angular momentum

Examples of angular momentum in the following topics:

• 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,
  • This is an expression for the law of conservation of angular momentum.

• 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.
  • Units for linear momentum are kg⋅m/s while units for angular momentum are kg⋅m2/s.
  • An object that has a large angular velocity ω, such as a centrifuge, also has a rather large angular momentum.

• Accretion Disks

  • The preceding section ignores an important aspect of accretion: the angular momentum of the accreta.
  • If the material starts with some net angular momentum it can only collapse so far before its angular velocity will be sufficient to halt further collapse.
  • First let's see why angular momentum can play a crucial role in accretion.
  • The initial specific angular momentum is $v b$.
  • If the material conserves angular momentum we can compare the centripetal acceleration with gravitational acceleration to give

• Gyroscopes

  • A gyroscope is a device for measuring or maintaining orientation based on the principles of angular momentum.
  • With the wheel rotating as shown, its angular momentum is to the woman's left.
  • The torque produced is perpendicular to the angular momentum, thus the direction of the angular momentum is changed, but not its magnitude.
  • This torque causes a change in angular momentum ΔL in exactly the same direction.
  • Figure (b) shows that the direction of the torque is the same as that of the angular momentum it produces.

• Angular Quantities as Vectors

  • The direction of angular quantities, such as angular velocity and angular momentum, is determined by using the right hand rule.
  • Angular momentum and angular velocity have both magnitude and direction and, therefore, are vector quantities.
  • The direction of angular momentum and velocity can be determined along this axis.
  • The right hand rule can be used to find the direction of both the angular momentum and the angular velocity.
  • The direction of angular velocity ω size and angular momentum L are defined to be the direction in which the thumb of your right hand points when you curl your fingers in the direction of the disk's rotation as shown.

• Wave Equation for the Hydrogen Atom

  • 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 mℓ (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 wavefunctions must be found.
  • Due to angular momentum conservation, states of the same ℓ but different mℓ have the same energy.

• Angular Momentum Transport

  • The specific angular momentum of material in circular orbit is given by the orbital velocity times the square of the radius,
  • Because matter is falling toward the centre the angular momentum flows inward
  • The viscous stress is proportional to the viscosity and the angular velocity gradient,

• Angular vs. Linear Quantities

  • The familiar linear vector quantities such as velocity and momentum have analogous angular quantities used to describe circular motion.
  • This type of motion has several familiar vector quantities associated with it, including linear velocity and momentum.
  • It has the same set of vector quantities associated with it, including angular velocity and angular momentum.
  • However, we can define an angular momentum vector which is constant throughout this motion.
  • The magnitude of the angular momentum is equal to the rate at which the angle of the particle advances:

• Relationship Between Torque and Angular Acceleration

  • Torque is equal to the moment of inertia times the angular acceleration.
  • Torque and angular acceleration are related by the following formula where is the objects moment of inertia and $\alpha$ is the angular acceleration .
  • Similar to Newton's Second Law, angular motion also obeys Newton's First Law.
  • Relationship between force (F), torque (τ), momentum (p), and angular momentum (L) vectors in a rotating system
  • Torque, Angular Acceleration, and the Role of the Church in the French Revolution

• Relationship Between Linear and Rotational Quantitues

  • The description of motion could be sometimes easier with angular quantities such as angular velocity, rotational inertia, torque, etc.
  • The velocity (i.e. angular velocity) is indeed constant.
  • This is the first advantage of describing uniform circular motion in terms of angular velocity.
  • As we use mass, linear momentum, translational kinetic energy, and Newton's 2nd law to describe linear motion, we can describe a general rotational motion using corresponding scalar/vector/tensor quantities:
  • For the description of the motion, angular quantities are the better choice.