State and prove the law of conservation of angular momentum

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state and prove the law of conservation of angular momentum.

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What is the Law of Conservation of Angular Momentum?

The law of conservation of angular momentum is a fundamental principle in physics, stating that when no external torque acts on a system, the total angular momentum of the system remains constant. This principle is integral to many areas of physics, including mechanics, astronomy, and quantum mechanics. Its applications are vast and can be observed in phenomena ranging from the spinning of a figure skater to the orbits of planets.

Statement of the Law

The law of conservation of angular momentum states that in the absence of an external torque, the total angular momentum of a closed system remains constant over time.

Mathematical Expression

Angular momentum, denoted as \mathbf{L}, is the rotational analog of linear momentum. It is defined as the cross-product of the position vector \mathbf{r} and the linear momentum \mathbf{p}:

\mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{r} \times m\mathbf{v}

where:

  • \mathbf{L} is the angular momentum
  • \mathbf{r} is the position vector
  • \mathbf{p} = m\mathbf{v} is the linear momentum of the mass m with velocity \mathbf{v}
  • m is the mass
  • \mathbf{v} is the velocity vector

When considering a rigid body rotating about an axis, the angular momentum can also be expressed as:

\mathbf{L} = I\mathbf{\omega}

where:

  • I is the moment of inertia
  • \mathbf{\omega} is the angular velocity

Conservation Equation

In an isolated system where no external torques are acting, the rate of change of angular momentum is zero:

\frac{d\mathbf{L}}{dt} = 0

This implies that:

\mathbf{L}_{\text{initial}} = \mathbf{L}_{\text{final}}

Proof of Conservation of Angular Momentum

To prove this law, we start by considering Newton’s Second Law in rotational form. The rotational analog to Newton’s second law states that the torque \mathbf{\tau} acting on a body is equal to the rate of change of its angular momentum:

\mathbf{\tau} = \frac{d\mathbf{L}}{dt}

Derivation for a Particle

For a particle moving in a path with respect to a fixed point, the torque is the rate of change of angular momentum with respect to time:

  1. Start from Torque Definition: Use the definition of torque for a single particle:
    $$ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} = \frac{d(\mathbf{r} \times m\mathbf{v})}{dt} $$

  2. Substitute Force: From Newton’s second law, \mathbf{F} = \frac{d(m\mathbf{v})}{dt}, thus:
    $$ \mathbf{\tau} = \mathbf{r} \times \frac{d(m\mathbf{v})}{dt} $$

  3. Expand Using Product Rule: Expand the derivative using the product rule:
    $$ \frac{d}{dt}(\mathbf{r} \times m\mathbf{v}) = \frac{d\mathbf{r}}{dt} \times m\mathbf{v} + \mathbf{r} \times \frac{d(m\mathbf{v})}{dt} $$

  4. Recognize Velocity and Zero Cross-Product: Note that \frac{d\mathbf{r}}{dt} = \mathbf{v}, so:
    $$ \mathbf{v} \times m\mathbf{v} = \mathbf{0} $$

    Thus:
    $$ \mathbf{\tau} = \mathbf{r} \times \frac{d(m\mathbf{v})}{dt} $$

  5. Conclude Constant Angular Momentum for Zero Torque: If \mathbf{\tau} = 0, then the derivative term must be zero:
    $$ \frac{d\mathbf{L}}{dt} = 0 \implies \mathbf{L} = \text{constant} $$

Generalization to a System of Particles

In a system of particles, the total angular momentum \mathbf{L}_{\text{total}} is the sum of the angular momentum of all particles:

\mathbf{L}_{\text{total}} = \sum_i \mathbf{L}_i = \sum_i (\mathbf{r}_i \times \mathbf{p}_i)

The total external torque on the system is:

\mathbf{\tau}_{\text{external}} = \frac{d}{dt} \left( \sum_i \mathbf{L}_i \right)

If \mathbf{\tau}_{\text{external}} = 0, then:

\frac{d\mathbf{L}_{\text{total}}}{dt} = 0 \implies \mathbf{L}_{\text{total}} = \text{constant}

This concludes the proof, and thus, the angular momentum of a closed system is conserved.

Examples of Angular Momentum Conservation

  1. Figure Skater: When a figure skater pulls in their arms during a spin, they reduce their moment of inertia (I) and increase their spin rate (\omega) to conserve angular momentum (L = I\omega).

  2. Planetary Orbits: Planets conserve angular momentum as they orbit the Sun. For example, when a planet moves closer to the Sun, it speeds up due to conservation of angular momentum.

  3. Gyroscopes: A gyroscope maintains its orientation due to angular momentum conservation, which is the principle behind navigation systems.

By understanding the applications and derivation of the law of conservation of angular momentum, we gain insights into the dynamics of rotating systems and their natural tendencies under isolated conditions. This knowledge is particularly crucial in engineering, astrophysics, and understanding everyday physical phenomena. @LectureNotes