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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Force | 2/11 | https://en.wikipedia.org/wiki/Force | reference | science, encyclopedia | 2026-05-05T09:33:04.607426+00:00 | kb-cron |
Sir Isaac Newton described the motion of all objects using the concepts of inertia and force. In 1687, Newton published his magnum opus, Philosophiæ Naturalis Principia Mathematica. In this work Newton set out three laws of motion that have dominated the way forces are described in physics to this day. The precise ways in which Newton's laws are expressed have evolved in step with new mathematical approaches.
=== First law ===
Newton's first law of motion states that the natural behavior of an object at rest is to continue being at rest, and the natural behavior of an object moving at constant speed in a straight line is to continue moving at that constant speed along that straight line. The latter follows from the former because of the principle that the laws of physics are the same for all inertial observers, i.e., all observers who do not feel themselves to be in motion. An observer moving in tandem with an object will see it as being at rest. So, its natural behavior will be to remain at rest with respect to that observer, which means that an observer who sees it moving at constant speed in a straight line will see it continuing to do so.
=== Second law ===
According to the first law, motion at constant speed in a straight line does not need a cause. It is change in motion that requires a cause, and Newton's second law gives the quantitative relationship between force and change of motion. Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object. A modern statement of Newton's second law is a vector equation:
F
=
d
p
d
t
,
{\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}},}
where
p
{\displaystyle \mathbf {p} }
is the momentum of the system, and
F
{\displaystyle \mathbf {F} }
is the net (vector sum) force. If a body is in equilibrium, there is zero net force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an unbalanced force acting on an object it will result in the object's momentum changing over time. In common engineering applications the mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of momentum,
F
=
d
p
d
t
=
d
(
m
v
)
d
t
,
{\displaystyle \mathbf {F} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} \left(m\mathbf {v} \right)}{\mathrm {d} t}},}
where m is the mass and
v
{\displaystyle \mathbf {v} }
is the velocity. If Newton's second law is applied to a system of constant mass, m may be moved outside the derivative operator. The equation then becomes
F
=
m
d
v
d
t
.
{\displaystyle \mathbf {F} =m{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}.}
By substituting the definition of acceleration, the algebraic version of Newton's second law is derived:
F
=
m
a
.
{\displaystyle \mathbf {F} =m\mathbf {a} .}
=== Third law ===
Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if
F
1
,
2
{\displaystyle \mathbf {F} _{1,2}}
is the force of body 1 on body 2 and
F
2
,
1
{\displaystyle \mathbf {F} _{2,1}}
that of body 2 on body 1, then
F
1
,
2
=
−
F
2
,
1
.
{\displaystyle \mathbf {F} _{1,2}=-\mathbf {F} _{2,1}.}
This law is sometimes referred to as the action-reaction law, with
F
1
,
2
{\displaystyle \mathbf {F} _{1,2}}
called the action and
−
F
2
,
1
{\displaystyle -\mathbf {F} _{2,1}}
the reaction. Newton's third law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. The third law means that all forces are interactions between different bodies. and thus that there is no such thing as a unidirectional force or a force that acts on only one body. In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero:
F
1
,
2
+
F
2
,
1
=
0.
{\displaystyle \mathbf {F} _{1,2}+\mathbf {F} _{2,1}=0.}
More generally, in a closed system of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the center of mass of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system. Combining Newton's second and third laws, it is possible to show that the linear momentum of a system is conserved in any closed system. In a system of two particles, if
p
1
{\displaystyle \mathbf {p} _{1}}
is the momentum of object 1 and
p
2
{\displaystyle \mathbf {p} _{2}}
the momentum of object 2, then
d
p
1
d
t
+
d
p
2
d
t
=
F
1
,
2
+
F
2
,
1
=
0.
{\displaystyle {\frac {\mathrm {d} \mathbf {p} _{1}}{\mathrm {d} t}}+{\frac {\mathrm {d} \mathbf {p} _{2}}{\mathrm {d} t}}=\mathbf {F} _{1,2}+\mathbf {F} _{2,1}=0.}
Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained.