Equations of motion in cylindrical coordinates

If you are not familiar with the cylindrical coordinate system - study the article coordinate systems

Radius vector

To describe motion in cylindrical coordinates, we use three parameters: r, θ and z. The position of the point in cylindrical coordinates will be written as follows:

r_p = r·u_r + θ·u_θ + z·u_z (1) radius-vector, u_i - unit vectors of axes

Here, the position of the point is expressed in the coordinates of the cylindrical coordinate system, these are the vectors u_r, u_θ and u_z. The vector u_r is always directed in the direction from the center to the point, the vector u_θ is always perpendicular to the vector u_r and is directed tangentially to a circle with radius r, the direction of the vector u_z does not depend on the position of the point.

Speed

To obtain the velocity vector, we need to differentiate the radius vector by time:

v = dr_p/dt (2.1)
v = dr·u_r/dt + dθ·u_θ/dt + dz·u_z/dt (2.2)

Since the vectors u_r and u_θ change when moving the material point, their time derivatives will not be zero, so we need to find their derivatives, for which we will translate them into Cartesian coordinate system:

u_r = cosθu_x + sinθu_y (3.1)
u_θ = -sinθu_x + cosθu_y (3.2)
u_z = u_z (3.3)

And differentiate by time:

du_r/dt = dcosθu_x/dt + dsinθu_y/dt = -θ'sinθu_x + θ'cosθu_y = θ'u_θ considering 3.2 du_θ/dt = -dsinθu_x/dt + dcosθu_y/dt = -θ'cosθu_x - θ'sinθu_y = -θ'u_r considering 3.1 du_z/dt = 0

Translating the radius vector into a Cartesian coordinate system, we get:

r = r·u_r + z·u_z

Substituting the obtained differentiation results, we obtain the formula of the velocity vector for cylindrical coordinates:

v = dr/dt = d(r·u_r + z·u_z)/dt =
    r'·u_r + r·u'_r + z'·u_z + z·u'_z =
    r'·u_r + rθ'·u_θ + z'·u_z velocity vector

Taking the coefficients of the unit vectors of the axes of cylindrical coordinates as corresponding to the velocity, we obtain:

v_r = r'
v_θ = r·θ'
v_z = z'

Acceleration

To obtain the acceleration formula, we differentiate the speed by time:

a = d²r_p/dt² = dv/dt = d(r'·u_r + rθ'·u_θ + z'·u_z)/dt
d(r'·u_r)/dt = (r'' - r'·θ')·u_r
d(θ'·u_θ)/dt = (θ'' + r'·θ')·u_θ
d(z'u_z)/dt = z''u_z components of the acceleration vector

Thus, the coefficients of the coordinates are:

a_r = r'' - rθ'²
a_θ = 2·r'θ' + rθ''
a_z = z''

If the motion occurs in two-dimensional coordinates r-θ, then to describe the motion, only the first two equations.

Centripetal and normal acceleration

The angle is determined by the offset points at a distance ds along the trajectory, the radial displacement is dr and the displacement in the direction of the tangent r dθ. Since these two components are mutually perpendicular, the angle θ can be defined from the equality:

tg θ = r dθ/ dr 

or

tg θ = r / dr/dθ

If the angle is positive, then it is measured from the radial line against the direction of the clock or in the positive direction of the angle. If the angle is negative, then it is measured in the opposite direction (clockwise).

For example, a cardioid described by the equation

r = a(1 + cos θ) 

dr / dθ = - a sin θ

when θ=30°, tg = a(1+ cos 30°)/(-a sen 30°) = -3.73², or θ = -75°, measured against the direction of the clock.

Algorithm for solving problems

Cylindrical coordinates are convenient to use for the analysis of systems in which the trajectories of motion are given relative to a radial line, or can be conveniently expressed in cylindrical coordinates. Having determined the coordinates of the point, the equations of motion can be applied to express the force in the form of acceleration components.

1. Set the inertial coordinate system r, θ, z and draw a diagram of a free body
2. Put the accelerations a_r, a_θ, a_z are directed along the positive direction axes r, θ, z if the directions are unknown
3. Determine all unknown quantities in the problem

4. Apply the equations of motion

5. Define r and time derivatives dr/dt, d²r/dt², dθ/dt, d²θ/dt², d²z/dz², then apply the acceleration equations
   • a_r= d²r/dt²- r dθ²/dt
   • a_θ = r d²θ/dt²+ 2 dr/dt dθ/dt
   • a_z= d²z/dt²
6. If any of the acceleration components is calculated with a minus sign, then this acceleration component is directed along the negative direction of the coordinate axis