“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.
随便看	compact disc recordable compact disc rewritable compact disc system compact disk compact galaxy compactification compaction compaction test compactness compactness theorem compact object compact operator compact source compact system camera compandor companion cell companion matrix company company director company law company taxation Compaq Computer Corporation comparability comparative advantage comparative biogeography