Conservation of Angular Momentum - AP Physics C: Mechanics Study Guide

Getting Started

Consider a spinning figure skater or a planet orbiting its star. What physical principles govern the stability and changes in their rotational motion? This chapter explores angular momentum, a fundamental quantity that, under specific conditions, remains unchanged, providing a powerful tool for analyzing rotating systems, from subatomic particles to galaxies. We will investigate the core question: what interaction causes a system's angular momentum to change, and when can we declare it to be conserved?

What You Should Be able to Do

After completing this section, you will be able to:

• Define a physical system and identify all external torques acting upon it relative to a chosen origin.

• Formulate the relationship between the net external torque on a system and the time rate of change of its angular momentum, $\sum {\vec{\tau }}_{ext}=\frac{d\vec{L}}{dt}$.

• Apply the principle of conservation of angular momentum to predict the final state (e.g., angular velocity) of a system after an internal change in its mass distribution (moment of inertia).

• Justify, using a rigorous torque analysis, whether angular momentum is a conserved quantity for a given physical situation and choice of system.

• Relate the total angular impulse delivered to a system to the total change in its angular momentum by integrating the torque over time.

Key Concepts & Mechanisms

This section analyzes the conservation of angular momentum through the lens of Dynamics as Cause, where torques are the direct cause of changes in rotational motion.

System & Preconditions

The first and most critical step in any conservation law analysis is to define the system: the collection of objects whose motion is being analyzed. Everything outside this boundary is the surroundings. An interaction between the system and its surroundings is an external force/torque, while an interaction between components within the system is an internal force/torque.

The primary precondition for the conservation of angular momentum is that the system must be rotationally isolated. This means that the vector sum of all external torques acting on the system about a specific point is zero.

$$\sum {\vec{\tau }}_{ext}=\vec{0}$$

This does not require the net external force to be zero. For example, the gravitational force of the Sun on an orbiting planet exerts no torque about the Sun's center (since $\vec{r}$ and $\vec{F}$ are parallel), so the planet's angular momentum is conserved even though an external force is constantly acting on it.

Key Steps / Relations

The relationship between torque and angular momentum is the rotational analog of Newton's Second Law.

1. Definition of Angular Momentum: For a single particle of mass $m$ and velocity $\vec{v}$ at a position $\vec{r}$ from an origin, the angular momentum $\vec{L}$ is defined as the cross product of its position vector and its linear momentum vector, $\vec{p}=m\vec{v}$.

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

   For a rigid body rotating about a fixed axis, this simplifies to $\vec{L}=I\vec{\omega }$, where $I$ is the moment of inertia and $\vec{\omega }$ is the angular velocity vector.

2. The Dynamical Cause of Change: We can find the relationship between torque and angular momentum by taking the time derivative of the definition of $\vec{L}$. Using the product rule for derivatives:

   $$\frac{d\vec{L}}{dt}=\frac{d}{dt}(\vec{r}\times \vec{p})=\left(\frac{d\vec{r}}{dt}\times \vec{p}\right)+\left(\vec{r}\times \frac{d\vec{p}}{dt}\right)$$

   The first term is $(\vec{v}\times m\vec{v})$, which is zero because the cross product of two parallel vectors is zero. The second term contains $\frac{d\vec{p}}{dt}$, which by Newton's Second Law is the net force $\sum \vec{F}$. This gives:

   $$\frac{d\vec{L}}{dt}=\vec{r}\times \sum \vec{F}$$

   The term $\vec{r}\times \sum \vec{F}$ is the definition of the net torque, $\sum \vec{\tau }$.

3. The Rotational Law of Motion: This leads to the fundamental differential relation for rotational dynamics:

   $$\sum \vec{\tau }=\frac{d\vec{L}}{dt}$$

   For a system of particles, internal torques (from forces between particles within the system) cancel out in action-reaction pairs due to Newton's Third Law. Therefore, only the sum of external torques causes a change in the total angular momentum of the system:

   $$\sum {\vec{\tau }}_{ext}=\frac{d{\vec{L}}_{sys}}{dt}$$

4. The Conservation Condition: If the precondition is met—that is, if the net external torque on the system is zero ($\sum {\vec{\tau }}_{ext}=\vec{0}$)—then the governing equation becomes:

   $$\frac{d{\vec{L}}_{sys}}{dt}=\vec{0}$$

   This differential equation implies that the total angular momentum of the system, ${\vec{L}}_{sys}$, must be a constant vector. This is the Principle of Conservation of Angular Momentum.

   $${\vec{L}}_{initial}={\vec{L}}_{final}$$

Outputs & Effects

When angular momentum is conserved, any internal change to the system's configuration must preserve the total value of $\vec{L}$. For a system whose moment of inertia $I$ can change, the relation $L=I\omega$ (for rotation about a principal axis) leads to a direct, inverse relationship between $I$ and $\omega$.

• Qualitative Behavior: If a spinning system reconfigures itself to decrease its moment of inertia (e.g., a figure skater pulling her arms in, or a star collapsing), its angular velocity must increase to keep the product $I\omega$ constant. Conversely, increasing the moment of inertia will decrease the angular velocity.

• Quantitative Results: For a system changing from an initial state (i) to a final state (f) with no net external torque, we can write:

  $$I_i{\omega }_i=I_f{\omega }_f$$

  This allows for the calculation of a final angular velocity if the change in the moment of inertia is known.

Regulation & Limits

• Validity Domain: This principle is universally valid. However, its application is most useful for systems where external torques are either zero or negligible compared to the effects of internal changes. It is particularly powerful in analyzing collisions and central force motion.

• System Selection is Crucial: The conservation of $\vec{L}$ depends entirely on the choice of the system boundary. For two colliding disks, if the system is defined as both disks, the torques they exert on each other are internal, and the system's total angular momentum is conserved. If the system is defined as only one disk, the torque from the other disk is external, and that single disk's angular momentum is not conserved.

• Energy is Not Necessarily Conserved: In many processes where angular momentum is conserved, mechanical energy is not. For example, when a skater pulls her arms in, she does positive work with her muscles. This work increases the system's rotational kinetic energy ($K_{rot}=\frac{1}{2}I{\omega }^2$). Since $L=I\omega$ is constant, we can write $K_{rot}=\frac{L^2}{2I}$. As she pulls her arms in, $I$ decreases, and thus $K_{rot}$ increases.

Key Models & Diagrams

The analysis of any rotational dynamics problem follows from identifying the torques on the chosen system. This determines which physical law governs the system's behavior over time.

Key Components & Evidence

• Angular Momentum ($\vec{L}$): The rotational analog of linear momentum. For a point particle, $\vec{L}=\vec{r}\times \vec{p}$; for a rigid body, $\vec{L}=I\vec{\omega }$. Its conservation is a fundamental principle. (Units: kg⋅m²/s).

• Torque ($\vec{\tau }$): The rotational analog of force, defined as $\vec{\tau }=\vec{r}\times \vec{F}$. It is the agent of change for angular momentum. (Units: N⋅m).

• Moment of Inertia ($I$): A scalar quantity that measures a body's resistance to a change in its angular velocity. It depends on the mass and its distribution relative to the axis of rotation. (Units: kg⋅m²).

• Angular Velocity ($\vec{\omega }$): A vector quantity describing the rate of rotation and the direction of the spin axis. (Units: rad/s).

• System: The defined collection of objects under analysis. The choice of system determines which forces/torques are internal versus external.

• External Torque: A torque exerted on the system by an agent in the surroundings. Only external torques can change a system's total angular momentum.

• Internal Torque: A torque exerted by one part of the system on another. By Newton's Third Law, internal torques always sum to zero.

• Rotational Second Law ($\sum {\vec{\tau }}_{ext}=d\vec{L}/dt$): The fundamental law connecting the cause (net external torque) to the effect (change in angular momentum).

• Angular Impulse (${\vec{J}}_{\tau }$): The integral of torque over a time interval, ${\vec{J}}_{\tau }=\int \vec{\tau }dt$. It equals the total change in angular momentum, $\Delta \vec{L}$.

Skill Snapshots

Causation

• Driver → Change: A sustained net external torque applied to a flywheel → causes a continuous change in the flywheel's angular momentum.

• Driver → Change: A student on a spinning stool extending their legs outward (increasing $I$) → causes their angular velocity to decrease, as dictated by $L_{const}=I\omega$.

• Driver → Change: The absence of net external torque on a planet-star system → causes the planet's angular momentum vector to remain constant throughout its orbit.

Comparison

• System vs. System: A system subject to a net external frictional torque will have a decreasing angular momentum, whereas an idealized, frictionless system will maintain a constant angular momentum.

• Conservation Law vs. Conservation Law: Conservation of linear momentum applies when $\sum {\vec{F}}_{ext}=0$, while conservation of angular momentum applies when $\sum {\vec{\tau }}_{ext}=0$. A system can have one quantity conserved but not the other.

• Energy vs. Momentum: In an inelastic rotational collision (e.g., a clay disk dropped on a spinning platter), the total angular momentum of the system is conserved (no external torques), but rotational kinetic energy is not conserved (work is done during the non-elastic deformation).

Change and Continuity Over Time (CCOT)

• Baseline: A satellite orbits a planet in a stable circular path with constant angular momentum ${\vec{L}}_0$.

• Change 1: It fires thrusters tangentially to slow down. This creates an external torque, causing its angular momentum to decrease to a new value ${\vec{L}}_1$.

• Change 2: With thrusters off, it settles into a new, lower orbit. In this new orbit, the net external torque about the planet is again zero.

• Continuity: The angular momentum is now constant at the new value ${\vec{L}}_1$. The governing law $\sum {\vec{\tau }}_{ext}=d\vec{L}/dt$ remains valid throughout the entire process, describing both the periods of change and the periods of constancy.

Common Misconceptions & Clarifications

1. Misconception: "Angular momentum is conserved in any collision."

   Clarification: Angular momentum is conserved only if the net external torque on the system of colliding objects is zero. If, for example, the collision involves significant friction with an external axle, the angular momentum of the colliding objects will not be conserved.

2. Misconception: "If angular momentum is conserved, rotational kinetic energy must also be conserved."

   Clarification: This is false. A figure skater pulling in her arms conserves angular momentum ($L=I\omega$), but her rotational kinetic energy ($K=L^2/2I$) increases because she does positive work to decrease her moment of inertia ($I$). Energy is only conserved if the collision or reconfiguration is perfectly elastic.

3. Misconception: "An object must be moving in a circle to have angular momentum."

   Clarification: Any object with linear momentum $\vec{p}$ has angular momentum $\vec{L}=\vec{r}\times \vec{p}$ about any origin from which the position vector $\vec{r}$ is not parallel to $\vec{p}$. For example, a meteor moving in a straight line has a constant, non-zero angular momentum about any point not on its path of travel.

4. Misconception: "Forces that pass through the center of mass produce no torque."

   Clarification: A force produces zero torque only if its line of action passes through the pivot point or origin about which torque is being calculated. This point may or may not be the center of mass. For a free object in space, it is often convenient to choose the center of mass as the origin, but it is not a requirement.

One-Paragraph Summary

Conservation of angular momentum is a foundational principle in physics, arising directly from the rotational formulation of Newton's laws. The core mechanism is that the time rate of change of a system's total angular momentum is precisely equal to the net external torque exerted upon it. Consequently, for any system defined such that the net external torque is zero, its total angular momentum vector remains constant. This holds true even if the system's internal configuration changes, leading to the inverse relationship between moment of inertia and angular velocity observed in phenomena like a spinning skater or a collapsing star. The principle's predictive power depends critically on the careful selection of a system boundary to isolate it from external torques, providing a robust analytical tool for situations where forces may be complex but their rotational effects are simple.