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?
In physics, angular momentum is the rotational equivalent of linear momentum. It is a vector quantity that quantifies the amount of rotation an object has around a specific axis. It depends on the object's mass, its velocity, and its distance from the axis of rotation. Essentially, it is a measure of an object's rotational inertia in motion.
2. What are the two main types of angular momentum?
Angular momentum is broadly categorised into two types based on the nature of the motion:
- Spin Angular Momentum: This is the momentum an object has due to its rotation about its own centre of mass. A prime example is the Earth spinning on its axis, which causes day and night.
- Orbital Angular Momentum: This is the momentum an object has due to its revolution around an external point or axis. For example, the Earth revolving around the Sun in its yearly orbit possesses orbital angular momentum.
The total angular momentum of a system is the vector sum of its spin and orbital angular momenta.
3. What are the primary formulas used to calculate angular momentum?
There are two key formulas to calculate angular momentum, depending on the situation:
- 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, with its magnitude given by L = mvr sin(θ).
- 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 standard properties of angular momentum are:
- SI Unit: The SI unit for angular momentum is kilogram metre squared per second (kg m²/s).
- Dimensional Formula: The dimensional formula is [M L² T⁻¹], representing mass, length squared, and time inverse.
5. How does the principle of conservation of angular momentum work with an example?
The principle of conservation of angular momentum states that if no net external torque acts on a closed system, its total angular momentum remains constant. A classic example is an ice skater performing a spin. When the skater extends their arms, their moment of inertia (I) increases, so their angular velocity (ω) decreases to keep the angular momentum (L = Iω) constant. When they pull their arms in, their moment of inertia decreases, causing their angular velocity to increase dramatically, resulting in a faster spin.
6. Why is angular momentum considered a vector quantity and not a scalar?
Angular momentum is a vector quantity because it has both magnitude and direction. The magnitude describes 'how much' rotation there is, while the direction specifies the axis and orientation of the rotation in space. This direction is crucial for understanding rotational dynamics. The direction is determined using the Right-Hand Thumb Rule: if you curl the fingers of your right hand in the direction of rotation, your thumb points along the axis in the direction of the angular momentum vector.
7. How are torque and angular momentum fundamentally related?
Torque is the rotational analogue of force and is fundamentally defined as the rate of change of angular momentum over time. This relationship is expressed by the formula τ = dL/dt, which is the rotational equivalent of Newton's second law for linear motion (F = dp/dt). This means that to change an object's angular momentum—to make it spin faster, slower, or change its axis of rotation—a net external torque must be applied.
8. What is the key difference between angular momentum and linear momentum?
The key difference lies in the type of motion they describe. Linear momentum (p = mv) is associated with an object's motion in a straight line (translational motion) and changes when a net external force is applied. In contrast, angular momentum (L = Iω) is associated with an object's rotational motion around an axis and changes only when a net external torque is applied. One describes the quantity of translational motion, while the other describes the quantity of rotational motion.
9. How does conservation of angular momentum explain why planets move faster when they are closer to the Sun?
A planet's orbit is an almost-isolated system where the Sun's gravity provides the central force but exerts very little torque. Therefore, the planet's angular momentum is conserved. According to the formula L = Iω (or L ∝ r²ω), as a planet moves along its elliptical orbit and gets closer to the Sun, its distance (r) decreases. To keep the angular momentum (L) constant, its angular velocity (ω) must increase. This results in the planet speeding up at its perihelion (closest point) and slowing down at its aphelion (farthest point).
10. What is the role of angular momentum in keeping a bicycle stable while it's moving?
The stability of a moving bicycle is largely due to the angular momentum of its spinning wheels. The wheels act as gyroscopes. A spinning object with significant angular momentum resists changes to its axis of rotation. For a bicycle wheel, its angular momentum vector points along the axle. To make the bicycle fall over, a torque must be applied to change the direction of this vector. The gyroscopic effect makes the wheel resist this tilting torque, which helps keep the bicycle upright and stable, especially at higher speeds.
