Notes on Relativity and Cosmology - Physics Department, UCSB

1.7. NEWTON’S OTHER LAWS 37
Newton’s Second Law: In any inertial frame,
(net force on an object) = (mass of object)(acceleration of object)
F = ma.
The phrase “in any inertial frame” above means that the acceleration must be
measured relative to an inertial frame of reference. By the way, part of your
homework will be to show that calculating the acceleration of one object in any
two inertial frames always yields identical results. Thus, we may speak about
acceleration ‘relative to the class of inertial frames.’
⋆ ⋆ ⋆ Note: We assume that force and mass are independent of the reference
frame.
On the other hand, Newton’s third law addresses the relationship between two
forces.
Newton’s Third Law: Given two objects (A and B), we have
(force from A on B at some time t) = - (force from B on A at some time t)
⋆⋆ This means that the forces have the same size but act in opposite directions.
Now, this is not yet the end of the story. There are also laws that tell us what
the forces actually are. For example, Newton’s Law of Universal Gravitation
says:
Given any two objects A and B, there is a gravitational force between them
(pulling each toward the other) of magnitude
F AB = G m Am B
d 2 AB
with G = 6.673 × 10 −11 Nm 2 /kg 2 .
Important Observation: These laws hold in any inertial frame. As a result, there
is no special inertial frame that is any different from the others. As a result,
it makes no sense to talk about one inertial frame being more ‘at rest’ than
any other. You could never find such a frame, so you could never construct
an operational definition of ‘most at rest.’ Why then, would anyone bother to
assume that a special ‘most at rest frame’ exists?
Note: As you will see in the reading, Newton discussed something called ‘Absolute
space.’ However, he didn’t need to and no one really believed in it. We
will therefore skip this concept completely and deal with all inertial frames on
an equal footing.
The above observation leads to the following idea, which turns out to be much
more fundamental than Newton’s laws.