Importance of keeping constant velocity

Question

In finding work done by a spring or gravity (i.e conservative forces), when an external agent does work, why is it done so that velocity i.e kinetic energy remains constant? I know from Work energy theorem that $W_{\mathrm{net}}=\Delta K$. So if there is no conservative force and velocity stays constant,work done by external force will be equal to work done by conservative force. But what is the need of keeping $K$ constant? Why can't an external agent apply a force greater or equal to that of conservative force?

Answer 1

But what is the need of keeping $K$ constant?

It is not necessary that the velocity (and thus kinetic energy) be constant when the external agent does work as long as there are no dissipative (friction) forces involved and that there is no change in velocity (kinetic energy) between the start and end of the spring being compressed or extended by an external agent.

What happens in between does not matter. Per the work energy theorem, the net work done on an object equals its change in kinetic energy. When the initial and final kinetic energy is the same, the net work done by the external agent and spring is zero.

Why can't an external agent apply a force greater or equal to that of conservative force?

It can. As long as the force is varied in such a way that the initial and final kinetic energy is the same.

The gravity analogy is lifting an object of mass $m$ from rest on the ground and bringing it to rest at a height $h$ above the ground. To do this the external agent force had to initially be greater than the force of gravity to get the object moving, and at the end be less than the force of gravity to bring it to a stop at height $h$. The change in KE is zero. The work done by the agent is $+mgh$ and the work done by gravity is $-mgh$ for a net work of zero. Gravity takes the energy given the object by the external agent and stores it as gravitational potential energy of the object-earth system.

Similarly, in the case of the spring mass system where the mass begins and ends at rest (or begins and ends with the same velocity) the external agent force does positive work of $+\frac{1}{2}kx^2 $ and the spring does negative work of $-\frac{1}{2}kx^2$ for a net work of zero. The work done by the elastic force of the spring takes the energy given the object by the external agent and stores it as elastic potential energy of the spring mass system.

Hope this helps.

Answer 2

I think that the sort of situation you are talking about is moving a mass/charge in a gravitational/electric field from one position to another and evaluating the work done by an external force to do that.
As to whether the external force needs to be equal in magnitude and opposite in direction depends on the situation and indeed there are many situations when the external force is zero and all the work done is by the gravitational/electric field.
For example consider a system which consists of a book and the Earth with no external forces present.
If the book is released it will fall downwards (and the Earth will rise upwards a very small amount) and the internal gravitational forces (force on book due to Earth and force on Earth due to book) will do work and increase the kinetic energy of the book (and the Earth).

Now consider the book alone as the system then the external force is the force on the book due to the Earth.
Again as the book falls it gains kinetic energy but this time the external force (gravitational attraction) does the work.
There may be another external force acting on the book, you applying a force equal in magnitude but opposite in direction to the gravitational attraction.
Because there is no net force on the book, no net work is done on the book and the book would fall at a constant velocity.

But what is the need of keeping K constant? Why can't an external agent apply a force greater or equal to that of conservative force?
There is no necessity for this but sometimes you do want to just consider changes which do not include a change in the kinetic energy of a mass.

Answer 3

Let us restrict ourselves to one dimension, all the comments made here are generalized to three dimensions too, but if you want a derivation then please check this video out: Derivation of work energy theorem in 3 dimensions with motivation.

The basic equation for our particle is $$F=ma \tag1$$ Here $F$ is the net force, we are not assuming that it's conservative or not.

If we are given the force as a function of displacement from the particle then by applying the chain rule to acceleration we get $$F=m\frac{dv}{dt}\frac{dx}{dx}=mv\frac{dv}{dx}$$ $$\int F\,dx=\int mv\,dv$$ $$\int F\, dx=\frac12m(v^2-u^2)=\Delta K \tag2$$

If the force doesn't depend on the path then the integral in equation $2$ is path-independent. This allows us to introduce the concept of potential energy which we define as follows $$\phi=-\int F\,dx \tag3$$

This $\phi$ can be interpreted as work done by the same force as in equation $2$ but in the opposite direction. But we don't really need this interpretation, in fact, the negative sign in equation $3$ is purely conventional.

If you still want to define this potential in terms of external force then you need to keep $\Delta K=0$ otherwise, the value that you will get will depend on the magnitude of external force applied, which defeats the purpose of potential, since it was defined at first place due the nice property that it is unique at each point in space (well more accurately the potential difference between two points is unique).

Answer 4

What does mean by conservative field is that the force term in absence of external force by the system on its constitutes particles is given by gradient of a function called as potential.

So whenever external force do work on the system against its force by means of kinetic energy of particles, then that energy is stored as potential energy and as work done have some displacement, that changes position of potential and shows change in potential energy.

In absence of external force, the stored potential energy can do work on the particles and bring back system to its initial position. In case of spring or gravity if floor and particles are rigid and there is negligible friction in case of spring and gravity, there is transferring between potential and kinetic energy for long, oscillation.

Now question is about when in conservative force, work done by external agency is through kinetic energy, but as kinetic energy is converted into potential energy. So speed of particles not remain constant through out process then why constant speed is used for expression of kinetic energy. Answer is constant speed is not used but maximum speed amount to maximum of kinetic energy upto whatever point, generally minimum of available kinetic enetgy.

Answer 5

I'm not sure, I understand you correctly, but velocity and kinetic energy do not stay constant under a conservative force. What does stay constant is the sum of kinetic and potential energy, but usually there is kinetic energy transferring into potential energy at any given time. In the case of a bouncing object on a spring, for instance, there are moments when the object comes to rest and the spring is fully compressed (KE=0, PE=max), and other moments where the spring is in its resting position and the object is moving very fast (KE=max, PE=0). Or in the case of gravity, an object would steadily lose potential energy and gain kinetic energy (speed up) as it falls toward Earth.

Hope that helps!

Answer 6

Maybe you just want to ask why whenever a external work is done then that external agent just matches the conservative force in opposite direction giving it no velocity from our side so that particle could not attain a velocity otherwise. The conservative force apply a give an acceleration and that external agent just opposes it so as particle is at quasi statically at rest and we have not given any velocity to it and it does not attain any tendency to move further or deviated.