Conservation of Angular Momentum

The law of conservation of angular momentum states that activities through external torque are important for changing angular momentum. Hence, when an object is in the spinning phase within a closed system, and no external torques are used, there will be no changes within angular momentum. In simpler terms, if the net torque equals zero, only then the angular momentum is conserved. The angular momentum is fully analogous to the linear momentum. It was initially represented as “Uniform circular motion and gravitation”. It has similar implications for conducting the rotation in a forward way. Both the linear and angular momentum are observed to be particles of atoms and sub-atoms

Angular momentum definition 

Angular momentum characterises the rotary inertia of objects within motion through an axis. This axis might or might not pass throughout the object. Let’s take an example of the earth’s orbit. It rotates due to its annual revolution of the sun and routine rotation on its axis. Angular momentum magnitude is equivalent to linear momentum. If any system is isolated from all external forces, its entire angular momentum will remain constant. This is referred to as conservation of angular momentum. 

The conserved quantity is commonly referred to as angular momentum. Thereby, in many formulas, angular momentum is denoted as L. It is said that like conservation of linear momentum, angular momentum is also constant at net torque (when it is zero). It considers the second law of Newton of rotational motion.

= dL/ dt, 

Net torque= 0, the formula is dL/ dt = 0

Therefore, if the changes in angular momentum (ΔL) is observed to be zero, it is a constant stage.  

The mathematical depiction for the law of conservation of angular momentum: 

L = conserved when = 0

Angular momentum of a point particular

To understand the law of conservation of angular momentum, it is important to consider a single point object for reference. When a particle moves in a circle, two vectors work on it- a velocity factor, which is tangentially directed, and the angular acceleration ‘w’ that works along the radius. Therefore, at any given point, the angular velocity and acceleration work perpendicular to each other, which propels the particle along a curved path. 

So, based on this, we can derive the formula for the law of conservation of angular momentum. Let’s consider a particle moving angularly but with linear momentum. Therefore, its momentum value will be described by mv (product of mass and velocity). Or, 

p = mv ………. (a)

As velocity is a vector unit, we need to consider its component along the perpendicular direction of the plane. Its value will be equal to vsinθ.

So, equation (a) can be written as: 

p = mvsinθ ………. (b)

When we consider the angular momentum of the particle, it is given by the equation of:

L = r X p ……. (c)

Here, r is a directional vector whose magnitude can be written as r because the angle between this vector and the direction of movement is 0. But we have to replace p with mvsinθ.  

Therefore, the entire equation of angular momentum can be written as: 

l = rmvsinθ …….. (d)

Angular momentum of a system of particles

When a system of particles is considered, the total angular momentum is conserved so that the resultant equals 0. The formula for representing the angular momentum of a system of particles can be described as:  

L = l1 + l2 + l3 + l4 + ………… + ln 

Here, n represents the number of particles in the system, ‘L’ is the total angular momentum, while ‘l’ is considered as the momentum of a single particle.

Since L = rmvsinθ

We can modify the angular momentum equation further like: 

L = r1mv1sinθ + r2mv2sinθ + r3mv3sinθ + …….. + rnmvnsinθ 

Difference between angular and linear momentum

To understand a particle’s motion in free space, we need to understand the relationship between angular and linear momentum. When a body moves in a straight line, its velocity acts along the axis, and therefore the angle is 0. So, the value of the vector is the same as that of the magnitude. That’s why linear momentum can be expressed as: 

p = mv

On the other hand, if a body moves in a curved path, its velocity will act in the tangential direction of the curved path. This is why an angle is created between the axis and the vector component. Therefore, angular momentum is the product of the particle’s mass, the magnitude of the velocity vector, and the angle between the vector and the axis. The equation used for representing it is: 

l = mvr sinθ

The SI unit of kgms-1 expresses linear momentum, while the SI unit for angular momentum is expressed as kg.m2.rad.s-1. 

Conclusion

The law of conservation of angular momentum is very important to consider while defining the angular momentum of a point object, a system of objects, and a rigid body. In a neutral state, the total momentum of a system is mainly given as zero. When there is a collision of particles, the change in the momentum is always conserved because the sum initial momentum of all particles is equal to the sum of the final momentum of the particles. Besides, it defines the Newtonian laws of motion and explains how they apply to rotational and curvilinear motion.