angular coordinate

(noun)

An angle measured from the polar axis, usually counter-clockwise. 

Related Terms

Examples of angular coordinate in the following topics:

  • Introduction to the Polar Coordinate System

    • The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth.
    • The radial coordinate is often denoted by $r$ or $ρ$ , and the angular coordinate by $ϕ$, $θ$, or $t$.
    • In many contexts, a positive angular coordinate means that the angle $ϕ$ is measured counterclockwise from the axis.  
    • Adding any number of full turns ($360^{\circ} $ or $2\pi$ radians) to the angular coordinate does not change the corresponding direction.
    • In green, the point with radial coordinate $3$ and angular coordinate $60$ degrees or $(3,60^{\circ})$.

  • Cylindrical and Spherical Coordinates

    • Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
    • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
    • The spherical coordinates (radius $r$, inclination $\theta$, azimuth $\varphi$) of a point can be obtained from its Cartesian coordinates ($x$, $y$, $z$) by the formulae:
    • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.
    • The dot is the point with radial distance $\rho = 4$, angular coordinate $\varphi = 130$ degrees, and height $z = 4$.

  • Wave Equation for the Hydrogen Atom

    • Although the resulting energy eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows generally from this isotropy of the underlying potential.
    • The angular momentum quantum number ℓ = 0, 1, 2, ... determines the magnitude of the angular momentum.
    • In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wavefunctions must be found.
    • Due to angular momentum conservation, states of the same ℓ but different mℓ have the same energy.
    • The wavefunction itself is expressed in spherical polar coordinates:

  • Description of the Hydrogen Atom

    • Although the resulting energy eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely, generally from this isotropy of the underlying potential.
    • This corresponds to the fact that angular momentum is conserved in the orbital motion of the electron around the nucleus.
    • Therefore, the energy eigenstates may be classified by two angular momentum quantum numbers, ℓ and m (both are integers).
    • The angular momentum quantum number ℓ = 0, 1, 2, ... determines the magnitude of the angular momentum.
    • In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wave functions must be found.

  • Angular Acceleration, Alpha

    • Angular acceleration is the rate of change of angular velocity, expressed mathematically as $\alpha = \Delta \omega/\Delta t$ .
    • Angular acceleration is the rate of change of angular velocity.
    • Angular acceleration is defined as the rate of change of angular velocity.
    • In equation form, angular acceleration is expressed as follows:
    • The units of angular acceleration are (rad/s)/s, or rad/s2.

  • Constant Angular Acceleration

    • Constant angular acceleration describes the relationships among angular velocity, angle of rotation, and time.
    • Simply by using our intuition, we can begin to see the interrelatedness of rotational quantities like θ (angle of rotation), ω (angular velocity) and α (angular acceleration).
    • Similarly, the kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
    • As in linear kinematics where we assumed a is constant, here we assume that angular acceleration α is a constant, and can use the relation: $a=r\alpha $ Where r - radius of curve.Similarly, we have the following relationships between linear and angular values: $v=r\omega \\x=r\theta $
    • Relate angle of rotation, angular velocity, and angular acceleration to their equivalents in linear kinematics

  • Hydrostatics

    • or that isentropic stars must have constant angular velocity on cylindrical surfaces.

  • Rotational Collisions

    • In a closed system, angular momentum is conserved in a similar fashion as linear momentum.
    • For objects with a rotational component, there exists angular momentum.
    • Angular momentum is defined, mathematically, as L=Iω, or L=rxp.
    • An object that has a large angular velocity ω, such as a centrifuge, also has a rather large angular momentum.
    • After the collision, the arrow sticks to the rolling cylinder and the system has a net angular momentum equal to the original angular momentum of the arrow before the collision.

  • Conservation of Angular Momentum

    • The law of conservation of angular momentum states that when no external torque acts on an object, no change of angular momentum will occur.
    • The conserved quantity we are investigating is called angular momentum.
    • The symbol for angular momentum is the letter L.
    • If the change in angular momentum ΔL is zero, then the angular momentum is constant; therefore,
    • (I: rotational inertia, $\omega$: angular velocity)

  • Quantum Numbers

    • This model describes electrons using four quantum numbers: energy (n), angular momentum (ℓ), magnetic moment (mℓ), and spin (ms).
    • This number therefore has a dependence only on the distance between the electron and the nucleus (i.e. the radial coordinate r).
    • The second quantum number, known as the angular or orbital quantum number, describes the subshell and gives the magnitude of the orbital angular momentum through the relation.
    • The magnetic quantum number describes the energy levels available within a subshell and yields the projection of the orbital angular momentum along a specified axis.
    • The fourth quantum number describes the spin (intrinsic angular momentum) of the electron within that orbital and gives the projection of the spin angular momentum (s) along the specified axis.