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“AT ANY [INSTANT] OF [TIME], THE [NET FORCE] ON A [BODY] IS EQUAL TO THE BODY’S [ACCELERATION] MULTIPLIED BY ITS [MASS] OR, EQUIVALENTLY, THE [RATE] AT WHICH THE BODY’S [MOMENTUM] IS CHANGING WITH [TIME]”
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Newton’s second law:
Explaining this one in words is hard, so for now I’ll stick to the mathematics.
Mathematically, it is usually taught as
In this form, it is incorrect in special relativity.
It can be fixed, though, simply by going back to Newton’s original definition (in modern notation) as
F = [“vector force” on object]
m = object’s [mass]
a = the resulting [acceleration]
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d = “delta” = change
p = “momentum”
t = “time”
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“momentum” is abbreviated “p” for “pulsus”
(“to push/strike”)
Using this form, we simply use the relativistic, rather than classical momentum, and the second law remains unchanged
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Second law
The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed
By “motion”, Newton meant the quantity now called momentum, which depends upon the amount of matter contained in a body, the speed at which that body is moving, and the direction in which it is moving
In modern notation, the momentum of a body is the product of its mass and its velocity:
p
m
v
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{\displaystyle \mathbf {p} =m\mathbf {v} \,,}where all three quantities can change over time. In common cases the mass
m
{\displaystyle m} does not change with time and the derivative acts only upon the velocity. Then force equals the product of the mass and the time derivative of the velocity, which is the acceleration:[22]
F
m
d
v
d
t
m
a
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{\displaystyle \mathbf {F} =m{\frac {d\mathbf {v} }{dt}}=m\mathbf {a} \,.}As the acceleration is the second derivative of position with respect to time, this can also be written
F
m
d
2
s
d
t
2
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{\displaystyle \mathbf {F} =m{\frac {d^{2}\mathbf {s} }{dt^{2}}}.}
Newton’s second law, in modern form, states that the time derivative of the momentum is the force:[23]: 4.1
F
d
p
d
t
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{\displaystyle \mathbf {F} ={\frac {d\mathbf {p} }{dt}}\,.}When applied to systems of variable mass, the equation above is only valid only for a fixed set of particles. Applying the derivative as in
F
m
d
v
d
t
+
v
d
m
d
t
(
i
n
c
o
r
r
e
c
t
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{\displaystyle \mathbf {F} =m{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}+\mathbf {v} {\frac {\mathrm {d} m}{\mathrm {d} t}}\ \ \mathrm {(incorrect)} }can lead to incorrect results.[24] For example, the momentum of a water jet system must include the momentum of the ejected water:[25]
F
e
x
t
d
p
d
t
−
v
e
j
e
c
t
d
m
d
t
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{\displaystyle \mathbf {F} {\mathrm {ext} }={\mathrm {d} \mathbf {p} \over \mathrm {d} t}-\mathbf {v} {\mathrm {eject} }{\frac {\mathrm {d} m}{\mathrm {d} t}}.}
A free body diagram for a block on an inclined plane, illustrating the normal force perpendicular to the plane (N), the downward force of gravity (mg), and a force f along the direction of the plane that could be applied, for example, by friction or a string
The forces acting on a body add as vectors, and so the total force on a body depends upon both the magnitudes and the directions of the individual forces.[23]: 58 When the net force on a body is equal to zero, then by Newton’s second law, the body does not accelerate, and it is said to be in mechanical equilibrium. A state of mechanical equilibrium is stable if, when the position of the body is changed slightly, the body remains near that equilibrium. Otherwise, the equilibrium is unstable.[15]: 121 [23]: 174
A common visual representation of forces acting in concert is the free body diagram, which schematically portrays a body of interest and the forces applied to it by outside influences.[26] For example, a free body diagram of a block sitting upon an inclined plane can illustrate the combination of gravitational force, “normal” force, friction, and string tension.[note 4]
Newton’s second law is sometimes presented as a definition of force, i.e., a force is that which exists when an inertial observer sees a body accelerating. This is sometimes regarded as a potential tautology — acceleration implies force, force implies acceleration. To go beyond tautology, an equation detailing the force might also be specified, like Newton’s law of universal gravitation. By inserting such an expression for
F
{\displaystyle \mathbf {F} } into Newton’s second law, an equation with predictive power can be written.[note 5]
Newton’s second law has also been regarded as setting out a research program for physics, establishing that important goals of the subject are to identify the forces present in nature and to catalogue the constituents of matter
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