Conservation of Angular Momentum

Law of Conservation of Angular Momentum

To understand the law of conservation of angular momentum, we need to understand:

• angular velocity
• rotational inertia
• angular momentum
• torque.

Angular Velocity

The angular velocity is the rate of rotation of an object. It is measured in radians per second, \( \mathrm{\frac{rad}{s}} \). We can find angular velocity using:

• the velocity in linear motion, whose units are in meters per second, \( \mathrm{\frac{m}{s}} \)
• the radius of the object rotating about an axis, whose units are in seconds, \( \mathrm{s} \)

This gives us

$$\omega= \frac{v}{r}$$

Radians are dimensionless; they’re the ratio of an arc length on a circle and that circle’s radius. And so, the units for angular velocity cancel to \( \frac{1}{s} \).

Rotational Inertia

Rotational inertia is an object’s resistance to change in angular velocity. An object with high rotational inertia is harder to rotate than an object with low rotational inertia. Rotational inertia depends on how we distribute the mass of an object or system. If we have an object with a point mass, \(m\), at a distance, \(r\), from the center of rotation, the rotational inertia is \( I=mr^2 \). The rotational inertia of an object increases when it moves further away from the center of rotation. Rotational inertia has units of \( \mathrm{kg\,m^2} \).

Angular Momentum

Angular momentum is the product of the angular velocity, \( \omega \), and rotational inertia, \( I \). We write angular momentum as \( L=I\omega \).

Angular momentum has units of \( \mathrm{\frac{kg\,m^2}{s}} \).Before assigning angular momentum to a particle, we need to define an origin or reference point.

Torque

We represent torque by the greek letter, \( \tau \).

If we have a distance, \( r \), from a pivot point to where force, \( F \) is applied, the magnitude of torque is \( \tau= rF\sin\theta. \) A different way of expressing torque is in terms of the perpendicular lever arm, \( r_{\perp} \), where \( r_{\perp} = r\sin\theta. \) This gives the torque as \( \tau=r_{\perp}F \). Torque has units of \( \mathrm{N\,m} \) where \( 1\,\mathrm{N\,m}=1\,\mathrm{\frac{kg\,m}{s^2}}. \)

Net External Torque and the Conservation of Angular Momentum

The net external torque is expressed as the change of angular momentum over the change in time. We write it as $$\tau_{\mathrm{net}}=\frac{\Delta{L}}{\Delta{t}}.$$ If the net external torque acting on a system is zero, the angular momentum remains constant over time for a closed/isolated system. This means that the change in angular momentum is zero or

$$\Delta{L}=\frac{\tau_{\mathrm{net}}}{\Delta{t}}=\frac{0}{\Delta{t}}=0$$

Another way to express this would be to consider two events in a system. Let’s call the angular momentum of the first event, \( L_1 \), and the angular momentum of the second event, \( L_2 \). If the net external torque acting on that system is zero, then

$$L_1=L_2$$

Note that we define angular momentum in terms of the moment of inertia with the following formula:

$$L = I\omega.$$

Using this definition, we can now write

$$I_1{\omega_{1}}= I_2{\omega_{2}}.$$

In some cases, the conservation of angular momentum is on one axis and not another. Say the net external torque on one axis is zero. The component of the angular momentum of the system along that particular axis will not change. This applies even if other changes take place in the system.

Some other things to take note of:

• Angular momentum is analogous to linear momentum. Linear momentum has an equation of \( p=mv \).

• The conservation of angular momentum is analogous to that of the conservation of momentum as well. The conservation of linear momentum is the equation \( p_1=p_2 \) or \( m_1v_1=m_2v_2. \)

• The equation \( \tau_{\mathrm{net}}=\frac{\Delta{L}}{\Delta{t}} \) is the rotational form of Newton's second law.

Define Conservation of Angular Momentum

The conservation of momentum in simple terms means that the momentum before is equal to the momentum after. More formally,

Conservation of Angular Momentum Formula

The formula \( {I_1}\omega_1={I_2}\omega_2 \) corresponds to the definition of conservation of angular momentum.

Conservation of Angular Momentum in Inelastic Collisions

An inelastic collision is a collision characterized by the loss of some kinetic energy. This loss is due to the conversion of some kinetic energy into other forms of energy. If the greatest amount of kinetic energy is lost, i.e., objects collide and stick together, we call it a perfectly inelastic collision. Despite the loss of energy, momentum is conserved in these systems. However, the equations we use throughout the article are slightly modified when discussing the conservation of angular momentum for perfectly inelastic collisions. The formula becomes

$$ {I_1}\omega_1 + {I_2}\omega_2= (I_1 +I_2)\omega$$

due to the objects colliding and sticking together. As a result, we now consider the two individual objects as a single object.

Conservation of Angular Momentum Examples

One can use the corresponding equations to solve problems involving the conservation of angular momentum. As we have defined angular momentum and discussed the conservation of angular momentum, let us work through some examples to gain a better understanding of momentum. Note that before solving a problem, we must never forget these simple steps:

1. Read the problem and identify all variables given within the problem.
2. Determine what the problem is asking and what formulas are needed.
3. Draw a picture if necessary to provide a visual aid.
4. Apply the necessary formulas and solve the problem.

Examples

Let us apply the conservation of angular momentum equations to a few examples.