在以角速度 [latex]boldsymbol{omega}[/latex] 旋转的参考系中, 离心力 作用于质量为 m、位置矢量为 r 的物体上的力 Fcf 由矢量公式给出:Fcf = -m ω × (ω × r)。该公式表明力的方向垂直于旋转轴且向外。

(图片仅供参考)
在以角速度 [latex]boldsymbol{omega}[/latex] 旋转的参考系中, 离心力 作用于质量为 m、位置矢量为 r 的物体上的力 Fcf 由矢量公式给出:Fcf = -m ω × (ω × r)。该公式表明力的方向垂直于旋转轴且向外。
The vector formulation of centrifugal force provides a complete description of its magnitude and direction. The formula [latex]\mathbf{F}_{\mathrm{cf}} = -m \boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r})[/latex] uses the vector cross product. Here, [latex]\boldsymbol{\omega}[/latex] is the angular velocity vector, which points along the axis of rotation. The term [latex]\boldsymbol{\omega} \times \mathbf{r}[/latex] represents the tangential velocity of the point. The second cross product, [latex]\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r})[/latex], results in a vector that points radially inward, representing the centripetal acceleration. The negative sign in the formula flips this direction, resulting in a force vector that points radially outward from the axis of rotation. The magnitude of this force can be simplified to [latex]m \omega^2 r_{\perp}[/latex], where [latex]r_{\perp}[/latex] is the perpendicular distance from the mass to the axis of rotation. This mathematical precision is crucial for analyzing motion in rotating systems, such as the dynamics of machinery, planetary atmospheres, and spacecraft. It is a key component in the transformation of Newton’s second law from an inertial frame to a rotating frame, which also includes the Coriolis force and the Euler force.
这种表述是旋转坐标系中位置矢量微分的直接结果。惯性系中的总加速度是旋转坐标系中观测到的加速度、向心加速度、科里奥利加速度和欧拉加速度之和。当我们将牛顿第二定律([latex]mathbf{F}_{mathrm{real}} = m mathbf{a}_{mathrm{inertial}}[/latex])重新整理到旋转坐标系中时,这些加速度项移到了方程的力一侧,并以负号的虚拟力的形式出现。因此,离心力是项[latex]-m(boldsymbol{omega} times (boldsymbol{omega} times mathbf{r}))[/latex]。
离心力公式
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