S2.2 Force and mass: Newton`s 2nd and 3rd laws
[A] Preamble
- We introduce the concept of force to quantify external influence on acceleration.
- We introduce the concept of mass to quantify inertia (resistance to acceleration).
- 2nd and 3rd laws together:
- allow us to define force and mass
- provide framework for explaining motion under given forces.
[B] The 2nd and 3rd laws
Key Point 2.2
Newton’s 2nd Law: The acceleration

![\[ \vec{F}=m\vec{a} \]](mastermathpng-2.png)
Key Point 2.3
Newton’s 3rd Law: The force

![\[ \vec{F}_{12} = - \vec{F}_{21} \]](mastermathpng-5.png)
Commentary
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Experiment 1
- Consider two bodies effectively isolated from all influences except one anothers.
- Empirical observation:
- magnitudes of accelerations have constant ratio
- directions of accelerations are always opposite
- Explicit formulation:
where
is acceleration of 1 due to 2
is acceleration of 2 due to 1
- γ is a positive constant
- Definition of mass: Associate with each body a property, mass, such that
γ=m2/m1
- Then mass of any body (m2) can be related to unit of mass (m1, say) by measuring γ.

Weighing astronauts
It might occur to you to wonder if you can ’do’ experiments like this. And you can; in fact it is one way that astronauts in (zero-gravity) orbit for long periods of time can keep an eye on their mass (not ’weight’). You can find out more about how NASA have done this experiment and see a photo of it being used.Experiment 2
- Consider one body, with variable m, subjected to ‘constant external influence.’
- Empirical observation:
- acceleration has constant direction
- magnitude varies inversely with m
Explicit formulation:
(for constant external influence)
- Definition of force:
Associate with the
‘external influence’ a force
defined by
- Return to
and substituteγ=m2/m1
- Then
implies

Reassurance?
At this point you probably feel that Newton’s Laws are not as straightforward as you thought. Indeed their logical structure is not at all what first meets the eye. But the ‘bottom lines’ (the equations!) are the same as ever. The main point now is that you learn how to apply them correctly.
The scale of forces in Nature
The range of mangnitudes of forces in Nature is huge. Here are some examples:- The gravitational pull of the Sun on the Earth : ~ 1022N
- The thrust of rocket engines : ~ 1010N
- The pull of a large locomotive : ~ 106N
- The force decelerating a car during sharp braking : ~ 104N
- The force between two protons in a nucleus : ~ 104N
- The gravitational pull on you : ~ 103N
- The gravitational pull on a 5p piece : ~ 10-2N
- The force between the electron and proton in a hydrogen atom : ~ 10-7N
- The force deflecting an atomic force microscope tip : ~ 10-12N
Learning Resources
![]() | HRW Chapter 5.4-6, 5.8 |
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