Tru Physics

Angular Momentum

—

by

Tru Physics
in

Angular momentum describes the rotational motion of objects. It plays a crucial role in understanding the behavior of spinning objects and the conservation laws that govern their interactions.

Note: If you’re trying to calculate angular momentum, check out our angular momentum calculators!

A spinning top is a common example of angular momentum in the classical sense of the term.
A spinning top is a common example of angular momentum in the classical sense of the term.

Definition and Basic Properties

Angular momentum, denoted by L, is a vector quantity that characterizes the rotational motion of a particle or a system of particles. For a single particle moving in a plane, the angular momentum is defined as the cross product of the position vector (r) and the linear momentum (p):

\vec{L}=\vec{r} \times \vec{p}

The direction of the angular momentum vector is perpendicular to the plane defined by the position and linear momentum vectors, following the right-hand rule.

For a system (or rigid body) of particles, the total angular momentum is the sum of the angular momenta of the individual particles:

\vec{L}_{total}=\sum \vec{\ell}_i

Angular momentum is a conserved quantity, meaning that the total angular momentum of a closed system remains constant if no external torques are acting on it. This principle, known as the conservation of angular momentum, has crucial implications for the stability of rotating systems, such as planets, stars, and even subatomic particles.

Moment of Inertia and Angular Velocity

To understand the rotational motion of rigid bodies, we introduce two essential concepts: the moment of inertia (I) and the angular velocity (\omega). The moment of inertia is a scalar quantity that depends on both the mass distribution and the axis of rotation of the object. It plays a role somewhat analogous to mass in linear motion problems, affecting the resistance of the body to changes in rotational motion just as mass affects the resistance of a body to changes in linear motion. For a point mass m at a distance r from the axis of rotation, the moment of inertia is given by:

I=mr^2

For a rigid body with multiple point masses, the total moment of inertia is the sum of the moments of inertia of the individual masses:

I_{total}=\sum m_i r_i^2

Angular velocity is a vector quantity that describes the rate of rotation of an object. The direction of the angular velocity vector is defined by the right-hand rule.

In analogy with Newton’s second law for linear motion (F = m a), we have the following equation for rotational motion:

\tau = I \alpha

where \tau is the torque (the rotational analogue of force) and \alpha is the angular acceleration.

For a rigid body, the relationship between angular momentum, moment of inertia, and angular velocity is given by:

L = I \omega

Angular Momentum in Quantum Mechanics

In quantum mechanics, angular momentum plays a vital role in describing the behavior of particles and their wave functions. There are two types of angular momentum in quantum mechanics: orbital angular momentum and spin angular momentum (an intrinsic property of particles, independent of their motion). These two can be summed into a total angular momentum.

Orbital Angular Momentum

L=\sum \ell_i

|\vec{L}| = \hbar \sqrt{\ell (\ell + 1)}

with a z-axis projection specified as:

L_z = m_{\ell} \hbar

Spin Angular Momentum

S=\sum s_i

|\vec{S}| = \hbar \sqrt{s(s+1)}

with a z-axis projection specified as:

S_z = m_s \hbar

Total Angular Momentum

J=L+S

|\vec{J}| = \hbar \sqrt{j(j+1)}

with a z-axis projection specified as:

J_z = m_j \hbar

Advanced Topics

At a higher level of study, angular momentum involves more advanced mathematical techniques and explores various applications in classical and quantum mechanics. Some of these topics include:

  1. Rigid Body Dynamics: The study of the motion of extended bodies in which the distance between particles remains fixed. This field uses the principles of angular momentum and torque to understand the behavior of rotating objects, such as gyroscopes and tops.
  2. Precession and Nutation: These are complex motions exhibited by rotating objects subject to external torques. Precession is the change in the orientation of the rotation axis, while nutation is a small oscillation superimposed on the precessional motion. These phenomena are essential for understanding the motion of celestial bodies, like the Earth.
  3. Clebsch-Gordan Coefficients: In quantum mechanics, Clebsch-Gordan coefficients are used to combine the angular momenta of two particles into a single total angular momentum. These coefficients are important in atomic and nuclear physics, particularly when studying the coupling of angular momenta in multi-electron atoms or the interaction between nucleons in atomic nuclei.
  4. Spherical Harmonics: Spherical harmonics are mathematical functions used to describe the angular dependence of wave functions in quantum mechanics. They are particularly important in solving the Schrödinger equation for systems with spherical symmetry, such as atoms and molecules.

Want to study more? Visit our Index here!

Do you prefer video lectures over reading a webpage? Follow us on YouTube to stay updated with the latest video content!

Comments

One response to “Angular Momentum”

  1. Angular Momentum Calculator – Tru Physics

    […] are several different ways to calculate angular momentum. Find the formula that you are using below and then use the corresponding angular momentum […]

    Reply

Have something to add? Leave a comment!