# How Newton's third law works in a system with external force on it?

I got a little confused about Newton's third law. So if there is a system of two particles that interact mutually with each other and also this system is itself as a whole is acted upon by some external force then rate of change of momentum of the system is no more zero so how will it be possible to show that still force on particle 1 due to particle 2 is same and opposite to force on 2 due to 1?

• I'm not sure what you mean by "how will it be possible to show". Is your concern the fact that the ratio of acceleration to mass for the two particles will not be identical because of the influence of the external force? – garyp Mar 24 '19 at 20:51

Let the system under consideration be particle $$1$$ with mass $$m_1$$ and particle $$2$$ with mass $$m_2$$ and the two equal magnitude and opposite direction forces be $$\vec F_{\text{on 1 due to 2}}$$ and $$\vec F_{\text{on 2 due to 1}}$$.
An external force $$\vec F_{\rm external}$$ is applied to mass $$1$$.
Newton’s third law states that the two internal forces stay the same with $$\vec F_{\text{on 1 due to 2}}=-\vec F_{\text{on 2 due to 1}}$$ and if the particles start moving the magnitude and direction of these two forces may change but they will still have equal magnitude and opposite direction.
With an external force applied to particle $$1$$ Newton’s third law is still valid.
The net force on particle $$1$$ will be $$\vec F_{\text{on 1 due to 2}}+ \vec F_{\rm external}$$ and the force on particle $$2$$ will be $$\vec F_{\text{on 2 due to 1}}$$.
The linear acceleration of the centre of mass of the system will be $$\dfrac{\vec F_{\rm external}}{m_1+m_2}$$ and there will also be an angular acceleration of the system if the external force is applying a torque about the centre of mass of the system.