Conservation of Angular Momentum

Introduction:

Angular momentum can be defined as the property of any rotating body given by moment of inertia times angular velocity. That is, it is the property of a rotating body given by the product of the moment of inertia and the angular velocity of the rotating object. It is clear that this is a vector quantity, along with magnitude, the direction is also considered. Any object or body moving with mass possesses momentum and the angular momentum is the property characterizing the rotary inertia of an object or system of objects in motion around an axis that may or may not pass through the object or system. Even, the Earth has orbital angular momentum because of its annual revolution about the Sun and spin angular momentum because of its daily rotation about its axis. We can say that the magnitude of the angular momentum of an orbiting object is equal to its linear momentum (times of the perpendicular distance r from the center of rotation to the line drawn in the direction of its instantaneous motion and passing through the object’s center of gravity.

The information about the conservation of angular momentum from various physics-related articles is available here. Angular momentum and its law of conservation are important topics in physics. Students who want to flourish in physics need to be well known about momentum to get deep knowledge about it to do well on their exams. The relationship between torque and angular momentum, conservation of angular momentum, and its applications are provided here to assist students in effectively understanding the respective topic. Continue to visit our website for additional physics help.

Overview:

Angular momentum is an important measure for studying dynamics on different temporal and spatial scales. We can conclude that the angular momentum in a closed system is constant in total but can be redistributed within that system and the transport of angular momentum is also done vertically, carrying angular momentum as part of the Hadley and other mean meridional circulations. So, the law of conservation of angular momentum then guarantees that the angular momentum of the particle is constant (again referred to as an origin at the center of the circle). The angular momentum is constant in magnitude and constant in direction (the motion is confined to a single plane, the plane of rotation).

The angular momentum of a rigid body can be considered as the product of the moment of inertia and the angular velocity and it is analogous to linear momentum. If there is no external torque on the object, then it is subject to the basic constraints of the conservation of angular momentum principle. The angular momentum and linear momentum are two examples of the parallels between linear and rotational motion. They both have the same form and are subject to the fundamental constraints of conservation laws, the conservation of momentum, and the conservation of angular momentum. If an object is spinning in a closed system and no external torques are applied to it, then it will have no change in angular momentum.

In general, the conservation of angular momentum describes the angular acceleration of an ice skater as she brings her arms and legs close to the vertical axis of rotation. The angular momentum is constant or conserved if the net torque is zero. The conserved quantity that we are examining is called angular momentum.

Conservation of Angular Momentum:

Angular momentum is the rotational analogue of linear momentum and it is denoted by I.

Now, the angular momentum of a particle in rotational motion is defined as,

l=r × p

It is a cross product of r, that is, the radius of the circle formed by the body in rotational motion, and p, that is the linear momentum of the body, the magnitude of a cross product of two vectors is always the product of their magnitude multiplied with the sine of the angle between them.

So, in the case of angular momentum, the magnitude is given by,

l= r × p sinθ⁡

Relationship between Torque and Angular Momentum:
The relationship between torque and angular momentum can be described as,

^→l=^→r ^→p

Differentiate L. H. S. and R. H. S.

d ^→l ⁄ d t=d ⁄ d t ( ^→r × ^→p)

By using the property of differentiation on cross products the expression can be written as,

d ^→l ⁄ d t =d r ⁄ d t × ^→p+ r d ^→p ⁄ d t

Here, d ^→r ⁄ d t is the change in displacement with time.

Thus, it is linear velocity ^→v.

Now,

d ^→l ⁄ d t =^→v × ^→p+ r d ^→p ⁄ d t

Here, p is linear momentum that is, mass times velocity.

Now,

d ^→l ⁄ d t= ^→v× m ^→v+ ^→r d^→p ⁄d t

When noticing the first term, there is ^→v ^→v .

The magnitude of the cross product is,

^→v × ^→ v sinθ⁡ , where the angle is 0.

Thus, the whole term becomes 0.
From newton’s second law, we know that d^→p ⁄ d t is force,

That is,

d ^→l ⁄ d t=^→r ^→F

We know that ^→r ^→F is torque.

Therefore,

d ^→l ⁄ d t= ^→τ.

So, the rate of change of angular momentum is torque.

Calculation of Conservation of Angular Momentum:

The angular momentum of any system is conserved as long as there is no net external torque 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 angular momentum.

Currently, there are two ways to calculate the angular momentum of any object.

If it is a point object in a rotation, then the angular momentum is equal to the radius times the linear momentum of the object.

That is,

^→ l=^→r ^→p

If there is an extended object, like earth, for example, the angular momentum is given by moment of inertia, that is, how much mass is in motion in the object and how far it is from the center, times the angular velocity.

^→l=^→I × ^→ω

However, in both cases, as long as there is no net force acting on it, the angular momentum before is equal to angular momentum after some given time, imagine rotating a ball tied to a long string, the angular momentum would be,

^→l=^→r ^→p=^→r m ^→v

Now, if we somehow decrease the radius of the ball by shortening the string while it is in rotation, the r will reduce. Then, according to the law of conservation of angular momentum L should remain the same. There is no way for mass to change.

Thus, ^→v should increase. In order to keep the angular momentum constant.

Therefore, this is the proof for the conservation of angular momentum.

Applications of Conservation of Angular Momentum:

The law of conservation of angular momentum has many applications such as:

• Electric generators,
• Aircraft engines.

Frequently Asked Question (FAQs):

Question 1: What is the conservation of angular momentum?

Answer: The angular momentum is constant for a system with no external torque.

Question 2: What are the applications of conservation of angular momentum?

Answer:A few applications of angular momentum are:

• Aircraft engine
• Electric generators

Question 3: Where can we find the center of mass of two particles having equal mass?

Answer: The center of mass of two particles having equal mass is found in midway between them.