What is Momentum?
Momentum is a vector quantity which in simple terms is defined as the product of mass and velocity. The momentum of a closed system, unless an external force is applied to the system, remains the same. This is known as the principle of conservation of momentum. It is a very important principle in mechanics as this forms the base of many scientific processes including the takeoff of rockets. Momentum is generally considered to be of two types, which are linear momentum and angular momentum. In linear momentum, we use the linear velocity and calculate the dynamics of the system in that frame of reference while in the case of angular momentum, we use angular momentum to understand the dynamics of a particular system. Both the linear momentum as well as angular momentum can be possessed by a body at the same time.
Angular Momentum
The property that characterizes the rotatory inertia of an object in motion about the axis which may or may not pass through that specified object is known as angular momentum. One of the best examples of angular momentum is the Earth’s rotation and revolution. For example, the annual revolution that the Earth carries out about the Sun reflects orbital angular momentum while its everyday rotation about its axis shows spin angular momentum.
Angular momentum is broadly categorized into:-
The spin angular momentum. (e.g., rotation)
The orbital angular momentum. (e.g. revolution)
The total angular momentum of a body is the sum of spin and orbital angular momentum.
It can be said that angular momentum is a vector quantity, i.e. it requires both magnitude and direction. The angular momentum possessed by a body going through orbiting motion is also said to be equal to its linear momentum. The angular momentum is also given as the product of mass (m) and linear velocity (v) of the object multiplied by the distance (r) perpendicular to the direction of its motion, i.e., mvr. But, in the case of a spinning body, the angular momentum is the summation of mvr for all the particles which make the object.
Some vital things to consider about angular momentum are:
Symbol = As the angular momentum is a vector quantity, it is denoted by symbol L.
Units = It is measured in SI base units: Kg m²s⁻¹.
Dimensional formula = M L² T⁻¹
Formula to calculate angular momentum (L) = mvr, where m = mass, v = velocity, and r = radius.
Angular Momentum Formula
The angular momentum of an object having mass (m) and linear velocity (v) with respect to a fixed point can be given as:
L = mvr sin θ
Or
\[\overrightarrow{L}\] = r x \[\overrightarrow{p}\] (in terms of vector product)
Where,
\[\overrightarrow{L}\] = Angular Momentum
v = linear velocity of the object
m = mass of the object
\[\overrightarrow{p}\] = linear momentum
r = radius, i.e., distance amid the object and the fixed point around which it revolves.
Moreover, angular momentum can also be formulated as the product of the moment of inertia (I) and the angular velocity (ω) of a rotating body. In this case, the angular momentum is derivable from the below expression:
\[\overrightarrow{L}\] = I x ω
Where,
L→is the angular momentum.
I is the rotational inertia.
ω is the angular velocity.
The direction of the angular momentum vector, in this case, is the same as the axis of rotation of the given object and is designated by the right-hand thumb rule.
Right-Hand Thumb Rule
The right-hand thumb rule gives the direction of angular momentum and states that if someone positions his/her hand in a way that the fingers come in the direction of r, then the fingers on that hand curl towards the direction of rotation, and thumb points towards the direction of angular momentum (L), angular velocity, and torque.
Angular Momentum and Torque
For a continuous rigid object, the total angular momentum is equal to the volume integral of angular momentum density over the entire object. Here, torque is defined as the rate of change of angular momentum. Torque is related to angular momentum in a way similar to how force is related to linear momentum. Now, when we know what the angular momentum and torque are, let's see how these two are related. To see this, we need to find out how objects in rotational motion get moving or spinning in the first position.
Let's take the example of a wind turbine. We all know that it's the wind that makes the turbine spins. But how is it doing so? Well, the wind is pushing the turbine's blade by applying force to blades at some angles and radius from the axis of rotation of the turbine. In simple words, the wind is applying torque to the turbine. Hence, it is torque that gets rotatable objects spinning when they are standing still. Moreover, if the torque is applied to an object which is already spinning in the same direction in which it is spinning, it upsurges its angular velocity. Hence, we can say that torque is directly proportional to the angular velocity of a rotating body. Since torque can change the angular velocity, it can also change the amount of angular momentum as the angular momentum depends on the product of the moment of inertia and angular velocity. This is how torque is related to angular momentum.
Consider a string is tied to a point mass. Now, if we apply torque on the same point mass, it would start rotating around the centre. Here, the particle of mass m would move with a perpendicular velocity V┴ to the radius r of the circle.
Now, the magnitude of \[\overrightarrow{L}\] will be:
L = rmv sin ϕ
= r p⊥
= rmv⊥
= r⊥p
= r⊥mv
Where,
Φ is the angle formed between \[\overrightarrow{r}\] and \[\overrightarrow{p}\]
p⊥ and v⊥ are the segments of \[\overrightarrow{p}\] and \[\overrightarrow{v}\] perpendicular to \[\overrightarrow{r}\] .
r⊥ is the perpendicular distance between the extension of \[\overrightarrow{p}\] and the fixed point.
Note: The equation or formula L = r⊥mv representing the angular momentum of an object changes only when you apply a net torque. Hence, if no torque is applied, then the perpendicular velocity of the object will alter according to the radius (the distance between the centre of the circle, and the centre of the mass of the body). It means velocity will be high for a shorter radius and low for a longer one.
FAQs on Angular Momentum
1. What is angular momentum in Physics?
Angular momentum is the rotational equivalent of linear momentum. It is a vector quantity that measures the rotational inertia of a body in motion about an axis. In simple terms, it quantifies the amount of rotation an object has. An object's angular momentum depends on its mass, velocity, and distance from the axis of rotation. A classic example is a spinning ice skater, who changes their speed of rotation by manipulating their angular momentum.
2. What are the two main types of angular momentum?
Angular momentum is generally categorised into two types based on the motion of the object:
- Spin Angular Momentum: This is the angular momentum an object has due to its rotation about its own centre of mass, or its own axis. For example, the Earth rotating on its axis daily exhibits spin angular momentum.
- Orbital Angular Momentum: This is the angular momentum an object has due to its revolution around an external point or axis. For example, the Earth revolving around the Sun annually possesses orbital angular momentum.
The total angular momentum of a system is the vector sum of its spin and orbital angular momentum.
3. What are the key formulas to calculate angular momentum?
There are two primary formulas to calculate angular momentum, depending on the context:
- For a point particle, angular momentum (L) is the cross product of its position vector (r) and its linear momentum (p). The formula is: L = r × p or in terms of magnitude, L = mvr sin(θ), where 'm' is mass, 'v' is velocity, and 'θ' is the angle between r and p.
- For a rotating rigid body, angular momentum (L) is the product of its moment of inertia (I) and its angular velocity (ω). The formula is: L = Iω.
4. What is the SI unit and dimensional formula for angular momentum?
The properties of angular momentum are defined as follows:
- SI Unit: The standard unit for angular momentum is kilogram metre squared per second (kg m²/s).
- Dimensional Formula: The dimensional formula for angular momentum is [M L² T⁻¹].
5. Can you explain angular momentum with a simple, real-world example?
A classic example of angular momentum is an ice skater performing a spin. When the skater's arms are extended, their mass is distributed farther from the axis of rotation, increasing their moment of inertia. To conserve angular momentum, their rotational speed (angular velocity) is slow. When they pull their arms in close to their body, they decrease their moment of inertia. Because angular momentum must be conserved (assuming no external torque from the ice), their angular velocity must increase, causing them to spin much faster.
6. What is the principle of conservation of angular momentum?
The principle of conservation of angular momentum states that if no external torque acts on a system, its total angular momentum remains constant. This means that for an isolated system, the initial angular momentum is equal to the final angular momentum (L_initial = L_final, or I₁ω₁ = I₂ω₂). This fundamental principle explains many phenomena, from the motion of planets in elliptical orbits (they speed up when closer to the sun) to the stability of a spinning top.
7. Why is angular momentum a vector quantity, and how is its direction determined?
Angular momentum is a vector quantity because it has both magnitude (how much rotation) and a specific direction (the axis of rotation). The direction is crucial for describing the plane and orientation of the rotation. The direction of the angular momentum vector is determined using the Right-Hand Thumb Rule. If you curl the fingers of your right hand in the direction of the object's rotation, your thumb will point in the direction of the angular momentum vector (L), which is along the axis of rotation.
8. How are torque and angular momentum fundamentally related?
Torque is the rotational equivalent of force and is fundamentally related to angular momentum. Specifically, torque is the rate of change of angular momentum over time. This relationship (τ = dL/dt) is analogous to Newton's second law for linear motion (F = dp/dt). This means that to change an object's angular momentum (i.e., to make it spin faster, slower, or change its axis of rotation), an external torque must be applied. If there is no net external torque, angular momentum is conserved.
9. What is the practical importance of understanding angular momentum in science and engineering?
The concept of angular momentum is critical in many fields:
- Astrophysics: It governs the motion of planets, stars, and galaxies. The formation of planetary systems and the stability of orbits are explained by the conservation of angular momentum.
- Engineering: It is used in the design of gyroscopes for navigation systems in aircraft and satellites. Flywheels in engines use this principle to store rotational energy and smooth out power delivery.
- Quantum Mechanics: At the atomic level, particles like electrons have intrinsic spin angular momentum, which is a key property determining chemical and magnetic behaviour.
10. Under what conditions is angular momentum not conserved?
Angular momentum is not conserved when a net external torque acts on the system. Torque is a rotational force that causes a change in angular momentum. For example, when you use a wrench to tighten a bolt, you are applying a torque, which changes the bolt's angular momentum from zero to a non-zero value. Similarly, friction from the air or at the axle of a spinning wheel acts as an external torque that gradually slows it down, thus reducing its angular momentum.
11. How does an object's shape or mass distribution affect its angular momentum?
An object's shape and mass distribution are encapsulated in a quantity called the moment of inertia (I). For a given angular velocity (ω), an object with a larger moment of inertia (mass distributed farther from the axis of rotation, like a hollow sphere) will have a greater angular momentum than an object with a smaller moment of inertia (mass concentrated near the axis, like a solid sphere of the same mass and radius). This is clear from the formula L = Iω. Therefore, changing an object's shape, like an ice skater pulling in their arms, directly changes its moment of inertia and, to conserve angular momentum, its rate of spin.
