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Angular Momentum Calculator

Calculate the angular momentum of an object using moment of inertia

Input Parameters

For Angular Momentum (L)

kg·m²
rad/s

For Linear Momentum (p)

kg
m/s

This calculator computes angular momentum, the rotational equivalent of linear momentum. [1, 2, 4] It is a conserved quantity, meaning it remains constant unless an external torque acts on the system. [2, 3]

  • Angular Momentum (L): The product of an object's moment of inertia and its angular velocity. [1]
  • Linear Momentum (p): The product of an object's mass and its linear velocity. [1]

L = I × ω

p = m × v

Enter parameters and click Calculate

About Angular Momentum Calculator

The Universe's Spin: A Master Guide to Our Angular Momentum Calculator

Watch an ice skater pull in their arms and spin faster. Observe a planet speed up as it gets closer to its star. See how a gyroscope resists being tilted. These seemingly unrelated phenomena are all governed by one of the most profound and fundamental conservation laws in the universe: the **Conservation of Angular Momentum**.

Just as linear momentum describes an object's "quantity of motion" in a straight line, **angular momentum** (`L`) describes its "quantity of rotational motion." It is the rotational equivalent of linear momentum and is a measure of how much effort is required to stop an object from spinning. More importantly, in a closed system, it is a quantity that never changes.

Welcome to the definitive guide to this cornerstone of physics. Our Angular Momentum Calculator is a tool designed to help you quantify this crucial value for both single particles and rotating rigid bodies. This comprehensive article will not only guide you through the calculator's use but will also take you on a deep dive into the physics, revealing the elegant principles that explain everything from the stability of a bicycle to the formation of galaxies.

What is Angular Momentum? Inertia in Rotation

Angular momentum is a measure of an object's rotational inertia. It depends not just on how fast an object is spinning, but also on its mass and how that mass is distributed relative to the axis of rotation.

Like its linear counterpart, angular momentum is a **vector quantity**. It has both a magnitude (how much rotational motion) and a direction. The direction is determined by the **right-hand rule**: if you curl the fingers of your right hand in the direction of the object's rotation, your thumb points in the direction of the angular momentum vector, which lies along the axis of rotation.

Our calculator can compute angular momentum for two primary scenarios.

Scenario 1: Angular Momentum of a Single Particle

Consider a single particle of mass `m` moving with a linear velocity `v` at a distance `r` from a central point. This could be a planet orbiting the sun or a ball being swung on a string. Its angular momentum `L` is defined as the cross product of its position vector `r` and its linear momentum vector `p` (`p = mv`).

The magnitude of the angular momentum for a particle in circular motion (or at the point of closest approach) is given by a simpler formula:

L = r * p = r * m * v
  • L is the angular momentum, measured in **kilogram-meters squared per second (kg·m²/s)**.
  • r is the **radius** of the circular path or the distance from the pivot point, in meters (m).
  • m is the **mass** of the particle in kilograms (kg).
  • v is the **linear (tangential) velocity** of the particle in meters per second (m/s).

Scenario 2: Angular Momentum of a Rotating Rigid Body

For a solid, extended object that is rotating (like a spinning wheel, a planet rotating on its axis, or a spinning top), we use a different but analogous formula. We need to know how the object's mass is distributed.

L = I * ω
  • L is the angular momentum (kg·m²/s).
  • I is the **Moment of Inertia** of the object. This is the rotational equivalent of mass. It measures an object's resistance to being spun up or slowed down. Its value depends on the object's mass and, critically, how that mass is shaped and distributed around the axis of rotation. It is measured in **kg·m²**.
  • ω (omega) is the angular velocity of the object, measured in radians per second (rad/s).

Notice the beautiful parallel between the linear and rotational formulas:

Linear Momentum

p = m * v

Angular Momentum

L = I * ω

How to Use the Angular Momentum Calculator

Step 1: Select the Scenario

Choose whether you are calculating for a **Single Particle** in orbit or a **Rotating Rigid Body**. This will present the correct set of inputs.

Step 2: Enter the Variables

For a Single Particle:

  • Enter the **Mass (m)** in kg.
  • Enter the **Velocity (v)** in m/s.
  • Enter the **Radius (r)** in meters.

For a Rotating Rigid Body:
  • Enter the **Moment of Inertia (I)** in kg·m². (You may need to calculate this first based on the object's shape).
  • Enter the **Angular Velocity (ω)** in rad/s. (Our calculator can help convert from common units like RPM).

Step 3: View the Result

The calculator will instantly compute the angular momentum `L` in its standard SI unit of kg·m²/s.

The Law of Conservation of Angular Momentum

This is where the concept becomes truly powerful. The law states that if no **external torque** acts on a rotating system, its total angular momentum (`L`) will remain constant. A torque is a rotational force—a twist or a turn.

If Net External Torque = 0, then L_initial = L_final

This means `I_initial * ω_initial = I_final * ω_final`. This single equation explains a vast range of phenomena.

Example: The Spinning Ice Skater

An ice skater begins a spin with her arms outstretched. She then pulls her arms in tight to her body.

  • • Initial State (Arms Out): Her mass is distributed far from her axis of rotation. This gives her a **large Moment of Inertia (I_initial)**. She spins with a certain angular velocity (ω_initial).
  • • Final State (Arms In): By pulling her arms in, she brings her mass closer to the axis of rotation. This **decreases her Moment of Inertia (I_final)**.
  • • Conservation in Action: Since no external torque is acting on her (we can ignore the tiny friction from the ice), her angular momentum `L` must stay the same. Because `L = I * ω` must be constant, and `I` has decreased, her angular velocity `ω` **must increase** to compensate. She spins dramatically faster.

Real-World Examples and Importance

Planetary Orbits

Planets move in elliptical orbits. As a planet gets closer to its star (`r` decreases), its angular momentum (`L=rmv`) must be conserved. To compensate for the smaller radius, its velocity (`v`) must increase. This is why planets move fastest at perihelion (closest approach) and slowest at aphelion (farthest point).

Gyroscopic Stability

A spinning top, a gyroscope, or even a rifle bullet is stable because of its angular momentum. The spinning gives it a large `L` vector. To change the direction of this vector (i.e., to make the object wobble or tumble), an external torque must be applied. The object's angular momentum gives it "gyroscopic inertia" that resists this change in orientation, keeping it stable.

Diving and Gymnastics

A diver leaves the board with a certain amount of angular momentum. By tucking into a tight ball (`I` decreases), they can increase their rotational speed (`ω` increases) to perform multiple somersaults. By extending their body before hitting the water (`I` increases), they slow their rotation (`ω` decreases) for a clean entry.

Galaxy Formation

A vast, slow-rotating cloud of gas and dust in space has an enormous amount of angular momentum due to its sheer size. As gravity pulls this cloud together to form a star and planets, its overall radius `r` decreases dramatically. To conserve angular momentum, its rotational speed `ω` must increase immensely, leading to the formation of a spinning protostellar disk from which planets are born.

Frequently Asked Questions (FAQ)

Q: How do I find the Moment of Inertia (I)?

The Moment of Inertia depends on the shape of the object. For simple shapes, there are standard formulas (e.g., for a solid sphere, `I = (2/5)mr²`; for a solid disk, `I = ½mr²`). For complex shapes like a car or a human body, it is usually determined experimentally or through computer-aided design (CAD) software.

Q: What is a "torque"?

Torque (`τ`) is the rotational equivalent of force. Just as a net force causes a change in linear momentum, a net torque causes a change in angular momentum (`τ = ΔL / Δt`). If you want to make something start spinning, stop spinning, or change its axis of spin, you must apply a torque.

A Pillar of Modern Physics

The conservation of angular momentum is a principle that holds true from the quantum level of electron spin to the cosmic scale of galactic rotation. It is a concept of breathtaking scope and elegance. Understanding it is essential for anyone serious about physics, astronomy, or engineering.

Use our calculator to quantify the "quantity of rotation" in various systems. Explore how changing mass distribution affects angular velocity. Calculate the immense angular momentum of our own planet and gain a new appreciation for the universal law that keeps our world spinning steadily on its axis.

Frequently Asked Questions

What is angular momentum?
Angular momentum (L) is a vector quantity that represents the rotational equivalent of linear momentum, calculated as L = Iω where I is moment of inertia and ω is angular velocity.
What are the SI units of angular momentum?
The SI units of angular momentum are kilogram-meter squared per second (kg·m²/s).
How is angular momentum related to linear momentum?
Angular momentum is the rotational analog of linear momentum, with L = r × p where r is the position vector and p is linear momentum.
What is the conservation of angular momentum?
The law of conservation of angular momentum states that the total angular momentum of a closed system remains constant in the absence of external torques.
How do I calculate angular momentum for a point particle?
For a point particle, L = r × p = m(r × v), where m is mass, r is position vector, and v is velocity.
What is the relationship between angular momentum and torque?
Torque equals the time derivative of angular momentum: τ = dL/dt.
How do I find angular momentum for a rigid body?
For a rigid body, L = Iω, where I is the moment of inertia and ω is the angular velocity.
What is the angular momentum of a rotating disk?
For a rotating disk, L = ½MR²ω, where M is mass, R is radius, and ω is angular velocity.
How does angular momentum change in a collision?
In an isolated system, total angular momentum is conserved during collisions, though it may be redistributed between objects.
What is the angular momentum of a spinning top?
For a spinning top, L = Iω, where I depends on the shape and mass distribution of the top.
How do I calculate angular momentum for orbital motion?
For orbital motion, L = mvr sin(θ), where m is mass, v is velocity, r is radius, and θ is the angle between r and v.
What is the relationship between angular momentum and kinetic energy?
Rotational kinetic energy equals L²/(2I), where L is angular momentum and I is moment of inertia.
How does angular momentum relate to precession?
Precession occurs when a torque is applied perpendicular to the angular momentum vector, causing the axis of rotation to change direction.
What is the angular momentum of a particle in a magnetic field?
For a charged particle in a magnetic field, the angular momentum can change due to the magnetic force acting as a torque.
How do I find angular momentum for a system of particles?
For a system of particles, total angular momentum equals the vector sum of individual angular momenta: L_total = ΣL_i.
What is the angular momentum of a gyroscope?
A gyroscope maintains its angular momentum direction due to conservation of angular momentum, resisting changes in orientation.
How does angular momentum affect stability?
Higher angular momentum increases the stability of rotating objects, making them more resistant to external torques.
What is the relationship between angular momentum and moment of inertia?
Angular momentum equals moment of inertia times angular velocity: L = Iω.
How do I calculate angular momentum for a rotating sphere?
For a rotating sphere, L = (2/5)MR²ω, where M is mass, R is radius, and ω is angular velocity.
What is the angular momentum of a planet orbiting the sun?
For planetary orbits, L = mvr, where m is planet mass, v is orbital velocity, and r is orbital radius.
How does angular momentum change with radius?
For a given angular velocity, angular momentum increases with the square of the radius due to moment of inertia dependence.
What is the angular momentum of a rotating rod?
For a rotating rod about its center, L = (1/12)ML²ω, where M is mass, L is length, and ω is angular velocity.
How do I find angular momentum for a rotating cylinder?
For a rotating cylinder, L = ½MR²ω, where M is mass, R is radius, and ω is angular velocity.
What is the relationship between angular momentum and angular acceleration?
Angular acceleration equals the time derivative of angular velocity: α = dω/dt, and torque equals Iα.
How does angular momentum affect the motion of a spinning object?
Angular momentum determines the object's resistance to changes in its rotational motion and direction.
What is the angular momentum of a rotating ring?
For a rotating ring, L = MR²ω, where M is mass, R is radius, and ω is angular velocity.
How do I calculate angular momentum for a rotating cube?
For a rotating cube about an axis through its center, L = (1/6)Ma²ω, where M is mass, a is side length, and ω is angular velocity.
What is the angular momentum of a satellite in orbit?
For a satellite in circular orbit, L = m√(GMr), where m is satellite mass, G is gravitational constant, M is central mass, and r is orbital radius.
How does angular momentum relate to the right-hand rule?
The direction of angular momentum follows the right-hand rule: curl fingers in direction of rotation, thumb points in L direction.
What is the angular momentum of a rotating cone?
For a rotating cone about its axis, L = (3/10)MR²ω, where M is mass, R is base radius, and ω is angular velocity.
How do I find angular momentum for a rotating ellipsoid?
For a rotating ellipsoid, L depends on the axis of rotation and the principal moments of inertia of the ellipsoid.
What is the relationship between angular momentum and angular velocity?
Angular momentum is directly proportional to angular velocity: L = Iω, where I is the proportionality constant (moment of inertia).
How does angular momentum change in a rotating system?
Angular momentum changes only when an external torque is applied to the system: ΔL = τΔt.
What is the angular momentum of a rotating hemisphere?
For a rotating hemisphere about its axis, L = (2/5)MR²ω, where M is mass, R is radius, and ω is angular velocity.
How do I calculate angular momentum for a rotating pyramid?
For a rotating pyramid, L depends on the axis of rotation and the mass distribution relative to that axis.
What is the angular momentum of a rotating torus?
For a rotating torus, L = M(R² + ¾r²)ω, where M is mass, R is major radius, r is minor radius, and ω is angular velocity.
How does angular momentum affect the behavior of spinning objects?
Angular momentum causes spinning objects to maintain their axis of rotation and resist external torques.
What is the relationship between angular momentum and angular displacement?
Angular momentum is related to angular displacement through the moment of inertia and angular velocity.
How do I find angular momentum for a rotating prism?
For a rotating prism, L depends on the axis of rotation and the geometric properties of the prism.
What is the angular momentum of a rotating sphere with a hole?
For a rotating sphere with a hole, L = (2/5)M(R² - r²)ω, where M is mass, R is outer radius, r is hole radius, and ω is angular velocity.
How does angular momentum relate to the parallel axis theorem?
The parallel axis theorem allows calculation of moment of inertia about any axis, which then determines angular momentum.
What is the angular momentum of a rotating cylinder with a hole?
For a rotating cylinder with a hole, L = ½M(R² - r²)ω, where M is mass, R is outer radius, r is hole radius, and ω is angular velocity.
How do I calculate angular momentum for a rotating irregular shape?
For irregular shapes, angular momentum is calculated by integrating over the mass distribution: L = ∫r × v dm.
What is the relationship between angular momentum and angular impulse?
Angular impulse equals the change in angular momentum: J = ΔL = τΔt.
How does angular momentum affect the motion of a spinning top?
Angular momentum causes a spinning top to precess around the vertical axis due to gravitational torque.
What is the angular momentum of a rotating cone with a hole?
For a rotating cone with a hole, L depends on the mass distribution and the specific geometry of the cone and hole.
How do I find angular momentum for a rotating composite object?
For composite objects, total angular momentum equals the sum of angular momenta of all components: L_total = ΣL_i.
What is the relationship between angular momentum and angular power?
Angular power equals torque times angular velocity: P = τω, which relates to the rate of change of angular momentum.
How does angular momentum affect the stability of rotating systems?
Higher angular momentum increases stability by making the system more resistant to external disturbances.
What is the angular momentum of a rotating ring with varying density?
For a ring with varying density, L = ∫ρ(r)r²ω dV, where ρ(r) is the density function and the integral is over the ring's volume.
How do I calculate angular momentum for a rotating object with changing mass?
For objects with changing mass, angular momentum can change due to both mass changes and external torques.
What is the relationship between angular momentum and angular frequency?
Angular momentum equals moment of inertia times angular frequency: L = Iω, where ω is angular frequency.
How does angular momentum affect the behavior of gyroscopic instruments?
Angular momentum enables gyroscopic instruments to maintain orientation and resist external torques.
What is the angular momentum of a rotating object in a gravitational field?
In a gravitational field, angular momentum can change due to gravitational torques acting on the object.
How do I find angular momentum for a rotating object with multiple axes?
For objects rotating about multiple axes, total angular momentum is the vector sum of angular momenta about each axis.
What is the relationship between angular momentum and angular acceleration in non-inertial frames?
In non-inertial frames, additional terms may appear in the angular momentum equation due to fictitious forces.
How does angular momentum affect the motion of spinning particles?
Angular momentum determines the quantum mechanical properties and behavior of spinning particles.
What is the angular momentum of a rotating object with time-dependent moment of inertia?
For objects with time-dependent moment of inertia, angular momentum can change even without external torques.
How do I calculate angular momentum for a rotating object in a fluid?
In a fluid, angular momentum can be affected by drag forces and fluid dynamics acting on the rotating object.
What is the relationship between angular momentum and angular velocity in relativistic systems?
In relativistic systems, angular momentum and angular velocity relationships are modified by relativistic effects.
How does angular momentum affect the behavior of rotating celestial bodies?
Angular momentum determines the rotation rates, stability, and orbital characteristics of celestial bodies.
What is the angular momentum of a rotating object with internal degrees of freedom?
For objects with internal degrees of freedom, total angular momentum includes both external and internal contributions.
How do I find angular momentum for a rotating object in an electromagnetic field?
In electromagnetic fields, angular momentum can change due to electromagnetic torques acting on the object.
What is the relationship between angular momentum and angular velocity in quantum systems?
In quantum systems, angular momentum is quantized and related to angular velocity through quantum mechanical principles.
How does angular momentum affect the stability of rotating machinery?
Angular momentum contributes to the stability and smooth operation of rotating machinery by resisting external disturbances.
What is the angular momentum of a rotating object with variable geometry?
For objects with variable geometry, angular momentum changes as the moment of inertia changes with the geometry.
How do I calculate angular momentum for a rotating object in a non-uniform gravitational field?
In non-uniform gravitational fields, angular momentum can change due to varying gravitational torques across the object.
What is the relationship between angular momentum and angular velocity in dissipative systems?
In dissipative systems, angular momentum can decrease over time due to energy losses, affecting angular velocity.
How does angular momentum affect the behavior of rotating objects in space?
In space, angular momentum conservation is crucial for maintaining orientation and stability of rotating objects.
What is the angular momentum of a rotating object with time-varying external torques?
With time-varying external torques, angular momentum changes according to the integral of torque over time.
How do I find angular momentum for a rotating object in a rotating reference frame?
In rotating reference frames, angular momentum calculations must account for the frame's own rotation and fictitious forces.
What is the relationship between angular momentum and angular velocity in coupled rotating systems?
In coupled rotating systems, angular momentum can be exchanged between components while total angular momentum is conserved.
How does angular momentum affect the dynamics of rotating objects in fluids?
In fluids, angular momentum affects the object's rotational dynamics through fluid-structure interactions and drag effects.

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