10 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Force | 7/11 | https://en.wikipedia.org/wiki/Force | reference | science, encyclopedia | 2026-05-05T09:33:04.607426+00:00 | kb-cron |
where
V
{\displaystyle V}
is the volume of the object in the fluid and
P
{\displaystyle P}
is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight. A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due to viscosity. For so-called "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction:
F
d
=
−
b
v
,
{\displaystyle \mathbf {F} _{\mathrm {d} }=-b\mathbf {v} ,}
where:
b
{\displaystyle b}
is a constant that depends on the properties of the fluid and the dimensions of the object (usually the cross-sectional area), and
v
{\displaystyle \mathbf {v} }
is the velocity of the object. More formally, forces in continuum mechanics are fully described by a stress tensor with terms that are roughly defined as
σ
=
F
A
,
{\displaystyle \sigma ={\frac {F}{A}},}
where
A
{\displaystyle A}
is the relevant cross-sectional area for the volume for which the stress tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the matrix diagonals of the tensor) as well as shear terms associated with forces that act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all strains (deformations) including also tensile stresses and compressions.
=== Fictitious ===
There are forces that are frame dependent, meaning that they appear due to the adoption of non-Newtonian (that is, non-inertial) reference frames. Such forces include the centrifugal force and the Coriolis force. These forces are considered fictitious because they do not exist in frames of reference that are not accelerating. Because these forces are not genuine they are also referred to as "pseudo forces". In general relativity, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry.
== Concepts derived from force ==
=== Rotation and torque ===
Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque of a force
F
{\displaystyle \mathbf {F} }
is defined relative to an arbitrary reference point as the cross product:
τ
=
r
×
F
,
{\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} ,}
where
r
{\displaystyle \mathbf {r} }
is the position vector of the force application point relative to the reference point. Torque is the rotation equivalent of force in the same way that angle is the rotational equivalent for position, angular velocity for velocity, and angular momentum for momentum. As a consequence of Newton's first law of motion, there exists rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's second law of motion can be used to derive an analogous equation for the instantaneous angular acceleration of the rigid body:
τ
=
I
α
,
{\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }},}
where
I
{\displaystyle I}
is the moment of inertia of the body
α
{\displaystyle {\boldsymbol {\alpha }}}
is the angular acceleration of the body. This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation. Equivalently, the differential form of Newton's second law provides an alternative definition of torque:
τ
=
d
L
d
t
,
{\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {dt} }},}
where
L
{\displaystyle \mathbf {L} }
is the angular momentum of the particle. Newton's third law of motion requires that all objects exerting torques themselves experience equal and opposite torques, and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques.
=== Yank === The yank is defined as the rate of change of force
Y
=
d
F
d
t
{\displaystyle \mathbf {Y} ={\frac {\mathrm {d} \mathbf {F} }{\mathrm {d} t}}}
The term is used in biomechanical analysis, athletic assessment and robotic control. The second ("tug"), third ("snatch"), fourth ("shake"), and higher derivatives are rarely used.
=== Kinematic integrals ===
Forces can be used to define a number of physical concepts by integrating with respect to kinematic variables. For example, integrating with respect to time gives the definition of impulse:
J
=
∫
t
1
t
2
F
d
t
,
{\displaystyle \mathbf {J} =\int _{t_{1}}^{t_{2}}{\mathbf {F} \,\mathrm {d} t},}
which by Newton's second law must be equivalent to the change in momentum (yielding the Impulse momentum theorem). Similarly, integrating with respect to position gives a definition for the work done by a force:
W
=
∫
x
1
x
2
F
⋅
d
x
,
{\displaystyle W=\int _{\mathbf {x} _{1}}^{\mathbf {x} _{2}}{\mathbf {F} \cdot {\mathrm {d} \mathbf {x} }},}
which is equivalent to changes in kinetic energy (yielding the work energy theorem). Power P is the rate of change dW/dt of the work W, as the trajectory is extended by a position change
d
x
{\displaystyle d\mathbf {x} }
in a time interval dt:
d
W
=
d
W
d
x
⋅
d
x
=
F
⋅
d
x
,
{\displaystyle \mathrm {d} W={\frac {\mathrm {d} W}{\mathrm {d} \mathbf {x} }}\cdot \mathrm {d} \mathbf {x} =\mathbf {F} \cdot \mathrm {d} \mathbf {x} ,}
so
P
=
d
W
d
t
=
d
W
d
x
⋅
d
x
d
t
=
F
⋅
v
,
{\displaystyle P={\frac {\mathrm {d} W}{\mathrm {d} t}}={\frac {\mathrm {d} W}{\mathrm {d} \mathbf {x} }}\cdot {\frac {\mathrm {d} \mathbf {x} }{\mathrm {d} t}}=\mathbf {F} \cdot \mathbf {v} ,}