# How can an external force acting on a particle be a Newton 3 pair with a conservative force?

If I particle is moved from a point A to a point B (both of which lie in a conservative force field - like a gravitational field for example), and to avoid confusion lets just say that B has a higher potential than A.

I understand that I am doing positive work on the particle - because I am transferring energy TO the particle. And in this case I understand that the conservative force must be doing negative work on the particle since $$\Delta U=-W=-\Delta T$$ where $\Delta T$ is the change in the kinetic energy of the particle, and if the particle is gaining potential $\implies$ $\Delta U$ is positive so negative work must be done.

But what I don't understand is how the force provided by me moving the particle from A to B forms a Newton 3 pair with the conservative force since Newton three pairs only apply to two particles that are interacting with each other and must be they must be a force of the same type - me moving a particle is not the same as the gravitational force on that particle (since the weight of the particle and the weight of the earth are Newton three pairs)

Are there other forces acting on the particle that link together the negative work done by the gravitational field and the positive work done by me that I have completely skipped out on? Or is my question completely wrong? Thanks!

Your reasoning about the work is nearly correct. You basically need to split up that equation into two equations and specify what work you are talking about.

With $\Delta U=-W$ , the work here is just referring to the work done by the conservative force. If the conservative force does positive work, then the particle is losing potential energy.

With $W=\Delta T$ , the work is the total work done on the particle. In our scenario this is the sum of the work done by us and the work done by the conservative force. This means that the particle can be gaining potential energy but maintain a constant kinetic energy if we supply an amount of work equal to the amount of potential energy gained.

As for the force pairs from Newton's third law, let's say that the conservative force is gravity from Earth. Then there is one force pair between the particle and the Earth (each exerts a gravitational force on the other). The other pair is between you and the particle. Whatever force you apply to the particle, the particle will push back with and equal but opposite force.

It seems like there is some confusion between these force pairs and work. If you are only interested in the work done on the particle, then you only focus on forces acting on that particle. Any forces that the particle exerts on other objects does not need to be considered in thinking about the work done just on the particle itself.

• so how would the force applied by me on the particle act has a newton third pair with gravity? – BigWig May 25 '18 at 16:13
• It doesn't. Those are two different interactions. You have the force of gravity between the particle and the Earth. You have the force of interaction between yourself and the particle. The force you apply and the gravitational force are two independent forces. – Aaron Stevens May 25 '18 at 16:14
• okay I can understand that, but would the positive work done by me be equal to the negative work done by the force then? since the particle is gaining potential as it moves up in space (away from the earth) – BigWig May 25 '18 at 16:16
• Not necessarily. This would be true only if the speed of the particle is not changing. Therefore, there is no net work being done, so the work by the 2 forces must cancel out. – Aaron Stevens May 25 '18 at 16:19
• @BigWig I fixed my answer to discuss these energy details. – Aaron Stevens May 26 '18 at 4:49