Laws Newton's laws of motion
1 laws
1.1 newton s first law
1.2 newton s second law
1.2.1 impulse
1.2.2 variable-mass systems
1.3 newton s third law
laws
newton s first law
explanation of newton s first law , reference frames (mit course 8.01)
the first law states if net force (the vector sum of forces acting on object) zero, velocity of object constant. velocity vector quantity expresses both object s speed , direction of motion; therefore, statement object s velocity constant statement both speed , direction of motion constant.
the first law can stated mathematically when mass non-zero constant, as,
∑
f
=
0
⇔
d
v
d
t
=
0.
{\displaystyle \sum \mathbf {f} =0\;\leftrightarrow \;{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=0.}
consequently,
an object @ rest stay @ rest unless force acts upon it.
an object in motion not change velocity unless force acts upon it.
this known uniform motion. object continues whatever happens doing unless force exerted upon it. if @ rest, continues in state of rest (demonstrated when tablecloth skilfully whipped under dishes on tabletop , dishes remain in initial state of rest). if object moving, continues move without turning or changing speed. evident in space probes continuously move in outer space. changes in motion must imposed against tendency of object retain state of motion. in absence of net forces, moving object tends move along straight line path indefinitely.
newton placed first law of motion establish frames of reference other laws applicable. first law of motion postulates existence of @ least 1 frame of reference called newtonian or inertial reference frame, relative motion of particle not subject forces straight line @ constant speed. newton s first law referred law of inertia. thus, condition necessary uniform motion of particle relative inertial reference frame total net force acting on zero. in sense, first law can restated as:
in every material universe, motion of particle in preferential reference frame Φ determined action of forces total vanished times when , when velocity of particle constant in Φ. is, particle @ rest or in uniform motion in preferential frame Φ continues in state unless compelled forces change it.
newton s first , second laws valid in inertial reference frame. reference frame in uniform motion respect inertial frame inertial frame, i.e. galilean invariance or principle of newtonian relativity.
newton s second law
explanation of newton s second law, using gravity example (mit ocw)
the second law states rate of change of momentum of body directly proportional force applied, , change in momentum takes place in direction of applied force.
f
=
d
p
d
t
=
d
(
m
v
)
d
t
.
{\displaystyle \mathbf {f} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}={\frac {\mathrm {d} (m\mathbf {v} )}{\mathrm {d} t}}.}
the second law can stated in terms of object s acceleration. since newton s second law valid constant-mass systems, m can taken outside differentiation operator constant factor rule in differentiation. thus,
f
=
m
d
v
d
t
=
m
a
,
{\displaystyle \mathbf {f} =m\,{\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}=m\mathbf {a} ,}
where f net force applied, m mass of body, , body s acceleration. thus, net force applied body produces proportional acceleration. in other words, if body accelerating, there force on it. application of notation derivation of g subscript c.
consistent first law, time derivative of momentum non-zero when momentum changes direction, if there no change in magnitude; such case uniform circular motion. relationship implies conservation of momentum: when net force on body zero, momentum of body constant. net force equal rate of change of momentum.
any mass gained or lost system cause change in momentum not result of external force. different equation necessary variable-mass systems (see below).
newton s second law approximation increasingly worse @ high speeds because of relativistic effects.
impulse
an impulse j occurs when force f acts on interval of time Δt, , given by
j
=
∫
Δ
t
f
d
t
.
{\displaystyle \mathbf {j} =\int _{\delta t}\mathbf {f} \,\mathrm {d} t.}
since force time derivative of momentum, follows that
j
=
Δ
p
=
m
Δ
v
.
{\displaystyle \mathbf {j} =\delta \mathbf {p} =m\delta \mathbf {v} .}
this relation between impulse , momentum closer newton s wording of second law.
impulse concept used in analysis of collisions , impacts.
variable-mass systems
variable-mass systems, rocket burning fuel , ejecting spent gases, not closed , cannot directly treated making mass function of time in second law; is, following formula wrong:
f
n
e
t
=
d
d
t
[
m
(
t
)
v
(
t
)
]
=
m
(
t
)
d
v
d
t
+
v
(
t
)
d
m
d
t
.
(
w
r
o
n
g
)
{\displaystyle \mathbf {f} _{\mathrm {net} }={\frac {\mathrm {d} }{\mathrm {d} t}}{\big [}m(t)\mathbf {v} (t){\big ]}=m(t){\frac {\mathrm {d} \mathbf {v} }{\mathrm {d} t}}+\mathbf {v} (t){\frac {\mathrm {d} m}{\mathrm {d} t}}.\qquad \mathrm {(wrong)} }
the falsehood of formula can seen noting not respect galilean invariance: variable-mass object f = 0 in 1 frame seen have f ≠ 0 in frame. correct equation of motion body mass m varies time either ejecting or accreting mass obtained applying second law entire, constant-mass system consisting of body , ejected/accreted mass; result is
f
+
u
d
m
d
t
=
m
d
v
d
t
{\displaystyle \mathbf {f} +\mathbf {u} {\frac {\mathrm {d} m}{\mathrm {d} t}}=m{\mathrm {d} \mathbf {v} \over \mathrm {d} t}}
where u velocity of escaping or incoming mass relative body. equation 1 can derive equation of motion varying mass system, example, tsiolkovsky rocket equation. under conventions, quantity u dm/dt on left-hand side, represents advection of momentum, defined force (the force exerted on body changing mass, such rocket exhaust) , included in quantity f. then, substituting definition of acceleration, equation becomes f = ma.
newton s third law
an illustration of newton s third law in 2 skaters push against each other. first skater on left exerts normal force n12 on second skater directed towards right, , second skater exerts normal force n21 on first skater directed towards left.
the magnitudes of both forces equal, have opposite directions, dictated newton s third law.
a description of newton s third law , contact forces
the third law states forces between 2 objects exist in equal magnitude , opposite direction: if 1 object exerts force fa on second object b, b simultaneously exerts force fb on a, , 2 forces equal in magnitude , opposite in direction: fa = −fb. third law means forces interactions between different bodies, or different regions within 1 body, , there no such thing force not accompanied equal , opposite force. in situations, magnitude , direction of forces determined entirely 1 of 2 bodies, body a; force exerted body on body b called action , , force exerted body b on body called reaction . law referred action-reaction law, fa called action , fb reaction . in other situations magnitude , directions of forces determined jointly both bodies , isn t necessary identify 1 force action , other reaction . action , reaction simultaneous, , not matter called action , called reaction; both forces part of single interaction, , neither force exists without other.
the 2 forces in newton s third law of same type (e.g., if road exerts forward frictional force on accelerating car s tires, frictional force newton s third law predicts tires pushing backward on road).
from conceptual standpoint, newton s third law seen when person walks: push against floor, , floor pushes against person. similarly, tires of car push against road while road pushes on tires—the tires , road simultaneously push against each other. in swimming, person interacts water, pushing water backward, while water simultaneously pushes person forward—both person , water push against each other. reaction forces account motion in these examples. these forces depend on friction; person or car on ice, example, may unable exert action force produce needed reaction force.
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