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S2.2 Force and mass: Newton`s 2nd and 3rd laws

[A] Preamble

[B] The 2nd and 3rd laws

Key Point 2.2

Newton’s 2nd Law: The acceleration $\vec{a}$ a body displays (with respect to an inertial reference frame) is related to the net external force $\vec{F}$ it experiences by
\[ \vec{F}=m\vec{a} \]
where m is its (inertial) mass. This is a vector equation.

Key Point 2.3

Newton’s 3rd Law: The force $\vec{F}_{12}$ exerted on body 1 by body 2 is equal in magnitude but opposite in direction to the force $\vec{F}_{21}$ exerted on body 2 by body 1:
\[ \vec{F}_{12} = - \vec{F}_{21} \]
The two forces constitute an ‘action-reaction pair’. They act on different bodies.

Commentary

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Experiment 1

 

Graphic - No title - massexperiment

  • 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:
    \[ {\vec{a}_{12} = - \gamma \vec{a}_{21} } \]
    where
    • $\vec{a}_{12}$ is acceleration of 1 due to 2
    • $\vec{a}_{21}$ 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

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

 

Graphic - No title - forceexperiment

  • Consider one body, with variable m, subjected to ‘constant external influence.’
  • Empirical observation:
    • acceleration has constant direction
    • magnitude varies inversely with m
  • Explicit formulation:

    \[ m\vec{a} = \mbox{{ \em constant vector}} \]

    (for constant external influence)

  • Definition of force: Associate with the ‘external influence’ a force $\vec{F}$ defined by
    \[ {\vec{F}=m\vec{a}} \]
  • Return to
    \[ \vec{a_{12}} = {- \gamma \vec{a_{21}}} \]
    and substitute
    γ=m2/m1
  • Then $ m_1\vec{a}_{12} = - m_2 \vec{a}_{21}$   implies
    \[ {\vec{F}_{12}= -\vec{F}_{21}} \]
 

Reassurance?

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 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

Textbook: HRW Chapter 5.4-6, 5.8
Self-Test Questions: