Why do we need to define total mechanical energy?

Question

The text I'm using (Taylor's Classical Mechanics) discusses potential energy, $PE$, and conservative forces. It gives potential energy as a reason to care about conservative forces, saying we can define PE such that $E=KE+PE$, where $E$ is the mechanical energy and $KE$ is the kinetic energy. What I don't understand is why we have decided to define total mechanical energy in the first place. Why have we decided to define mechanical energy as the sum of kinetic and potential energy? Why is this a useful quantity? I understand how $PE$ and $KE$ are useful on their own. Can we really say that any lost $KE$ or $PE$ is conserved by becoming the other?

Answer 1

Any conservative force can be described as the gradient of a potential energy function, i.e. $\vec{F} = -\nabla U$, or in one dimension, $F_x = - \frac{dU}{dx}$.

We might then obtain $$-\int_{x_1}^{x_2} F_x dx = \Delta U$$ Now since $\int_{x_1}^{x_2} F_x dx$ is the work $W_{c}$ done by the conservative force on the body, we obtain the following relation $$\Delta U = -W_{c}$$ You might then recall the work-energy theorem,

$$W_1 + W_2 + ... + W_n = \Delta E_k$$

Suppose two forces act on a body, one of which is conservative. For the work done by this force, we might substitute $W$ for $-\Delta U = U_1 - U_2$ in the statement of the work energy theorem:

$W_1 + W_2 = W_1 + (U_1 - U_2) = {E_{k}}_2 - {E_{k}}_1$

and we might rearrange this to

$W_1 + {E_{k}}_1 + U_1 = {E_{k}}_2 + U_2$

which is perhaps a more familiar statement of the conservation of energy. Note that if we had no non-conservative forces acting (i.e. no $W_1$ term), this simply results in the sum $E_k + U$ being constant.

So we might then say that mechanical energy is conserved in the absence of non-conservative forces! It is also straightforward to rearrange the above relation to obtain $W_{nc} = \Delta E_k + \Delta U$; the work done by non-conservative forces equals the change in mechanical energy!

As an aside, you might be interested to read about the Hamiltonian, which is an operator defined as $H = U + T$ where $U$ is potential energy and $T$ is kinetic energy. Classically, this represents the total energy of the system.

Answer 2

All forms of energy are either potential energy (energy of position) or kinetic energy (energy of motion). Mechanical energy generally refers to potential and kinetic energy at the macroscopic level, meaning the energy associated with objects that we can observe without the aid of microscopes. The total mechanical energy $E$ is conserved (is a constant) for systems not subject to dissipative (non conservative) forces, like kinetic friction forces, or

$$E_{mech}=\Delta KE +\Delta PE = Constant$$

The usefulness of this equation is that it allows us to analyze the motion of objects such as the motion of a mass on an ideal springs, the motion of a bob of an ideal (frictionless) pendulum, falling objects in a vacuum, objects sliding down frictionless incline planes, and so forth.

Can we really say that any lost 𝐾𝐸 or 𝑃𝐸 is conserved by becoming the other?

We can only say that for a system that is not subject to dissipate (non conservative) forces such as kinetic friction. For all real systems some of the mechanical energy will be lost due to friction. For example a real pendulum will gradually slow down losing both gravitational potential and kinetic energy that is lost in the form of heat. Total energy is still conserved, but part of it is no longer mechanical energy. The lost mechanical mechanical becomes an increase in the microscopic internal energy of those parts subjected to friction and is typically reflected in an increase in temperature of those parts. In order to account for the conservation of the total energy of an isolated system the equation becomes

$$E_{total}=\Delta KE +\Delta PE + \Delta U$$

Where $\Delta U$ is the change in internal (microscopic) energy of the system. The change in the internal microscopic energy of the system is the sum of the changes in microscopic kinetic and potential energy.

I guess I am just not convinced of why, in the case of a conservative force, any lost KE or PE is conserved as the other. Can you offer any explanation for why that is?

Conservative forces are called "conservative" because when they do work total mechanical energy (PE + KE) is conserved. Non conservative force dissipate mechanical energy into other energy forms (heat, sound, deformation, etc.).

Refer to the figures below, taken from the Lumen Physics website:

https://courses.lumenlearning.com/physics/chapter/7-5-nonconservative-forces

Figure (a) shows how mechanical energy is conserved for the case of an object falling onto and rebounding from an ideal spring. The conservative forces involved are gravity and the force of an ideal spring. Note that the total energy of the object varies from being completely gravitational potential energy before falling, to a combination of gravitational potential energy and kinetic energy on the way to and from the spring, to being completely spring potential energy when the object stops with the spring compressed. The total mechanical energy is constant at all times.

In figure (b) the object impacts the ground and comes to a stop. Unlike the spring force in figure (a), the force exerted by the ground is not a conservative force because it dissipates the kinetic energy the object has just prior to impact in the form of heat, sound and deformation energy. Thus mechanical energy is not conserved.

Hope this helps.