What is the dimension of kinetic energy

Mechanical work in one dimension

Subsections

You move an object along the route with the constant force that's the way to do the job

   

done. If the force is not constant, the distance is divided into infinitesimally small sections and receives the work for each section

   

The individual parts of the work are additive, so you get the definition of the work

The mechanical work is
(3.56)


Acceleration work or kinetic energy

We now ask ourselves: what is the effort to get a constant mass from the impulse on the impulse bring to? The effort, the acceleration work, depends on two variables

  • . With the change in momentum, we also change the speed or the mass or both.
  • The effort must depend on the route.

We call the effort the kinetic energy. Using the definition of the work equation (3.34) we write:

(3.57)

From the experiment and the definition of the momentum we know that or is. But now is also

   

and therefore

   

At the same time, the integration limits change from , to ,. So we have

(3.58)

that is, the work to get a constant mass from 0 to the momentum is to bring . This work must be viewed as kinetic energy. It is in the movement of the masses. If the mass is variable, the mass can always be viewed temporarily as constant and the kinetic energy can be calculated using the above procedure.

kinetic energy
(3.59)

The unit of kinetic energy is:

Potential energy

By potential energy we understand the possibility of doing work, excluding the energy that is in motion. Is work in the physical sense

(3.60)

So we are only looking at the component of forcerunning along the path element lies.

Now the force that the system brings up is the force that we have to work against . The energy stored in the system is therefore

(3.61)

Thus the potential energy is defined by

(3.62)

The unit of potential energy is:

Energy conservation of mechanical systems in one dimension

We consider a system whose energy is constant.

(3.63)

It is the as yet unspecified internal energy of a particle. For mass points is .

The constancy of all energy means that their time derivative must be zero

(3.64)

This equation is an expression of Hamilton's principle that the total energy is constant. In detail you have

(3.65)

Let us now assume that the internal energy is constant (e.g. mass points). Then

(3.66)

One-dimensional special case: linear in

We consider a one-dimensional problem and assume that

be. Then is the equation of motion

or with )

Is rewritten
and with
(3.67)

This equation of motion is also known as Newton's 2nd axiom or Newton's 2nd law of force. The derivation shows, however, that this law, useful as it may sometimes be, is not a fundamental law, but a law derived from the symmetry relationships of space Emmy Noether, [Noe18].


Work and performance

Example lever

The great way Force, that is, work, is retained with the lever.


It is the way along the railway!

so



Example:

Circular path

The unity of work is

In the general three-dimensional case, the work depends from the running track from.


Example:

Air resistance

   

When the acceleration is constant, applies

   

Then

   


Example:

In terms of sliding friction, we have


This means that, as expected, the work is proportional to the distance covered.

When calculating the work, time does not matter. If we want to take into account the time in which a job is done, we speak of performance.

Definition of performance
(3.72)

Equation (3.50) can be rewritten with the definition of work:


We used for the transformation that the derivative according to the upper limit (the lower is constant here) of an integral is the integrand itself. We get rewritten

(3.74)

The unit of performance is

(3.75)

Potential energy and forces

From the definition of potential energy we see that

(3.76)

The proof is:

   
  (3.77)

Balance and stability

In order to investigate the stability of an equilibrium position, we consider the three possible courses of the potential energy with a spatial coordinate



Equilibrium and potential energy




  1. In the event of a deflection, there is a restoring force: we have a stable equilibrium

    Condition is: , or

  2. With a deflection there is an increasing force towards the outside: we have an unstable equilibrium

    Condition is: , or

  3. In the event of a deflection, the mass is still in equilibrium: we have an indifferent equilibrium

    Condition is: , or

In 3 dimensions a mass point is in equilibrium if is.

Othmar Marti
Experimental physics
Ulm University