When Is Angular Momentum Not Conserved: When, Why, How, Detailed Facts And FAQs

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In this article, we’ll discuss “when is angular momentum not conserved.”

Angular momentum is a physical quantity analogous to linear momentum. It is the inherent property of the body or system of particles, which specifies the rotary inertia about an axis (may or may not pass through the body). When external torque acts on the body, the angular momentum is not conserved.

Torque–turning effect of FORCE. Just as in linear kinematics, force is responsible for accelerating or decelerating the body. In the same way, in rotational motion, torque is responsible for rotating the body or system of particles about an axis.

Principle of Conservation of Angular Momentum

As the principle signifies, if the resultant (net) external torque acting on an object or system of particles is zero. Then the total angular momentum is unchanged or conserved. In other words, the net angular momentum of the system does not change w.r.t time.

The Newton’s second law in angular form can be written as ,

τ=dL/dt

As per the conservation law, for an isolated system , the external torque is zero.

dL/dt=0

Now, it shows,

L = constant

The above principle can also be written as,

 the net angular momentum at some time ti = the net angular momentum at some time tf

Li =Lf

So, if the  external torque acts on the body the angular momentum of the body changes. The final momentum and the initial momentum of the system will not be the same.

Lets see an example in which the angular momentum is not conserved.

What is an example of momentum not being conserved

The body’s angular momentum changes when there is net torque acting on the body, causing an increase or decrease in angular velocity. I.e., Merry-go-round.

when is angular momentum not conserved
when is angular momentum not conserved “DSCN10871” by subewl is licensed under CC BY-SA 2.0

Suppose you’re sitting at the center of the merry-go-round, and it was spinning at its axis with some angular speed ω1. After a few minutes, you decided to jump outside of it. Now, According to the conservation of momentum, angular velocity needs to be increased as the moment of inertia (M.O.I depends upon mass and its distribution, and here mass decreases as you jump out of the MGR) decreases. 

l1ω1=l2ω2

But, we observe that the angular velocity decreases of the merry-go-round, which evidently shows that the angular momentum is not conserved.

Frequently asked questions: FAQs

Question: what is the formula of angular momentum?

There are various formulas of angular momentum.

L=r*p

L=Iω

L=∫τdt

Question: When is the angular momentum conserved?

When there is no external torque acting on the body or the resultant torque is zero, then it is said to be that the angular momentum is conserved or remains constant of the body or  the system.

Question: What are the examples of angular momentum being conserved?

Let’s see a few examples related in which it is evidently seen that the angular momentum is being conserved.

  • Neutron Star: At the end of life, a massive star’s core can collapse into a tiny and super dense object which is known as a neutron star. Being super dense and diminutive in size, it rotates rapidly. And the rapid rotation exhibits the consequence of the law of conservation of angular momentum. As the star collapses, its moment of inertia decreases (M.O.I– is the product of mass and distance squared), leading to an increase in its angular speed.
when is angular momentum not conserved
“Neutron Star” by Kevin M. Gill is licensed under CC BY 2.0
  • Helicopters have two propellers: Have you wondered why helicopters have two propellers? Is one propeller not enough to move it? Suppose if a helicopter has only one propeller on its head and no external torque acting on it, the angular momentum will remain constant. Initially, the angular momentum is zero as the propeller is not rotating. To conserve angular momentum, The helicopter would start spinning in the opposite direction at its axis as soon as the propeller rotates. Hence, one more propeller is provided on its tail to prevent spinning on its axis.
when is angular momentum not conserved
“XP353 Whirlwind HAR10 Boulmar 1978” by cvtperson is licensed under CC BY-SA 2.0
  • Tornado: The inner layer of a tornado, also known as a whirlwind, has high speed as its surface area is less. As no external torque is acting, angular momentum remains unchanged. Due to less surface area, the moment of inertia decreases, making the whirlwind rotate rapidly on its axis.
  • Cat landing safely: Well, a cat is very intelligent. It knows how to apply conservation of angular momentum while falling. Whenever a cat falls from heights, it stretches its body and the tail to increase the moment of inertia which ultimately results in a decrease in its speed(as no external torque is acting, so total angular momentum remains unchanged) which makes the cat to land safely.
  • A person carrying heavy weight standing on a rotating platform: Suppose, a person is standing on platform by carrying heavy weight in both of his hands. We observe that when his arms are stretched horizontally, his angular velocity is less as soon as he takes his arms closer to its body(it decreases M.O.I), which increases angular velocity.
  • Revolution of planets around Sun: The orbits of the planets around the sun are primarily elliptical. As it goes around the sun, its moment of inertia keeps on changing as M.O.I depends upon mass distribution. As they come closer to the sun, the moment of inertia decreases (as distance decreases), and the angular velocity increases (the consequence of conservation of angular momentum).why it happen? Because when the planet is at a far distance, the moment of inertia becomes more so angular velocity decreases as no external torques are acting, angular momentum remains constant.
when is angular momentum not conserved
“solar system” by Philippe Put is licensed under CC BY 2.0

Question: What is the moment of inertia?

Moment of inertia is a physical quantity which signifies the mass distribution from its axis of rotation. It is the quantity which is product of mass and square of the distance from its axis.

Question: What is the dimensional formula of angular momentum and torque?

The dimensional formula of angular momentum is M1L2T-1 and of torque is M1L2T–2.

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