# Conservation of Mechanical Energy Requirement

• fog37
In summary, the conservation of mechanical energy occurs when the net work done by non-conservative forces is zero, while conservation of total momentum happens when the external net force is zero. Non-conservative forces can produce zero work if they act perpendicular to the object's path or if their algebraic signs produce a net zero work. A conservative force can be expressed as a scalar potential function, but a time-dependent force cannot be considered conservative because its curl with respect to space coordinates is not zero.
fog37
Hello Forum,
Conservation of mechanical energy ME= KE+PE happens when the net work done by the non conservative forces is zero. Conservation of total momentum, instead, happens when the external net force is zero (or close to zero).

In the case of mechanical energy, the non-conservative forces are also external forces to the system and can be nonzero. If the other external forces are conservative, the ME is conserved but not momentum.
But what we care about is the work they produce. How can nonconservative forces produce zero work? I guess their work may be finite but negligibly small. Maybe the total sum of the work done by the nonconservative force is zero...

Is that correct?

fog37

First you should clarify the definitions. A conservative force is given by a time-independnet scalar potential,
$$\vec{F}(\vec{x})=-\vec{\nabla} V(\vec{x}).$$
Then you take Newton's 2nd Law, multiply it by ##\dot{\vec{x}}## and integrate over time to get the energy-conservation law
$$\frac{m}{2} \dot{\vec{x}}^2+V(\vec{x})=\text{const}.$$
Then there are forces that do no work. An example is the motion in an external magnetic field. The Lorentz force is always perpendicular to ##\dot{\vec{x}}## and thus it doesn't contribute to the work.
$$\vec{F} \cdot \dot{\vec{x}}=q (\dot{\vec{x}} \times \vec{B}) \cdot \dot{\vec{x}}=0.$$

Sure, so there are two types of forces: conservative and non-conservative. Both types of force can perform work. What matters for the mechanical energy to be conserved is that the net work done by the nonconservative forces is zero. How can that happen? If the nonconservative forces act perpendicularly to the object's path or if the algebraic signs of the works produce a net zero work...

As you mention, conservative forces admit a potential function V(x), which is a function of position. Can a time-dependent force be conservative? I don't think so. But why? What is the rigorous argument to explain that?

Thanks!
fog37

Conservative force means that we can define a scalar potential function for that force. To be able to define a scalar function for a vector field the curl (wrt space coordinates) of the vector field should be 0 because curl of a gradient should be always 0. Since force can be written as gradient when we take curl of it, we should obtain 0.

Sure, wherever the curl F = 0 we can express F= Delta V, i.e. we can use a scalar potential V.

but what if the force F was F(t), i.e. a function of time?

I think you cannot talk about curl of a time dependent function (without fixing time) because now your force is four vector but curl is defined on 3D. Also notice that if you define a potential when you follow different paths in space between two points, you may not reach same potential due to time dependence which makes your force path dependent . So I don't think a time dependent force is conservative. However, in a fixed time your force can be considered as a conservative force if the curl of it wrt space coordinates 0.

## What is the Conservation of Mechanical Energy Requirement?

The Conservation of Mechanical Energy Requirement is a fundamental principle in physics that states that energy cannot be created or destroyed, only transferred or converted from one form to another.

## Why is the Conservation of Mechanical Energy Requirement important?

This principle is important because it allows us to accurately predict the behavior of objects and systems in motion. It also helps us understand the relationship between different forms of energy, such as kinetic and potential energy.

## How is the Conservation of Mechanical Energy Requirement applied?

The Conservation of Mechanical Energy Requirement is applied by considering the total amount of mechanical energy (kinetic + potential) in a system. This total energy remains constant as long as there are no external forces acting on the system.

## What are some examples of the Conservation of Mechanical Energy Requirement in action?

Some examples of the Conservation of Mechanical Energy Requirement include a pendulum swinging back and forth, a roller coaster moving along its track, and a bouncing ball. In all of these cases, the total mechanical energy remains constant throughout the motion.

## Are there any exceptions to the Conservation of Mechanical Energy Requirement?

In some cases, external forces such as friction or air resistance can affect the total mechanical energy of a system. However, these forces are typically small and can be accounted for in calculations. Additionally, the Conservation of Mechanical Energy Requirement does not apply in situations involving nuclear reactions or extreme gravitational forces.

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