What is Conservation of Angular Momentum?
It is a very important quantity in Physics that is because it is a conserved quantity that is the total momentum or the total angular momentum of a closed system that remains constant.
In three dimensions the angular momentum that is for a point which is a particle is a pseudovector r × p the product which is a cross product of the particle's position vector r that is relative to some origin vector along with its momentum vector. The latter is denoted as that is p = mv in mechanics which is Newtonian.
This is the definition that can be applied to each point in continua like fluids and solids or physical fields.
Conservation of Angular Momentum
Unlike the discussed momentum the angular momentum does depend on where the origin is chosen since we know that the particle's position is measured from it. Just like for velocity which is angular velocity there are two special types of angular momentum they are the spin angular momentum and orbital angular momentum. The angular momentum which is the spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate.
The total angular momentum which is of an object is the sum of the spin that is orbital angular momenta. The angular momentum which is orbital to the vector which is of a point particle is always parallel and directly proportional to the orbital angular velocity vector that is denoted by ω of the particle. where the constant which is of proportionality that really depends on both the mass of the particle and its distance from origin.
Angular Momentum Vector
The spin which is of angular momentum vector of a body which is rigid is proportional but not always parallel to the spin angular velocity vector that is denoted by Ω that are making the constant of proportionality a second-rank that tensor rather than a scalar.
The Angular momentum is a quantity extensive that is the total angular momentum of any composite system is the sum of the angular momenta which is of its constituent parts. We will discuss that for a continuous rigid body the total momentum or the angular momentum is the volume integral of angular momentum density that is the angular momentum which is per unit volume in the limit as volume shrinks to zero over the entire body.
State Law of Conservation of Angular Momentum
It is the analogue which is rotational of linear momentum that it is denoted by l and angular momentum of a particle which is in rotational motion is defined as:
l = r × p
This whole thing or the whole process is a cross product of r that is the radius of the circle which is really formed by the body in rotational motion and p that is the linear momentum of the body. The magnitude which is of a cross product that is of two vectors which is is always the product of their magnitude that multiplied with the sine of the angle which is between them and therefore in the case of angular momentum the magnitude is given by:
l = r p sin θ
The Law of conservation which is of angular momentum that has many applications, including the following:
The Electric generators.
The Aircraft engines, etc.
To learn more about that to the conservation of momentum or the angular momentum and the other which is related topics too with the help of interactive video lessons.
The Angular momentum which we have earlier talked of a system that is conserved as long as there is no net external torque which is acting on the system. The earth has been rotating on its axis from the time the solar system was formed due to the law of conservation of momentum or we can say the angular momentum.
There are two ways in which we can calculate the angular momentum which is of any object if it is a point object in a rotation then our angular momentum is equal to the times radius that the linear momentum of the object.
If we talk about this whole thing the we have an extended object which is like our earth for example we can see that the angular momentum is given by moment of inertia that is how much mass is in motions in the object and how far it is from the centre times the angular velocity.
But in both cases which we are discussing as long as there is no net force acting on it. the angular momentum which is before and is equal to angular momentum after some given time and imagine rotating a ball which is tied to a long string the angular momentum.
Now when we somehow are able to decrease the radius of the ball by shortening the string while it is in rotation that the r which is the radius will reduce now according to the law of conservation which is of angular momentum denoted by L should remain the same there is no way for mass to change so therefore:
We should increase to keep the angular momentum constant so that this is the proof for the conservation which is of angular momentum.
The Angular momentum basically depends on the rotational velocity of an object but also its rotational inertia. When an object changes its shape (rotational inertia), its angular velocity will also change if there is no torque which is external.
An example which is when an ice skater changes and spins her rotation velocity by that are holding her arms outwards or pulling them inwards.
(image will be uploaded soon)
When she pulls her arms in her inertia which is rotational is reduced. Since there is no external net torque which is on the ice skater her angular momentum really remains constant which is because her angular velocity increases the magnitude.
FAQs on Conservation of Angular Momentum
1. What is the principle of conservation of angular momentum?
The principle of conservation of angular momentum states that if the net external torque acting on a system is zero, its total angular momentum remains constant. In simpler terms, for an isolated system, the amount of rotational motion it has will not change unless an external turning force is applied. This is a fundamental law in physics, analogous to the conservation of linear momentum.
2. What is the formula for angular momentum, and how does it express the conservation law?
The angular momentum (L) of a rigid body is given by the product of its moment of inertia (I) and its angular velocity (ω). The formula is:
L = Iω
The law of conservation is expressed mathematically as: If the net external torque is zero, then L remains constant. This means the initial angular momentum is equal to the final angular momentum. Therefore, we can write:
I₁ω₁ = I₂ω₂
This equation shows that if a system's moment of inertia (I) changes, its angular velocity (ω) must change in the opposite way to keep the total angular momentum (L) conserved.
3. What is the essential condition for the conservation of angular momentum?
The single most important condition for angular momentum to be conserved is that the net external torque acting on the system must be zero (τ_ext = 0). Torque is the rotational equivalent of force. If there is no external twisting force applied to a rotating system, its angular momentum will not change over time. Internal forces and torques within the system do not affect the total angular momentum.
4. How does Newton's second law for rotation lead to the principle of angular momentum conservation?
Newton's second law for rotational motion states that the net external torque (τ_ext) is equal to the rate of change of angular momentum (L). This is expressed as:
τ_ext = dL/dt
The principle of conservation is a direct consequence of this law. If there is no net external torque on the system, then τ_ext = 0. Substituting this into the equation gives:
dL/dt = 0
In calculus, if the derivative of a quantity with respect to time is zero, it means that the quantity itself is not changing; it is a constant. Therefore, L = constant. This is the mathematical derivation of the law of conservation of angular momentum from Newton's laws.
5. What are some common real-world examples of the conservation of angular momentum?
The conservation of angular momentum can be observed in many situations. Some key examples include:
- An Ice Skater's Spin: When a skater pulls their arms and legs close to their body, their moment of inertia decreases, causing their angular velocity (spin speed) to increase dramatically.
- A Planet's Orbit: A planet moves faster when it is closer to the Sun (perihelion) and slower when it is farther away (aphelion). This is because the gravitational force provides no torque, so its angular momentum is conserved.
- A Diver's Somersault: By tucking their body into a tight ball, a diver reduces their moment of inertia, allowing them to rotate quickly. They straighten out just before hitting the water to increase their moment of inertia and slow their rotation for a smooth entry.
- Helicopter Rotors: A helicopter has a smaller tail rotor to produce a counter-torque, preventing the main body of the helicopter from spinning in the opposite direction to the large main rotor, thus conserving the total angular momentum of the system.
6. Why does an ice skater spin faster when she pulls her arms in?
An ice skater's spin is a classic demonstration of L = Iω, where L (angular momentum) is conserved because the torque from the ice is negligible. When her arms are outstretched, her mass is distributed farther from her axis of rotation, giving her a large moment of inertia (I). To maintain a constant L, she has a relatively low angular velocity (ω). When she pulls her arms in, she brings her mass closer to the axis of rotation, which significantly decreases her moment of inertia. Since her angular momentum (L) must remain constant, a decrease in I must be compensated by a proportional increase in her angular velocity (ω), causing her to spin much faster.
7. Is rotational kinetic energy also conserved when angular momentum is conserved? Explain the difference.
No, rotational kinetic energy is not necessarily conserved even when angular momentum is. The rotational kinetic energy is given by KE = (1/2)Iω². Using the ice skater example, while her angular momentum (L = Iω) remains constant, her kinetic energy changes. We can express kinetic energy as KE = L² / (2I). When the skater pulls her arms in, her moment of inertia (I) decreases. Since L is constant, her kinetic energy (KE) must increase. This additional energy is not created from nowhere; it comes from the internal work she does with her muscles to pull her arms inward against the rotational forces.
8. What is the key difference between the conservation of linear momentum and angular momentum?
The key difference lies in the conditions required for each principle:
- Conservation of Linear Momentum: Applies when the net external force on a system is zero. It relates to an object's motion in a straight line (translational motion).
- Conservation of Angular Momentum: Applies when the net external torque on a system is zero. It relates to an object's rotational motion around an axis.
A system can have a net external force acting on it but zero net external torque. For example, a planet orbiting the sun has a constant gravitational force (a net force) but zero torque, so its linear momentum changes continuously while its angular momentum is conserved.
