pressure) will undergo an equal change, in its own motion, toward the contrary part. The
changes made by these actions are equal, not in the velocities but in the motions of the
bodies; that is to say, if the bodies are not hindered by any other impediments. For, as the
motions are equally changed, the changes of the velocities made toward contrary parts are
reciprocally proportional to the bodies. This law takes place also in attractions, as will be
proved in the next scholium” (Fairlie and Cayley 1965).
In the above, as usual, motion is Newton’s name for momentum, hence his careful
distinction between motion and velocity.
Newton used the third law to derive the law of conservation of momentum
(Cohen 1995), but from a deeper perspective, however, conservation of momentum
is the more fundamental idea (derived via Noether’s theorem from Galilean invariance), and holds in cases where Newton’s third law appears to fail, for instance,
when force fields as well as particles carry momentum, and in quantum mechanics:
F 12 ¼ F 21 ,
An illustration of Newton’s third law in which two people push against each
other. The first person on the left exerts a normal force F 12 on the second person
directed towards the right, and the second person exerts a normal force F 21 on the
first person directed toward the left.
The magnitudes of both forces are equal, but they have opposite directions, as
dictated by Newton’s third law.
The third law states that all forces between two objects exist in equal magnitude
and opposite direction: if one object A exerts a force F A on a second object B, then B
simultaneously exerts a force F B on A, and the two forces are equal in magnitude and
opposite in direction: F A ¼ ÀF B . The third law means that all forces are interactions
between different bodies, or different regions within one body, and thus that there is
no such thing as a force that is not accompanied by an equal and opposite force. In
some situations, the magnitude and direction of the forces are determined entirely by
one of the two bodies, say Body A; the force exerted by Body A on Body B is called
the “action,” and the force exerted by Body B on Body A is called the “reaction.”
This law is sometimes referred to as the action-reaction law, with F A called the
“action” and F B the “reaction.” In other situations, the magnitude and directions of
the forces are determined jointly by both bodies, and it is not necessary to identify
one force as the “action” and the other as the “reaction.” The action and the reaction
are simultaneous, and it does not matter which is called the action and which is called
the reaction; both forces are part of a single interaction, and neither force exists
without the other (Anon 2020; Bernard Cohen 1967).
The two forces in Newton’s third law are of the same type (e.g., if the road exerts
a forward frictional force on an accelerating car’s tires, then it is also a frictional
force that Newton’s third law predicts for the tires pushing backward on the road).
From a conceptual standpoint, Newton’s third law is seen when a person walks:
they push against the floor, and the floor pushes against the person. Similarly, the
tires of a car apply a shear force on the road, while the road pushes back on the tires
with a reaction shear force—the tires and road simultaneously push against each
10
2 Stress and Strain in Continuum
changes made by these actions are equal, not in the velocities but in the motions of the
bodies; that is to say, if the bodies are not hindered by any other impediments. For, as the
motions are equally changed, the changes of the velocities made toward contrary parts are
reciprocally proportional to the bodies. This law takes place also in attractions, as will be
proved in the next scholium” (Fairlie and Cayley 1965).
In the above, as usual, motion is Newton’s name for momentum, hence his careful
distinction between motion and velocity.
Newton used the third law to derive the law of conservation of momentum
(Cohen 1995), but from a deeper perspective, however, conservation of momentum
is the more fundamental idea (derived via Noether’s theorem from Galilean invariance), and holds in cases where Newton’s third law appears to fail, for instance,
when force fields as well as particles carry momentum, and in quantum mechanics:
F 12 ¼ F 21 ,
An illustration of Newton’s third law in which two people push against each
other. The first person on the left exerts a normal force F 12 on the second person
directed towards the right, and the second person exerts a normal force F 21 on the
first person directed toward the left.
The magnitudes of both forces are equal, but they have opposite directions, as
dictated by Newton’s third law.
The third law states that all forces between two objects exist in equal magnitude
and opposite direction: if one object A exerts a force F A on a second object B, then B
simultaneously exerts a force F B on A, and the two forces are equal in magnitude and
opposite in direction: F A ¼ ÀF B . The third law means that all forces are interactions
between different bodies, or different regions within one body, and thus that there is
no such thing as a force that is not accompanied by an equal and opposite force. In
some situations, the magnitude and direction of the forces are determined entirely by
one of the two bodies, say Body A; the force exerted by Body A on Body B is called
the “action,” and the force exerted by Body B on Body A is called the “reaction.”
This law is sometimes referred to as the action-reaction law, with F A called the
“action” and F B the “reaction.” In other situations, the magnitude and directions of
the forces are determined jointly by both bodies, and it is not necessary to identify
one force as the “action” and the other as the “reaction.” The action and the reaction
are simultaneous, and it does not matter which is called the action and which is called
the reaction; both forces are part of a single interaction, and neither force exists
without the other (Anon 2020; Bernard Cohen 1967).
The two forces in Newton’s third law are of the same type (e.g., if the road exerts
a forward frictional force on an accelerating car’s tires, then it is also a frictional
force that Newton’s third law predicts for the tires pushing backward on the road).
From a conceptual standpoint, Newton’s third law is seen when a person walks:
they push against the floor, and the floor pushes against the person. Similarly, the
tires of a car apply a shear force on the road, while the road pushes back on the tires
with a reaction shear force—the tires and road simultaneously push against each
10
2 Stress and Strain in Continuum