You have $$\dot{\hat{\rho}}=(-\sin{\phi}\hat{x}+\cos{\phi}\hat{y})\dot{\phi}$$But you already showed that $$\hat{\phi}=(-\sin{\phi}\hat{x}+\cos{\phi}\hat{y})$$Mason Smith said:
That makes perfect sense. Thank you so much for the insight, Chestermiller!Chestermiller said:
Cylindrical coordinates are a coordinate system used to describe the position of a point in three-dimensional space. They consist of a radial distance from a fixed point, an angle from a fixed axis, and a height or distance from a fixed plane.
Unit vectors in cylindrical coordinates are vectors that have a magnitude of 1 and are used to specify the direction of a point in the coordinate system. In cylindrical coordinates, the unit vectors are denoted as er, eθ, and ez, representing the radial, angular, and height directions respectively.
To convert from cylindrical coordinates to Cartesian coordinates, we use the following equations:
x = r cos(θ)
y = r sin(θ)
z = z
where r is the radial distance, θ is the angle, and z is the height or distance from the fixed plane.
To find the time derivatives in cylindrical coordinates, we use the chain rule. The time derivatives of the position coordinates are:
𝑟̇ = 𝑟̇0 cos(θ) - 𝑟0𝜃̇ sin(θ)
𝜃̇ = 𝑟0𝜃̇ cos(θ) + 𝑟̇0 sin(θ)
𝑧̇ = 𝑧̇0
where 𝑟̇0, 𝜃̇, and 𝑧̇0 are the time derivatives of the radial, angular, and height coordinates, respectively.
Cylindrical coordinates are useful because they can simplify the mathematical equations needed to describe certain physical phenomena. They are particularly useful for problems involving cylindrical objects or systems with cylindrical symmetry. They also allow for a more intuitive understanding of complex three-dimensional phenomena.
