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Notices

Kinematics

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  • Average vectors of velocity ($\vec{v}$) and acceleration ($\vec{a}$) of a point: $$\langle\vec{v} \rangle = \frac{\Delta \vec{r}}{\Delta t}, \langle\vec{a} \rangle = \frac{\Delta \vec{v}}{\Delta t},$$where $\Delta\vec{r}$ is the displacement vector (an increment of a radius vector).
  • Velocity and acceleration of a point: $$\vec{v} = \frac{d \vec{r}}{d t}, \vec{a} = \frac{d \vec{v}}{d t}.$$
  • Acceleration of a point expressed in projections on the tangent and the normal to a trajectory: $$a_{\parallel} = \frac{d v_{\parallel}}{d t}, a_{\perp} = \frac{v^2}{R},$$ where $R$ is the radius of curvature of the trajectory at the given point.
  • Distance covered by a point: $$s = \int v dt,$$ where $v$ is the modulus of the velocity vector of a point.
  • Angular velocity and angular acceleration of a solid body: $$\vec{\omega} = \frac{d \vec{\phi}}{dt}, \vec{\alpha} = \frac{d\vec{\omega}}{dt}.$$
  • Relations between linear and angular quantities for a rotating solid body: $$ \vec{v} = \vec{\omega}\times \vec{r}, \omega_{\perp} = \omega^2 R, |\omega_\parallel| = \alpha,$$ where $\vec{r}$ is the radius vector of the considered point relative to an arbitrary point on the rotation axis.

 

 

 

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