Jean Buridan the discoverer/inventor of momentum also discussed angular momentum. Just as linear momentum is conserved when no net external forces act, angular momentum is constant or conserved when the net torque is zero.

What is the Conservation of angular momentum?

Angular momentum is the rotational analog of linear momentum which is a vector quantity and is defined as the product of the mass of an object, m, and its velocity i.e., v whereas angular momentum is the property of any rotating object given by moment of inertia times angular velocity.

Linear momentum is conserved when the net torque is zero and we can see this by considering Newton’s 2nd law for the rotational motion:

τ→=dL/dt

where τ= torque.

For the situation where the net torque is zero dL/dt=0

If the change in angular momentum i.e., ΔL is zero, then the angular momentum is constant; therefore,

L= constant (when net τ= 0)

This is an expression of the law of conservation of angular momentum.

Difference between Angular and Linear momentum

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

p = mv

m= mass,

v= velocity.

On the other hand, if a body moves in a curved path, its velocity will act tangentially along the curved path. This is why the angle is not zero here and 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 its velocity vector, and the angle between the axis and the vector. The equation used for representing it is written as

l = mvr sinθ

The SI unit of linear momentum is kgm/s whereas the SI unit for angular momentum is expressed as kg.m2.rad.s-1.

Angular momentum of a point-sized object

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

So, based on this, the formula can be driven 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

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

Let this be the first equation

So, the equation first can be written as:

p = mvsinθ

Let this be the second equation

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

L = r X p

Let this be the third equation

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 zero. But here we have to replace p with mvsinθ.

Therefore, we can write the entire equation of angular momentum as:

l = rmvsinθ

Angular momentum of a system of particles

When a system of particles is taken into account, the total angular momentum is conserved so that the resultant is equal to zero. The formula for representing the angular momentum of a system of particles is described below

L = l1 + l2 + l3 + l4 + 15 +………… + ln

Here, n= number of particles in the system,

L= total angular momentum, while

L= the momentum of a single particle.

Since L = rmvsinθ

We can modify the angular momentum equation further as

L = r1mv1sinθ + r2mv2sinθ + r3mv3sinθ + r4mv4sinθ …….. + rnmvnsinθ

Applications of Conservation of Angular Momentum

One of the applications is that an ice skater can speed up a pirouette by starting with the arms out and bringing them in.

Conservation of angular momentum is the reason why helicopters either have to have a tail rotor or a second main rotor spinning in the opposite direction e.g., chinook helicopter because otherwise, the body would counterrotate with the rotor to give a total angular momentum equal to 0.

Spinning objects want to keep their orientation and this is the basis of gyroscopes. That’s why satellites are spinning, a bullet is spinning, an American football is spinning, and a frisbee is spinning. That’s also why a motorcycle or bike is much more stable once it has some speed than at a very low speed (and that is something every kid learns the hard way when they learn how to ride a bike). It’s because the spinning wheels have angular momentum (which is horizontal if the bike or motorcycle is upright) and it wants to stay that way (hence the bike or motorcycle wants to stay upright). That’s why we can ride a bike with our hands off the steering wheel.

Conservation of angular momentum entails conservation of rotational energy and that is how a flywheel is used to temporarily store energy.

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