- Source: Central force
- Source: Central Force
In classical mechanics, a central force on an object is a force that is directed towards or away from a point called center of force.: 93
F
→
=
F
(
r
)
=
|
F
(
r
)
|
r
^
{\displaystyle {\vec {F}}=\mathbf {F} (\mathbf {r} )=\left\vert F(\mathbf {r} )\right\vert {\hat {\mathbf {r} }}}
where
F
→
{\textstyle {\vec {F}}}
is the force, F is a vector valued force function, F is a scalar valued force function, r is the position vector, ||r|| is its length, and
r
^
=
r
/
‖
r
‖
{\textstyle {\hat {\mathbf {r} }}=\mathbf {r} /\|\mathbf {r} \|}
is the corresponding unit vector.
Not all central force fields are conservative or spherically symmetric. However, a central force is conservative if and only if it is spherically symmetric or rotationally invariant.: 133–38
Properties
Central forces that are conservative can always be expressed as the negative gradient of a potential energy:
F
(
r
)
=
−
∇
V
(
r
)
, where
V
(
r
)
=
∫
|
r
|
+
∞
F
(
r
)
d
r
{\displaystyle \mathbf {F} (\mathbf {r} )=-\mathbf {\nabla } V(\mathbf {r} )\;{\text{, where }}V(\mathbf {r} )=\int _{|\mathbf {r} |}^{+\infty }F(r)\,\mathrm {d} r}
(the upper bound of integration is arbitrary, as the potential is defined up to an additive constant).
In a conservative field, the total mechanical energy (kinetic and potential) is conserved:
E
=
1
2
m
|
r
˙
|
2
+
1
2
I
|
ω
|
2
+
V
(
r
)
=
constant
{\displaystyle E={\tfrac {1}{2}}m|\mathbf {\dot {r}} |^{2}+{\tfrac {1}{2}}I|{\boldsymbol {\omega }}|^{2}+V(\mathbf {r} )={\text{constant}}}
(where 'ṙ' denotes the derivative of 'r' with respect to time, that is the velocity,'I' denotes moment of inertia of that body and 'ω' denotes angular velocity), and in a central force field, so is the angular momentum:
L
=
r
×
m
r
˙
=
constant
{\displaystyle \mathbf {L} =\mathbf {r} \times m\mathbf {\dot {r}} ={\text{constant}}}
because the torque exerted by the force is zero. As a consequence, the body moves on the plane perpendicular to the angular momentum vector and containing the origin, and obeys Kepler's second law. (If the angular momentum is zero, the body moves along the line joining it with the origin.)
It can also be shown that an object that moves under the influence of any central force obeys Kepler's second law. However, the first and third laws depend on the inverse-square nature of Newton's law of universal gravitation and do not hold in general for other central forces.
As a consequence of being conservative, these specific central force fields are irrotational, that is, its curl is zero, except at the origin:
∇
×
F
(
r
)
=
0
.
{\displaystyle \nabla \times \mathbf {F} (\mathbf {r} )=\mathbf {0} .}
Examples
Gravitational force and Coulomb force are two familiar examples with
F
(
r
)
{\displaystyle F(\mathbf {r} )}
being proportional to 1/r2 only. An object in such a force field with negative
F
(
r
)
{\displaystyle F(\mathbf {r} )}
(corresponding to an attractive force) obeys Kepler's laws of planetary motion.
The force field of a spatial harmonic oscillator is central with
F
(
r
)
{\displaystyle F(\mathbf {r} )}
proportional to r only and negative.
By Bertrand's theorem, these two,
F
(
r
)
=
−
k
/
r
2
{\displaystyle F(\mathbf {r} )=-k/r^{2}}
and
F
(
r
)
=
−
k
r
{\displaystyle F(\mathbf {r} )=-kr}
, are the only possible central force fields where all bounded orbits are stable closed orbits. However, there exist other force fields, which have some closed orbits.
See also
Classical central-force problem
Particle in a spherically symmetric potential
Notes
References
GHQ Central Force was a home command of the British Army during the First World War.
Central Force, based in London, was formed on 5 August 1914 under Sir Ian Hamilton, who had the title of Commander-in-Chief Home Army. This was a retitling of the post of Inspector-General of the Home Forces and did not imply command over all forces based in Great Britain. Central Force supported the British Expeditionary Force in France and was responsible for Territorial Force troops charged with coastal defence. Subordinate formations were First Army based at Bedford, Second Army at Aldershot (later at Tunbridge Wells) and Third Army at Luton. Also attached to Central Force were the 1st and 2nd Mounted Divisions, the West Riding Division and the Northumbrian Division.
Sir Leslie Rundle became C-in-C Home Army in March 1915 when Hamilton left to take command of the Mediterranean Expeditionary Force. GHQ Central Force was redesignated as GHQ Home Forces in December 1915 when Sir John French was appointed Commander-in-Chief, Home Forces.
References
Mitchinson. Defending Albion: Britain's Home Army 1908-1919. pp. 80, 123.
External links
Hamilton, Ian, Correspondence relating to service as Commander-in-Chief Central Force, Home Defence, 1914-15, Liddell Hart Centre for Military Archives
Kata Kunci Pencarian:
- Star Wars: The Force Awakens
- Area 51
- Angkatan Udara Amerika Serikat
- MediaCorp TV12 Central
- Central Acquisition Radar (3D-CAR)
- Canadian Expeditionary Force
- Central Aircraft Manufacturing Company
- Kanada Tengah
- Angkatan Udara Israel
- Britania Raya
- Central force
- Central Force
- Classical central-force problem
- Central Reserve Police Force
- Central Industrial Security Force
- Central-force problem
- Conservative force
- Central Air Force Museum
- Ninth Air Force
- Central Armed Police Forces
No More Posts Available.
No more pages to load.
