# 2.1.2: Energy and Work

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In the Wikipedia definition quoted at the beginning it is stated that: *energy... ...must be transferred to an object in order to perform work on......the object*. Let’s recall what “work” means in physics. Actually, physics recognizes several different types of work – but one of them, the **mechanical ****work**, is of special importance. Suppose that a body is at rest at a initial position \(x_i\). Then, a force \(F\)* *is exerted to it for some time, and as the result of it the body is displaced to a new “final” position \(x_f\). We will call the difference

\[\Delta x = x_f - x_i \notag \]

the *displacement*, and the product of \(\Delta x\)* *and the force \(F\):

\[\Delta

W = F \cdot \Delta x\]

is the mechanical work performed in the process of shifting the body.

Performed by whom or by what? Well, we don’t need to specify, it’s enough to assume that there was an *object *capable of exerting the force \(F\)* *on the body. Initially, this object contained certain amount of energy, \(E_i\) and after the work was performed, this amount changed to a final lower value \(E_f). So, the difference:

\[\Delta E = E_f - E_i\]

is the energy the object “paid” for performing the work. This is obviously a negative number, right? And now we have reached a very important point – namely, a declaration that is known as the *Work-Energy Theorem*, or the *Work-Energy Principle*, or even by longer name: *The Principle of **Equiva**lence** of Work and Energy *– and it simply states that:

\[\Delta W = -\Delta E\]

(note that \Delta E* *is a negative number, so the minus in the above equation is needed to obtain a positive work). In other words, the above principle simply states that for performing work, the “performer” has to pay an equal price in energy. It also works the other way around: for increasing the energy of an object by ∆*E*, one has to “pay the same price” in work.