information:classification
Particle trajectories in cylindrical-polar coordinates

It is often convenient to formulate the trajectory of a particle r (t) = (x (t), y (t), z (t)) using polar coordinates in the X – Y plane. In this case, its velocity and acceleration take a convenient form.

Recall that the trajectory of a particle P is defined by its coordinate vector r measured in a fixed reference frame F. As the particle moves, its coordinate vector r (t) traces its trajectory, which is a curve in space, given by: r (t) = x (t) x ^ + y (t) y ^ + z (t) z ^, {\displaystyle \mathbf {r} (t)=x(t){\hat {\mathbf {x} }}+y(t){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where x̂, ŷ, and ẑ are the unit vectors along the x, y and z axes of the reference frame F, respectively.

Consider a particle P that moves only on the surface of a circular cylinder r (t) = constant, it is possible to align the z axis of the fixed frame F with the axis of the cylinder. Then, the angle θ around this axis in the x – y plane can be used to define the trajectory as, r (t) = r cos (θ (t)) x ^ + r sin (θ (t)) y ^ + z (t) z ^, {\displaystyle \mathbf {r} (t)=r\cos(\theta (t)){\hat {\mathbf {x} }}+r\sin(\theta (t)){\hat {\mathbf {y} }}+z(t){\hat {\mathbf {z} }},} where the constant distance from the center is denoted as r, and θ (t) is a function of time.

The cylindrical coordinates for r (t) can be simplified by introducing the radial and tangential unit vectors, r ^ = cos (θ (t)) x ^ + sin (θ (t)) y ^, θ ^ = − sin (θ (t)) x ^ + cos (θ (t)) y ^. {\displaystyle {\hat {\mathbf {r} }}=\cos(\theta (t)){\hat {\mathbf {x} }}+\sin(\theta (t)){\hat {\mathbf {y} }},\quad {\hat {\mathbf {\theta } }}=-\sin(\theta (t)){\hat {\mathbf {x} }}+\cos(\theta (t)){\hat {\mathbf {y} }}.} and their time derivatives from elementary calculus: d r ^ d t = ω θ ^. {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {r} }}}{{\text{d}}t}}=\omega {\hat {\mathbf {\theta } }}.} d 2 r ^ d t 2 = d (ω θ ^) d t = α θ ^ − ω r ^. {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {r} }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(\omega {\hat {\mathbf {\theta } }})}{{\text{d}}t}}=\alpha {\hat {\mathbf {\theta } }}-\omega {\hat {\mathbf {r} }}.}

d θ ^ d t = − θ r ^. {\displaystyle {\frac {{\text{d}}{\hat {\mathbf {\theta } }}}{{\text{d}}t}}=-\theta {\hat {\mathbf {r} }}.} d 2 θ ^ d t 2 = d (− θ r ^) d t = − α r ^ − ω 2 θ ^. {\displaystyle {\frac {{\text{d}}^{2}{\hat {\mathbf {\theta } }}}{{\text{d}}t^{2}}}={\frac {{\text{d}}(-\theta {\hat {\mathbf {r} }})}{{\text{d}}t}}=-\alpha {\hat {\mathbf {r} }}-\omega ^{2}{\hat {\mathbf {\theta } }}.}

Using this notation, r (t) takes the form, r (t) = r r ^ + z (t) z ^. {\displaystyle \mathbf {r} (t)=r{\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} In general, the trajectory r (t) is not constrained to lie on a circular cylinder, so the radius R varies with time and the trajectory of the particle in cylindrical-polar coordinates becomes: r (t) = r (t) r ^ + z (t) z ^. {\displaystyle \mathbf {r} (t)=r(t){\hat {\mathbf {r} }}+z(t){\hat {\mathbf {z} }}.} Where r, θ, and z might be continuously differentiable functions of time and the function notation is dropped for simplicity. The velocity vector v is the time derivative of the trajectory r (t), which yields: v P = d d t (r r ^ + z z ^) = v r ^ + r ω θ ^ + v z z ^ = v (r ^ + θ ^) + v z z ^. {\displaystyle \mathbf {v} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(r{\hat {\mathbf {r} }}+z{\hat {\mathbf {z} }}\right)=v{\hat {\mathbf {r} }}+r\mathbf {\omega } {\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}=v({\hat {\mathbf {r} }}+{\hat {\mathbf {\theta } }})+v_{z}{\hat {\mathbf {z} }}.}

Similarly, the acceleration a, which is the time derivative of the velocity v, is given by: a P = d d t (v r ^ + v θ ^ + v z z ^) = (a − v θ) r ^ + (a + v ω) θ ^ + a z z ^. {\displaystyle \mathbf {a} _{P}={\frac {\text{d}}{{\text{d}}t}}\left(v{\hat {\mathbf {r} }}+v{\hat {\mathbf {\theta } }}+v_{z}{\hat {\mathbf {z} }}\right)=(a-v\theta){\hat {\mathbf {r} }}+(a+v\omega){\hat {\mathbf {\theta } }}+a_{z}{\hat {\mathbf {z} }}.}

The term − v θ r ^ {\displaystyle -v\theta {\hat {\mathbf {r} }}} acts toward the center of curvature of the path at that point on the path, is commonly called the centripetal acceleration. The term v ω θ ^ {\displaystyle v\omega {\hat {\mathbf {\theta } }}} is called the Coriolis acceleration.