11.3 Conservation of Angular Momentum
3 min read•june 24, 2024
conservation is a fundamental principle in physics, describing how rotating objects maintain their spin. It's crucial for understanding everything from figure skaters to planetary motion.
When no act, stays constant. This concept explains why objects speed up when they contract and slow down when they expand, playing a key role in celestial mechanics and everyday rotational motion.
Conservation of Angular Momentum
Angular velocity in varying systems
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- Angular momentum () remains constant in an no external torques act upon it
- Formula for angular momentum: , represents , represents
- changes cause to adjust maintaining constant angular momentum
- Increasing leads to proportional decrease in ( pulling arms inward)
- Decreasing leads to proportional increase in (spinning figure skater extending arms outward)
- Initial and final states relate via equation:
- Allows calculation of angular velocity change given moment of inertia variation (, )
- The is crucial in determining the moment of inertia and subsequent angular velocity changes
Rotational energy with changing parameters
- Formula for :
- with varying moment of inertia affects rotational energy
- Increasing decreases and reduces (slowing spinning playground as children move outward)
- Decreasing increases and raises (speeding spinning playground merry-go-round as children move inward)
- Magnitude of moment of inertia change determines extent of rotational energy change
- Larger changes yield greater changes (significant collapse of a , drastic expansion of a star)
- Altering moment of inertia requires work input or output
- Work done accounts for rotational kinetic energy change (pushing children inward on playground merry-go-round adds energy)
Conservation of angular momentum in astronomy
- Stars and planets approximate rigid bodies with fixed mass
- Angular momentum conserved in these astronomical systems (no external torques, ignoring gravitational influences)
- Stellar or planetary contraction (radius reduction) decreases moment of inertia
- Decreasing increases conserving angular momentum (collapsing spinning molecular cloud rotates faster)
- Shrinking object exhibits faster rotation ( spins rapidly after )
- Stellar or planetary expansion (radius growth) increases moment of inertia
- Increasing decreases conserving angular momentum (expanding rotating star slows down)
- Expanding object exhibits slower rotation ( spins slower as it dramatically expands)
- from contracting gas cloud:
- Cloud contraction increases rotation rate conserving angular momentum
- Enhanced rotation flattens cloud into disk (conservation of angular momentum)
- Planets form from disk material inheriting cloud's angular momentum (orbits in same direction as original cloud rotation)
Rotational dynamics and angular momentum
- is the rotational equivalent of force and can change angular momentum
- (moment of inertia) determines an object's resistance to changes in rotational motion
- results from applied torque and is inversely proportional to rotational inertia
- is a change in the orientation of the rotational axis, often seen in spinning tops or gyroscopes
Key Terms to Review (35)
Angular acceleration: Angular acceleration is the rate of change of angular velocity over time. It describes how quickly an object is rotating or spinning.
Angular Acceleration: Angular acceleration is the rate of change of angular velocity with respect to time. It describes the rotational analog of linear acceleration, quantifying the change in the rotational motion of an object around a fixed axis or point.
Angular momentum: Angular momentum is a measure of the quantity of rotation of an object and is a vector quantity. It is given by the product of the moment of inertia and angular velocity.
Angular Momentum: Angular momentum is a fundamental concept in physics that describes the rotational motion of an object. It is the measure of an object's rotational inertia and its tendency to continue rotating around a specific axis. Angular momentum is a vector quantity, meaning it has both magnitude and direction, and it plays a crucial role in understanding the behavior of rotating systems across various topics in physics.
Angular velocity: Angular velocity is the rate at which an object rotates around a fixed axis. It is measured in radians per second (rad/s).
Angular Velocity: Angular velocity is a measure of the rate of change of the angular position of an object. It describes the speed of rotation or the change in the orientation of an object around a fixed axis or point. This concept is fundamental in understanding the motion of objects undergoing circular or rotational motion.
Axis of Rotation: The axis of rotation is the imaginary line around which an object or system rotates. It is the fixed point or line that an object pivots or spins around as it undergoes rotational motion.
Collapsing Spinning Ice Skater: The collapsing spinning ice skater is a classic example used to illustrate the principle of conservation of angular momentum. It demonstrates how an object's rotational speed can change dramatically when its moment of inertia is altered by changing the distribution of its mass.
Conservation of Angular Momentum: Conservation of angular momentum is a fundamental principle in physics that states the total angular momentum of a closed system remains constant unless an external torque is applied. This principle is essential in understanding the behavior of rotational motion and the dynamics of spinning objects.
Core Collapse Supernova: A core collapse supernova is a type of supernova that occurs when the core of a massive star collapses under its own gravity, triggering a catastrophic explosion that can outshine an entire galaxy. This event is closely tied to the concept of conservation of angular momentum, as the collapsing core spins faster and faster, profoundly impacting the dynamics of the explosion.
Expanding Rotating Nebula: An expanding rotating nebula is a vast cloud of interstellar gas and dust that is expanding outward while also rotating around a central axis. This phenomenon is a crucial step in the formation of new stars and planetary systems within the universe.
External Torques: External torques refer to the rotational forces that act on an object from outside the system. These torques can cause the object to rotate or change its angular momentum, which is a crucial concept in the study of rotational dynamics and the conservation of angular momentum.
I₁ω₁ = I₂ω₂: The equation I₁ω₁ = I₂ω₂ represents the principle of conservation of angular momentum, which states that the angular momentum of a system is constant unless an external torque is applied. This relationship is a fundamental concept in the study of rotational dynamics and is crucial for understanding the behavior of rotating systems.
Isolated System: An isolated system is a physical system that does not exchange any matter with its surroundings, but may exchange energy. It is a closed system that is completely separated from its environment, allowing for the study of the system's internal processes and the conservation of certain physical quantities.
Kg⋅m²: kg⋅m² is a unit of angular momentum, which is a measure of the rotational motion of an object. It represents the product of an object's mass (in kilograms) and the square of its distance from the axis of rotation (in meters squared). This unit is crucial in understanding the conservation of angular momentum, a fundamental principle in classical mechanics.
Kr = ½Iω²: The equation Kr = ½Iω² represents the kinetic energy of rotation, where Kr is the rotational kinetic energy, I is the moment of inertia of the rotating object, and ω is the angular velocity of the object. This equation is a fundamental principle in the context of the conservation of angular momentum.
L = Iω: L = Iω is the equation that represents the relationship between angular momentum (L), moment of inertia (I), and angular velocity (ω). This equation is a fundamental principle in the conservation of angular momentum, which describes how the total angular momentum of a closed system remains constant unless an external torque is applied.
Law of conservation of angular momentum: The law of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of the system remains constant. This principle is crucial in understanding rotational dynamics.
Merry-Go-Round: A merry-go-round, also known as a carousel, is a type of amusement park ride consisting of a rotating circular platform with seats for riders. It is a classic example of a system that exhibits conservation of angular momentum, a fundamental principle in rotational dynamics.
Molecular Cloud: A molecular cloud is a type of interstellar cloud that is primarily composed of molecular hydrogen (H2) and other molecules. These clouds are the birthplaces of stars, where gravity causes the gas and dust to collapse and form new stellar systems. The study of molecular clouds is crucial in understanding the process of star formation and the evolution of galaxies.
Moment of inertia: Moment of inertia is a measure of an object's resistance to changes in its rotational motion about a fixed axis. It depends on the mass distribution relative to the axis of rotation.
Moment of Inertia: The moment of inertia is a measure of an object's resistance to rotational acceleration. It is a scalar quantity that depends on the mass and distribution of an object's mass about a given axis of rotation. The moment of inertia is a crucial concept in the study of rotational dynamics, as it determines how an object will respond to applied torques.
Neutron star: A neutron star is a highly dense remnant of a massive star that has undergone a supernova explosion and collapsed under gravity. Composed mostly of neutrons, it exhibits incredibly strong gravitational and magnetic fields.
Neutron Star: A neutron star is an extremely dense, collapsed stellar remnant that forms when a massive star runs out of fuel and undergoes gravitational collapse. These objects are characterized by their incredibly dense, neutron-degenerate matter and their strong gravitational and magnetic fields.
Precession: Precession is the gradual change in the orientation of the rotational axis of a rotating body. It occurs due to an external torque acting on the body.
Precession: Precession is the phenomenon where the axis of rotation of a spinning object, such as a gyroscope or a planet, slowly changes direction over time. This change in the orientation of the rotational axis occurs without any external torque being applied to the object.
Rad/s: Radians per second (rad/s) is a unit of angular velocity, which measures the rate of change of an object's angular position over time. It is a fundamental unit in the study of rotational motion and is used to quantify the speed of rotating or spinning objects.
Red Giant Star: A red giant star is an aging, large, and luminous star that has exhausted the hydrogen in its core and has expanded in size, often becoming hundreds of times larger than the Sun. This expansion is a result of the star's transition into a later stage of its life cycle.
Rigid Body: A rigid body is an idealized object that maintains its shape and size regardless of the forces acting upon it. It is a fundamental concept in classical mechanics that simplifies the analysis of the motion and behavior of objects.
Rotating Platform: A rotating platform is a surface or structure that rotates around a fixed axis, often used in physics experiments to study the principles of angular momentum and rotational dynamics. The rotation of the platform allows for the observation and analysis of objects or systems undergoing rotational motion.
Rotational Inertia: Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to changes in its rotational motion. It is the rotational equivalent of linear inertia, which is a measure of an object's resistance to changes in its linear motion.
Rotational kinetic energy: Rotational kinetic energy is the energy an object possesses due to its rotation. It is given by $$KE_{rot} = \frac{1}{2} I \omega^2$$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.
Solar System Formation: Solar system formation is the process by which the Sun and its orbiting planets, moons, asteroids, and other celestial bodies were created from a giant molecular cloud approximately 4.6 billion years ago. This formation is closely tied to the concept of conservation of angular momentum, which governs the rotational motion and orientation of the objects within the solar system.
Spinning Figure Skater: A spinning figure skater is an athlete who performs rotational movements on ice skates, demonstrating their mastery of angular momentum. These spins are a core component of figure skating routines, showcasing the skater's balance, control, and technical prowess.
Torque: Torque is a measure of the rotational force applied to an object, which causes it to rotate about an axis. It is influenced by the magnitude of the force applied, the distance from the axis of rotation, and the angle at which the force is applied, making it crucial for understanding rotational motion and equilibrium.