10.5 Angular Momentum and Its Conservation
3 min read•june 18, 2024
is the rotational equivalent of in physics. It's crucial for understanding spinning objects, from figure skaters to planets. This concept helps explain why objects rotate faster when their mass distribution changes.
is a fundamental principle in physics. It states that when no external acts on a system, its total remains constant. This principle explains various phenomena, from the motion of celestial bodies to the tricks performed by gymnasts.
Angular Momentum
Angular vs linear momentum
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- Angular momentum () is the rotational equivalent of ()
- Linear momentum calculated using [p = mv](https://www.fiveableKeyTerm:p_=_mv), is mass and is velocity (bullet fired from a gun)
- Angular momentum calculated using , is and is (spinning flywheel)
- Both angular and linear momentum are vector quantities
- Linear momentum points in the same direction as linear velocity (arrow fired from a bow)
- Angular momentum points perpendicular to the plane of rotation, determined by the (spinning top)
- Conservation principles apply to both angular and linear momentum
- In the absence of external forces, linear momentum is conserved (colliding billiard balls)
- In the absence of external torques, angular momentum is conserved (orbiting planets)
- Mass affects linear momentum, while affects angular momentum
- Moment of inertia depends on mass distribution and shape of the object (hula hoop vs solid disk)
Torque effects on angular momentum
- Torque () is the rotational analogue of force in linear systems
- Torque causes a change in angular momentum over time: (opening a door)
- The net external torque acting on a system determines the change in angular momentum
- If net external torque is zero, angular momentum is conserved (spinning )
- Torque is a vector quantity, with its direction determined by the
- The direction of torque is perpendicular to the plane formed by the position vector and the force vector (turning a wrench)
- The magnitude of torque depends on the magnitude of the force and the perpendicular distance from the to the line of action of the force: (pushing a merry-go-round)
- is the rotational analog of linear impulse and represents the change in angular momentum due to torque applied over time
Conservation of Angular Momentum
Conservation of angular momentum
- When no external torque acts on a system, total angular momentum is conserved: (spinning figure skater)
- For a system with multiple objects, the total angular momentum is the sum of individual angular momenta: (solar system)
- When the moment of inertia changes, must change to conserve angular momentum:
- Figure skater starts spinning with arms extended
- Figure skater pulls arms inward, decreasing moment of inertia
- Angular velocity increases to conserve angular momentum
- When two objects collide and stick together, the total angular momentum before and after the collision remains constant: (two asteroids colliding and merging)
- In the absence of external torques, the angular momentum of a system remains constant, even if internal forces or torques are present (binary star system orbiting each other)
Rotational Dynamics
- (moment of inertia) is a measure of an object's resistance to rotational acceleration
- represents the angle through which an object rotates about its axis of rotation
- is the energy associated with an object's rotation about an axis
- Centripetal force is the force that keeps an object moving in a circular path, directed toward the center of rotation
Key Terms to Review (36)
Angular acceleration: Angular acceleration is the rate of change of angular velocity with respect to time. It is a vector quantity, often measured in radians per second squared ($\text{rad/s}^2$).
Angular Acceleration: Angular acceleration is the rate of change of angular velocity with respect to time. It describes the rotational equivalent of linear acceleration, representing the change in the speed of rotation or the change in the direction of rotation of an object around a fixed axis.
Angular Displacement: Angular displacement is a measure of the change in the angular position of an object about a fixed axis or point of rotation. It describes the amount of rotation an object undergoes, typically expressed in units of radians or degrees.
Angular Impulse: Angular impulse is a measure of the change in angular momentum of an object or system over a given time interval. It represents the rotational equivalent of linear impulse and is a fundamental concept in the study of rotational dynamics.
Angular momentum: Angular momentum is the rotational analog of linear momentum, representing the quantity of rotation of an object. It is a vector quantity given by the product of an object's moment of inertia and its angular velocity.
Angular Momentum: Angular momentum is a measure of the rotational motion of an object around a fixed axis. It describes the object's tendency to continue rotating and the amount of torque required to change its rotational state. This concept is fundamental in understanding the dynamics of rotating systems and is crucial in various areas of physics, from the motion of satellites to the behavior of subatomic particles.
Angular velocity: Angular velocity is the rate of change of the rotation angle with respect to time. It is usually 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 rotating around a fixed axis or point. It describes the speed of rotational motion and is a vector quantity, indicating both the magnitude and direction of the rotation.
Axis of Rotation: The axis of rotation is an imaginary line about which an object rotates or pivots. This concept is fundamental to understanding rotational motion and its associated dynamics, kinematics, and conservation principles.
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 governs the dynamics of rotational motion, the behavior of colliding extended bodies, and the unique properties of gyroscopic systems.
Euler: Euler is a renowned Swiss mathematician who made significant contributions to various fields of mathematics, including the study of rotational motion and angular momentum. His work has become fundamental in understanding the principles governing the dynamics of rotating systems.
Figure Skater Spin: A figure skater spin is a rotational maneuver performed by a skater on the ice, where they rotate rapidly around a vertical axis while balancing on one foot. This action is an excellent demonstration of angular momentum, as it illustrates how the skater can change their rotational speed by adjusting their body position. By bringing their arms and leg closer to their body, the skater decreases their moment of inertia, leading to an increase in angular velocity, showcasing the conservation of angular momentum.
Gyroscope: A gyroscope is a device that uses the principles of angular momentum to maintain orientation and stability. It operates based on the conservation of angular momentum, allowing it to resist changes in its axis of rotation. This property makes gyroscopes essential in navigation systems, stability control, and various technological applications.
L = Iω: The equation L = Iω represents the relationship between angular momentum (L), moment of inertia (I), and angular velocity (ω). Angular momentum is a measure of the rotational motion of an object and depends on both how much mass is concentrated and how fast the object is spinning. Understanding this relationship is crucial for analyzing rotational systems in physics, particularly when discussing conservation laws and the effects of external forces.
L = r × p: The equation L = r × p represents the relationship between angular momentum (L), the position vector (r), and linear momentum (p) of an object. In this equation, L indicates the angular momentum, which is a measure of the rotational motion of an object, while r is the distance from the axis of rotation to the point where linear momentum is measured, and p represents the linear momentum of the object, defined as the product of its mass and velocity. This equation helps to understand how angular momentum is conserved in systems where no external torques are acting.
Lagrange: Lagrange is a fundamental concept in physics that describes the motion of a system using a function called the Lagrangian, which is the difference between the kinetic and potential energies of the system. This approach provides a powerful and elegant way to analyze the dynamics of complex systems, particularly in the context of angular momentum and its conservation.
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 fundamental in analyzing rotational motion and interactions.
Linear momentum: Linear momentum is the product of an object's mass and its velocity. It is a vector quantity, possessing both magnitude and direction.
Linear Momentum: Linear momentum is a vector quantity that describes the motion of an object. It is defined as the product of an object's mass and its velocity, and it represents the object's resistance to changes in its motion.
Moment of inertia: Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It depends on both the mass of the object and how that mass is distributed 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 quantifies how an object's mass is distributed about its axis of rotation and determines the object's rotational dynamics, including angular acceleration, angular momentum, and rotational kinetic energy.
P = mv: The equation $p = mv$ represents the relationship between an object's momentum (p), its mass (m), and its velocity (v). Momentum is a fundamental concept in physics that describes the quantity of motion possessed by an object, and it is conserved in closed systems.
Parallel Axis Theorem: The parallel axis theorem is a principle used in rotational motion that allows us to calculate the moment of inertia of a rigid body about any axis parallel to an axis through its center of mass. This theorem states that the moment of inertia about the new axis is equal to the moment of inertia about the center of mass plus the product of the mass of the object and the square of the distance between the two axes. This concept is crucial in understanding how objects behave when they rotate around different axes.
Perpendicular Axis Theorem: The perpendicular axis theorem states that the moment of inertia of a planar lamina about an axis perpendicular to its plane is equal to the sum of the moments of inertia about two orthogonal axes lying in the plane of the lamina. This theorem is useful in simplifying calculations for the moment of inertia, especially when dealing with shapes that can be easily divided into simpler components.
Precession: Precession is the slow, circular motion of the axis of a spinning object, such as a gyroscope or a planet, around another axis due to a torque being applied to it. This phenomenon is observed in various contexts, including the motion of spinning tops, the orientation of the Earth's axis, and the behavior of gyroscopic devices.
Right-hand rule: The right-hand rule is a mnemonic used to determine the direction of angular momentum vectors. It states that if you curl the fingers of your right hand in the direction of rotation, your thumb points in the direction of the angular momentum vector.
Right-Hand Rule: The right-hand rule is a mnemonic device used to determine the direction of various vector quantities in physics, such as magnetic fields, angular momentum, and the force on a moving charge in a magnetic field. It is a simple and intuitive way to visualize the relationship between these vectors and their associated directions.
Rigid Body Rotation: Rigid body rotation is a type of rotational motion where an object rotates around a fixed axis without any deformation or change in shape. This concept is fundamental to understanding angular momentum and its conservation in physics.
Rotational inertia: Rotational inertia, also known as the moment of inertia, is a measure of an object's resistance to changes in its rotational motion about an axis. It depends on the object's mass distribution relative to the axis of rotation.
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 quantifies how difficult it is to change the rotational motion of an object around a fixed axis or point.
Rotational kinetic energy: Rotational kinetic energy is the energy possessed by a rotating object due to its angular motion. It is given by the formula $KE_{rot} = \frac{1}{2}I\omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.
Rotational Kinetic Energy: Rotational kinetic energy is the energy possessed by an object due to its rotational motion. It is the energy an object has by virtue of being in a state of rotation, and it depends on the object's rotational inertia and angular velocity.
SI unit of torque: The SI unit of torque is the newton-meter (Nm), which measures the rotational force applied to an object. Torque quantifies the tendency of a force to rotate an object about an axis.
Torque: Torque is the rotational equivalent of force, representing the ability to cause an object to rotate about a specific axis or pivot point. It is the product of the force applied and the perpendicular distance between the axis of rotation and the line of action of the force, and it plays a crucial role in the study of rotational motion and equilibrium.
τ = dL/dt: The term τ = dL/dt represents the rate of change of angular momentum, where τ is the torque acting on an object and dL/dt is the derivative of the object's angular momentum with respect to time. This relationship is fundamental in understanding the principles of angular momentum and its conservation.
τ = rF sin θ: The equation τ = rF sin θ represents the formula for calculating the torque acting on an object. Torque is a measure of the rotational force that causes an object to rotate around a specific axis or pivot point. This equation is fundamental in understanding the second condition for equilibrium and the conservation of angular momentum.