“circular motion”的概念、定义、翻译、参考文献-科学参考

circular motion

Mathematics

Motion of a particle in a circular path. Suppose that the path of the particle P is a circle in the Cartesian plane, with centre at the origin O and radius r₀. Let r, v, and a be the position vector, velocity, and acceleration of P. If P has polar coordinates (r₀, θ‎), then

$\begin{array}{l} r = r_{0} (i cos θ + j sin θ), \\ v = \dot{r} = r_{0} (- \dot{θ} i sin θ + \dot{θ} j cos θ), \\ a = \ddot{r} = r_{0} (- \ddot{θ} i sin θ - {\dot{θ}}^{2} i cos θ + \ddot{θ} j cos θ - {\dot{θ}}^{2} j sin θ) . \end{array}$

Let e_r = i cos θ‎ + j sin θ‎ and e_θ‎ = −i sin θ‎ + j cos θ‎, so that e_r and e_θ‎ are rotations of i and j by θ‎ anticlockwise. Then the previous equations become

$\begin{matrix} r = r_{0} e_{r}, & v = \dot{r} = r_{0} \dot{θ} e_{θ}, \\ a = \ddot{r} = & - r_{0} {\dot{θ}}^{2} e_{r} + r_{0} \ddot{θ} e_{θ} . \end{matrix}$

If the particle, of mass m, is acted on by a force F, where $F = F_{1} e_{r} + F_{2} e_{θ}$ , then the equation of motion $m \ddot{r} = F$ gives $- m r_{0} {\dot{θ}}^{2} = F_{1}$ and $m r_{0} \ddot{θ} = F_{2}$ . If the transverse component F₂ of the force is zero, then $\dot{θ} =$ constant and the particle has constant speed.

See also angular acceleration, angular velocity, radial and transverse components.

单词	circular motion
释义	circular motion Mathematics Motion of a particle in a circular path. Suppose that the path of the particle P is a circle in the Cartesian plane, with centre at the origin O and radius r₀. Let r, v, and a be the position vector, velocity, and acceleration of P. If P has polar coordinates (r₀, θ‎), then $\begin{array}{l} r = r_{0} (i cos θ + j sin θ), \\ v = \dot{r} = r_{0} (- \dot{θ} i sin θ + \dot{θ} j cos θ), \\ a = \ddot{r} = r_{0} (- \ddot{θ} i sin θ - {\dot{θ}}^{2} i cos θ + \ddot{θ} j cos θ - {\dot{θ}}^{2} j sin θ) . \end{array}$ Let e_r = i cos θ‎ + j sin θ‎ and e_θ‎ = −i sin θ‎ + j cos θ‎, so that e_r and e_θ‎ are rotations of i and j by θ‎ anticlockwise. Then the previous equations become $\begin{matrix} r = r_{0} e_{r}, & v = \dot{r} = r_{0} \dot{θ} e_{θ}, \\ a = \ddot{r} = & - r_{0} {\dot{θ}}^{2} e_{r} + r_{0} \ddot{θ} e_{θ} . \end{matrix}$ If the particle, of mass m, is acted on by a force F, where $F = F_{1} e_{r} + F_{2} e_{θ}$ , then the equation of motion $m \ddot{r} = F$ gives $- m r_{0} {\dot{θ}}^{2} = F_{1}$ and $m r_{0} \ddot{θ} = F_{2}$ . If the transverse component F₂ of the force is zero, then $\dot{θ} =$ constant and the particle has constant speed. See also angular acceleration, angular velocity, radial and transverse components.
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