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Conservation of Angular Momentum

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Angular momentum is a kind of momentum in which an object orbiting a point has a rotational quantity of motion that is different from linear momentum. This new quantity is called angular momentum and is represented by L. There are some interesting situations in which the angular momentum of a spinning object is conserved but the object changes its rotational speed. Near the end of a performance, many ice skaters go into a spin. The spin usually starts out slowly and then gets faster and faster. This might appear to be a violation of the law of conservation of angular momentum but is in fact a beautiful example of its validity.

Skating

What is Angular Momentum?

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Similar to the momentum in the case of translational motion, one introduces an angular momentum for rotations of a rigid body about a fixed axis. Angular momentum is a vector quantity that points along the rotation axis. Angular momentum, L, the product of the moment of inertia I relative to the rotation axis X and the angular velocity $\omega$.SI unit of angular momentum is kilogram meter squared per second (kgm2/s). 1 kgm2/s is the angular momentum of a body with moment of inertia 1 kgm2 that rotates with angular velocity 1 rad/s.

Definition

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The angular momentum of a system does not change under certain circumstances. The law of conservation of angular momentum is analogous to the conservation law for linear momentum. The difference is that the interaction that changes the angular momentum is a torque rather than a force.
If the net external torque acting on a system is zero, the total vector angular momentum of the system remains constant. Along with conservation of linear momentum, conservation of angular momentum is a general result that is valid for a wide range of systems. It holds true in both the relativistic limit and in the quantum limit. No exceptions have ever been found.
Note that the net external force need not be zero for angular momentum to be conserved. There can be a net external force acting on the system as long as the force does not produce a torque. This is the case for projectile motion because the force of gravity can be considered to act at the object's center of mass. Therefore, even though a thrown baton follows a projectile path, it continuous to spin with the same angular momentum around its center of mass. There is no net torque on the baton.

Equation

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If the angular momentum is conserved, the initial angular momentum is equal to the final angular momentum. Angular momentum can be calculated as the product of moment of inertia and angular velocity. So the equation of the angular momentum conservation is,
$I_{1}\omega_{1}$ = $I_{2}\omega_{2}$

Principle 

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The principle behind the conservation of angular momentum tells that if no net torques are involved, the angular momentum is conserved. The total angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero.

Examples

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Some of the common example of conservation of angular momentum is mentioned below:

A spinning skater:
Choosing the skater as the system, we can apply the conservation principle provided that the net external torque produced by air resistance and by friction between the skates and the ice is negligibly small. If the skater stretch his arms towards his body and then rotate, the moment of inertia decreases and the angular velocity increases. SO the angular moment is conserved. Thus, the consequence of pulling his arms and leg inward is that he spins with a larger angular velocity.

A satellite in an elliptical orbit:
An artificial satellite is placed in to an elliptical orbit about the earth, the only force of any significance that acts on the satellite is the gravitational force of the earth. However, at any instant, this force is directed toward the center of the earth and passes through the axis about which the satellite instantaneously rotates. Therefore, the gravitational force exerts no torque on the satellite. Consequently, the angular momentum of the satellite remains constant at all times.

Changing the axis of a bow:
An object's angular momentum can be different depending on the axis about which it rotates. Much more angular momentum is required when playing violin near the bow's handle, called the frog, as in the panel on the right; not only are most of the atoms in the bow are at greater distances, r, from the axis of rotation, but the ones in the tip also have more momentum, P. It is difficult for the player to quickly transfer a large angular momentum into the bow and then transfer it back out just as quickly. This is one of the reasons that string players tend to stay near the middle of the bow as much as possible.

Problems

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Let us discuss the problems of conservation of angular momentum.

Solved Examples

Question 1: Calculate the angular momentum of a rotating object whose moment of inertia is 10kgm2 and the angular velocity 0.5rad/s?

Solution:
 
It is given that,
I = 10kgm2 and $\omega$ = 0.5rad/s
Angular momentum formula is,
L = I$\omega$
L = 10$\times$0.5 = 5kgm2/s

 

Question 2: Calculate the final angular velocity of an object whose initial and final moment of inertia is 15kgm2 and 25kgm2 respectively. The initial angular velocity is given as 20rad/s?

Solution:
 
Given that,
$I_{1}$ = 15kgm2 , $I_{2}$ = 25kgm2 and $\omega_{1}$ = 20rad/s
Equation for conservation of angular momentum is,
$I_{1}\omega_{1}$ = $I_{2}\omega_{2}$

So,$\omega_{2}$ = $\frac{I_{1}\omega_{1}}{ I_{2}}$

$\omega_{2}$ = $\frac{15\times20}{25}$ = 12rad/s