axis, then $$\vec F = F\cos\theta\hat\imath + F\sin\theta\hat\jmath\implies W = (F\cos\theta)\cdot s + 0\cdot s = Fs\cos\theta.$$

So, we finally have the general definition of work.

If a constant force $\vec F$ acts on an object and displaces by $\vec s$, then $$W = \vec F \cdot \vec s.$$

What's nicer is that this definition satisfies an assumption we made about work: the total work must be the sum of all the work done by individual forces. We can easily prove this using our revised definition.

Proof: The net force can be written as $$\Sigma \vec F = \vec F_1 + \vec F_2 + \vec F_3 + \cdots + \vec F_n.$$ Dot product both sides by $\vec S$. By distributive property, we have $$W_{\text{net}} = W_1 + W_2 + W_3+ \cdots + W_n.$$

Work is intuitively the amount of success that one has in exerting a force on an object. For example, if a person exerts a force on an object, and the object accelerates in that same direction, positive work has been done. On the other hand, if the object accelerates in the opposite direction, negative work has been done, since the person has clearly failed.

More formally, work is the dot product of the force exerted and the distance traveled.

If $W$ is the work, $\vec{F}$ is the force vector, and $\vec{d}$ is the displacement vector, $$W=\vec{F}\cdot\vec{d}$$

If the force and displacement vectors are not uniform, one would have to take a line integral along the curve $C$, where $C$ is the path that the object takes and $\vec{dr}$ is the differential of the displacement vector: $$W=\int_C\vec{F}\cdot\vec{dr}$$

Energy (abbreviated as $E$) is the ability to cause a change. There are many different types of energy and many different ways to calculate energy, depending on the situation.

For mechanics, one must know two main types of energy:

1. potential energy: the energy stored in an object

2. kinetic energy: the energy in an object's motion

So what is the connection between work and energy? The answer lies in the Work-Energy Theorem:

The net work performed on an object is equal to the change in kinetic energy of the object.

This also means that the dimension of work is the same as the dimension for energy. One can derive this dimension directly from the definition: $$[E]=[W]=\left[\vec{F}\right]\cdot\left[\vec{d}\right]$$ $$[E]=[W]=\frac{[M](L]}{[T]^2}\cdot [L)$$ $$[E]=[W]=\frac{[M](L]^2}{[T)^2}$$ This means that the standard SI unit for work and energy is $\text{Nm}=\frac{\text{kg}\cdot m^2}{s^2}$. This unit is called the Newton-meter, or more commonly the Joule for convenience, and it is abbreviated $J$.