Angular Momentum and Conservation of Angular Momentum: Definition, Formulas, Conditions, 3 Worked Examples, and 8 Applications in Sports, Astronomy, and Quantum Physics

What Is Angular Momentum?

Angular momentum is the quantity of rotational motion of an object about an axis or point, measured in kg·m²/s. For a rigid body, angular momentum equals rotational inertia multiplied by angular velocity. Examples include a spinning bicycle wheel, a planet orbiting the Sun, and a figure skater in a spin.

For a reliable reference, see OpenStax develops angular momentum for rigid bodies and its relation to rotational motion.

What Is the Formula for Angular Momentum?

The formula for angular momentum of a rigid body is L = Iω, where I is rotational inertia in kg·m² and ω is angular velocity in rad/s. The table below lists 6 formulas by situation. Rows represent situations. Columns show the formula and the meaning of the symbols. Physics 1 Formula Sheet: Equations, Constants, and Exam Rules for 8 Units

Does a Particle Moving in a Straight Line Have Angular Momentum?

Yes, a particle moving in a straight line has angular momentum about any point that does not lie on its path. The magnitude is L = mvd, where d is the perpendicular distance from the point to the path. A 2.0 kg particle moving at 3.0 m/s along a line 4.0 m from the origin has L = 2.0 × 3.0 × 4.0 = 24 kg·m²/s. Angular momentum depends on the chosen reference point.

What Is the Unit of Angular Momentum?

The SI unit of angular momentum is the kilogram meter squared per second (kg·m²/s), which equals the joule second (J·s). The unit also equals N·m·s. The Planck constant has the same unit, which links classical and quantum angular momentum.

Which Direction Does Angular Momentum Point?

Angular momentum points along the rotation axis, as given by the right-hand rule. Follow these 3 steps:

1. Curl the fingers of the right hand in the direction of rotation.
2. Extend the thumb along the axis.
3. Read the direction of L from the thumb.

A wheel spinning counterclockwise as seen from above has L pointing upward.

What Is Rotational Inertia in the Angular Momentum Formula?

Rotational inertia, also called moment of inertia, measures how mass is distributed about an axis. Mass farther from the axis gives a larger value. The table below lists 7 common values. Rows represent shapes. Columns show the axis and the formula.

How Are Torque and Angular Momentum Related?

Net torque equals the rate of change of angular momentum: τ_net = ΔL/Δt. For a rigid body with constant rotational inertia, this relation becomes τ_net = Iα. The product of torque and time is the angular impulse, written ΔL = τΔt. Formula for Change in Momentum: Final Momentum Minus Initial Momentum, Equal to Mass Times Change in Velocity and to Impulse

A constant torque of 5.0 N·m acting for 3.0 s changes angular momentum by 5.0 × 3.0 = 15 kg·m²/s.

What Is Conservation of Angular Momentum?

The law of conservation of angular momentum states that the total angular momentum of a system stays constant when the net external torque on the system is zero. The equation is L_initial = L_final. For a rigid body with changing rotational inertia, the equation is I₁ω₁ = I₂ω₂.

What Are the Conditions for Conservation of Angular Momentum?

Angular momentum is conserved under 4 conditions:

• The net external torque about the chosen axis equals zero.
• The system includes every object that exerts torque on another object in it.
• Internal forces act as action-reaction pairs along the line between objects.
• The reference axis stays the same before and after the event.

Linear momentum may change while angular momentum stays constant. A pivot exerts an external force on a rotating rod, so linear momentum changes. The pivot force has zero lever arm about the pivot, so angular momentum about the pivot is conserved.

Why Is Angular Momentum Conserved?

Angular momentum is conserved because zero net torque gives zero rate of change of L. From τ_net = ΔL/Δt, setting τ_net = 0 gives ΔL = 0. Internal torques cancel in pairs by Newton’s third law.

Emmy Noether proved in 1918 that every conserved quantity corresponds to a symmetry of the laws of physics. Conservation of angular momentum corresponds to rotational symmetry, meaning the laws of physics are the same in every direction in space.

Is Kinetic Energy Conserved When Angular Momentum Is Conserved?

No, rotational kinetic energy changes when rotational inertia changes. The relation is K = L²/(2I). With L constant, a smaller I gives a larger K. The skater in Example 1 below gains 266 J by pulling in the arms. Collisions in which objects stick together lose kinetic energy, as in Example 2.

How Do You Solve Conservation of Angular Momentum Problems?

Equate the total angular momentum before and after the event, about the same axis, in 6 steps:

1. Define the system and the axis.
2. Confirm that the net external torque equals zero.
3. Write L_initial for every object, with signs for direction.
4. Write L_final for the combined or changed system.
5. Set L_initial = L_final and solve for the unknown.
6. Check the unit (kg·m²/s) and the sign of the answer.

Example 1: How Does a Skater Increase Spin Speed?

A skater has I = 4.0 kg·m² and spins at 1.5 rev/s with arms out. Pulling in the arms reduces I to 1.6 kg·m². Find the new spin rate.

• Conserve L: I₁ω₁ = I₂ω₂.
• Solve: ω₂ = (4.0 × 1.5) ÷ 1.6 = 3.75 rev/s, equal to 23.6 rad/s.
• Compare energy: K₁ = ½ × 4.0 × (9.42)² ≈ 178 J and K₂ = ½ × 1.6 × (23.6)² ≈ 444 J.
• Result: spin rate rises 2.5 times, and the skater supplies about 266 J of work.

Example 2: What Happens When a Child Steps onto a Spinning Disk?

A disk with M = 200 kg and R = 2.0 m spins at 2.0 rad/s. A 40 kg child at rest steps onto the rim. Find the new angular speed.

• Disk inertia: I = ½MR² = ½ × 200 × 2.0² = 400 kg·m².
• Initial L: 400 × 2.0 = 800 kg·m²/s, because the child at rest contributes zero.
• Final inertia: 400 + 40 × 2.0² = 560 kg·m².
• Solve: ω = 800 ÷ 560 ≈ 1.43 rad/s.
• Energy check: K falls from 800 J to about 571 J, a loss of about 229 J.

Example 3: What Happens When a Bullet Hits a Pivoted Rod?

A 0.010 kg bullet moving at 400 m/s embeds in the free end of a 2.0 kg rod, 1.0 m long, pivoted at the other end. Find the angular speed after impact.

• Initial L about the pivot: mvL = 0.010 × 400 × 1.0 = 4.0 kg·m²/s.
• Final inertia: (1/3) × 2.0 × 1.0² + 0.010 × 1.0² ≈ 0.677 kg·m².
• Solve: ω = 4.0 ÷ 0.677 ≈ 5.9 rad/s.
• Energy check: K falls from 800 J to about 11.8 J.

What Are Examples of Conservation of Angular Momentum?

Conservation of angular momentum appears in 8 common situations. The table below lists each example. Rows represent examples. Columns show what changes and the result.

How Does Conservation of Angular Momentum Explain Kepler’s Second Law?

Kepler’s second law states that a planet sweeps equal areas in equal times. Gravity from the Sun acts along the line to the planet, so it exerts zero torque about the Sun. Angular momentum L = mr² dθ/dt stays constant, and the swept-area rate dA/dt = L/(2m) stays constant. Johannes Kepler published the law in 1609.

Using NASA planetary fact sheet values, Earth’s orbital angular momentum is about 2.66 × 10⁴⁰ kg·m²/s. Earth’s spin angular momentum is about 5.9 × 10³³ kg·m²/s, which is about 4.5 million times smaller.

How Does Conservation of Angular Momentum Explain the Moon Moving Away from Earth?

Tides transfer angular momentum from Earth’s spin to the Moon’s orbit. Earth passed angular momentum through tidal friction to the Moon’s orbit, which lengthened Earth’s day and moved the Moon outward. Apollo Lunar Laser Ranging measures the present recession rate at 3.8 cm per year. The change in day length is about 2 milliseconds per century. Tidal analysis places the solar day near 13 hours about 3.2 billion years ago.

How Does Conservation of Angular Momentum Explain Fast-Spinning Neutron Stars?

A collapsing star core spins faster as its radius shrinks. For a uniform sphere of constant mass, ω₂ = ω₁(R₁/R₂)². A radius reduced by a factor of 10 raises the spin rate by a factor of 100. Hessels and colleagues reported a pulsar spinning at 716 Hz in the journal Science in 2006.

What Is the Difference Between Angular Momentum and Linear Momentum?

Linear momentum measures motion along a line, and angular momentum measures rotation about an axis or point. The table below compares 7 quantities. Rows represent quantities. Columns show the linear and angular versions.

What Is the Difference Between Angular Momentum and Torque?

Angular momentum is a state of rotation, and torque is the cause of its change. Angular momentum has the unit kg·m²/s, and torque has the unit N·m. A constant torque changes angular momentum at a steady rate. Zero torque leaves angular momentum unchanged.

What Is Angular Momentum in Quantum Mechanics?

Angular momentum in quantum mechanics is quantized in units of the reduced Planck constant ħ. According to NIST CODATA 2018, ħ = 1.054571817 × 10⁻³⁴ J·s. Niels Bohr proposed in 1913 that electron orbital angular momentum in hydrogen equals nħ, with n = 1, 2, 3.

Three facts summarize the quantum case:

• Orbital angular momentum has magnitude √(l(l+1)) ħ, with l = 0, 1, 2.
• Electron spin is ½ħ, and photon spin is 1ħ.
• Total angular momentum is conserved in atomic transitions and particle decays.

What Are Common Mistakes with Angular Momentum Problems?

Six mistakes cause most errors:

• Using different axes before and after the event
• Applying conservation when an external torque acts
• Mixing rev/s and rad/s
• Conserving kinetic energy in a sticking collision
• Omitting the rotational inertia of added masses
• Ignoring direction signs for opposite spins

What Are Angular Momentum and Conservation of Angular Momentum in Summary?

Angular momentum is L = Iω for rigid bodies and L = mvr sin θ for particles, measured in kg·m²/s. Net external torque changes angular momentum through τ_net = ΔL/Δt. Conservation of angular momentum holds when net external torque equals zero, giving I₁ω₁ = I₂ω₂. Skaters, divers, planets, the Earth-Moon system, and pulsars follow this law. Kinetic energy changes when rotational inertia changes.

People Also Ask About Angular Momentum and Conservation of Angular Momentum

What is angular momentum in simple terms?

Angular momentum is the quantity of rotational motion of an object about an axis or point.

What is the formula for angular momentum?

The formula is L = Iω for a rigid body and L = mvr sin θ for a particle.

What is the unit of angular momentum?

The SI unit is kg·m²/s, which equals J·s.

Is angular momentum a vector or a scalar?

Angular momentum is a vector, and the right-hand rule gives its direction.

What is the law of conservation of angular momentum?

The total angular momentum of a system stays constant when the net external torque is zero.

When is angular momentum not conserved?

Angular momentum changes when a net external torque acts on the system.

Why does a skater spin faster when pulling in the arms?

Pulling in the arms lowers rotational inertia, so angular speed rises to keep angular momentum constant.

Is kinetic energy conserved when angular momentum is conserved?

No, kinetic energy changes when rotational inertia changes, according to K = L²/(2I).

What is the relationship between torque and angular momentum?

Net torque equals the rate of change of angular momentum, τ_net = ΔL/Δt.

Does a particle in a straight line have angular momentum?

Yes, it has L = mvd about any point at perpendicular distance d from its path.

How does angular momentum explain Kepler’s second law?

Gravity from the Sun exerts zero torque about the Sun, so a planet’s angular momentum and swept-area rate stay constant.

What symmetry gives conservation of angular momentum?

Rotational symmetry gives conservation of angular momentum, according to Noether’s theorem of 1918.

Maurice Cotterell

MEd., BSc. QTS/OCT – Physics Teacher at Davenant Foundation School
The University of the West Indies

My passion as a teacher and leader in education is to contribute to students’ enhanced learning by creation of a positive teaching/learning environment, where they are motivated to develop strong critical thinking, analytical and problem-solving skills.
As acting Vice-Principal and head of Physics, I have amassed five years of experience in leading meetings, implementing strategies to maintain discipline and school ethos, liaise with stakeholders in planning for future development of the institution; successfully negotiated the implementation of various projects for student development as well as supervising effective execution of such. For the past 19 years, I have incorporated discovery learning within the classroom through practical activities, problem solving and cooperative learning and have realized that students retention in Physics and Mathematics have increased significantly.

I am always cognizant that students have different learning styles, and as such teaching strategies during planning and delivery must take this crucial factor into consideration and be tailored to meet such differences.