## Law of Conservation of Angular Momentum

In physics, the linear momentum, p = mv, has a rotational analog. It is called angular momentum (rarely, **moment of momentum** or **rotational momentum**), L, and for an object rotating about a fixed axis with angular velocity ω, it is defined as:

*L = Iω*

It is an important quantity in physics because it is a conserved quantity:

*The angular momentum of an isolated system remains constant in both magnitude and direction.*

When there is zero net torque acting on an object, and the object is rotating about a fixed axis or about an axis through its center of mass whose direction doesn’t change, we can write:

**I _{1}ω_{1} = I_{2}ω_{2} = constant**

Like conservation of energy and of linear momentum, this principle is a **universal conservation law**, valid at all scales from atomic and nuclear systems to the motions of galaxies. The **angular momentum is a vector quantity** and therefore it can be generalized as:

*The vector sum of the angular momenta of the parts of an isolated system is constant.*

This puts a strong constraint on the types of rotational motions which can occur in an isolated system. If one part of the system is given an angular momentum in a given direction, then some other part or parts of the system must simultaneously be given exactly the same angular momentum in the opposite direction.

**The law of conservation of angular momentum** explains the angular acceleration of an ice skater as she brings her arms and legs closer to the vertical axis of rotation. By bringing part of the mass of her body closer to the axis she decreases her body’s moment of inertia. Because angular momentum is the product of **moment of inertia (I)** and **angular velocity (ω)**, and the angular momentum remains constant according to this law, the angular velocity of the skater must increase.

I_{1}ω_{1} = I_{2}ω_{2}

## Example: Neutron Star Rotation

A **neutron star** is the collapsed core of a large star (usually of a red giant). Neutron stars are the smallest and densest stars known to exist, but they are **rotating extremely rapidly**. This rapid rotation is a direct consequence of the **law of conservation of angular momentum**. As the star’s core collapses, its rotation rate must increase, because of conservation of angular momentum, hence newly formed neutron stars must rotate at up to several hundred times per second. Some neutron stars emit beams of electromagnetic radiation that make them detectable as pulsars.

For example:

Assume a neutron star of a radius of **7 x 10 ^{5 }km**, which collapses under its own gravitation to a radius of

**10 km**. This star is rotating at a frequency of 1.0 revolution every 30 days. Assume that the star is a homogenous sphere at all times, and loses no mass.

From the law of conservation of angular momentum:

I_{1}ω_{1} = I_{2}ω_{2}

where the subscripts 1 and 2 refer to initial star and neutron star, respectively. The moment of inertia of a sphere about its central axis is:

I = ⅖ m_{1}r_{1}^{2}

therefore

## Conservation of Angular Momentum in Fluid Mechanics

**Euler’s turbomachine equation**, or sometimes called **Euler’s pump equation**, plays a central role in** turbomachinery** as it connects the **specific work Y** and the geometry and velocities in the impeller. The equation is based on the concepts of **conservation of angular momentum** and **conservation of energy**.

The **Euler’s turbomachine equations** are:

**Shaft torque: T _{shaft} = ρQ(r_{2}V_{t2} – r_{1}V_{t1})**

**Water horsepower: P _{w} = ω . T_{shaft } = ρQ(u_{2}V_{t2} – u_{1}V_{t1})**

**Pump head: H = P _{w} / ρgQ = (u_{2}V_{t2} – u_{1}V_{t1})/g**

where

**r**and_{1}**r**are the_{2}**diameters of the impeller**at the inlet and outlet respectively.**u**and_{1}**u**are the_{2}**absolute velocities of the impeller**(u_{1}= r_{1}. ω) at the inlet and outlet respectively.**V**and_{t1}**V**are the_{t2}**tangential velocities of the flow**at the inlet and outlet respectively.

**Euler’s turbomachine equations** can be used to predict the impact of changing the impeller geometry on the head. It does not matter when we deal with a pump or with a turbine. If torque and angular velocity are **of like sign**, work is being done on the fluid (a pump or compressor). If torque and angular velocity are **of opposite sign** work is being extracted from the fluid (a turbine). Thus for the design aspect of turbines and pumps, the Euler equations are extremely useful.