Angular Speed

# Angular Momentum

We turn a bicycle upside down so that it stands on the balance and the seat. We grab the front wheel and turn it with a jerk of the hand. The wheel, which is initially at rest, begins to rotate with a certain angular velocity after the impact of our hands. During the rotation, all points on the circumference move with uniform acceleration.

If we grab the wheel a bit closer to the axis of rotation instead of at the rim, rather at the tip (see Figure 1), we would need a slightly higher force to produce the same final angular velocity. This makes us think that we need the torque of a force for a certain speed of rotation of the wheel. We cause the torque with the force of our hands; in this case, the lever is equal to the distance between the force and the axis of rotation.

With the application of torque of a force, with which we act on the system for a certain time , we change the speed of rotation of the wheel. We say that we have changed the angular momentum with the torque of the force.

Figure 1: We turn the wheel with a torque

In this chapter, we will learn about two new physical quantities that are closely related:

• moment of inertia, and

• angular momentum.

## Moment of inertia

The moment of inertia has similar importance in angular momentum as the mass in the case of a linear momentum. Due to mass, a body persists in motion if no force acts on it. The same applies to the moment of inertia in rotation: a body persists in rotation if no torque acts on it.

Let's first learn about the moment of inertia of a point-like body moving around a circle of radius . We then extend the knowledge from this simple example to a rotating rigid body.

### Moment of inertia of a point body

Two forces act on a point body that rotates with uniform acceleration (Figure 2):

Figure 2: Forces in uniformly accelerated rotation

• The centripetal force , which is the force that keeps the body in rotation. It is equal to the product of the mass and the centripetal acceleration :

where the centripetal acceleration is given as:

Since the angular velocity changes with time (the body rotates with uniform acceleration), the centripetal acceleration also changes with time; if the angular velocity increases, the centripetal acceleration also increases.

• The second force is the tangential force and according to Newton's second law, it causes the body to move with tangential acceleration (linear acceleration) :

 We note that that tangential acceleration is the product of the angular acceleration and radius : Let's multiply both sides of the equation by : We note that the moment of inertia of a body is given as: We note also that the torque is given as the product of the tangential force and the radius : Let's note again that the tangential acceleration is given as the product of the angular acceleration and radius : or:

The expressions we obtained are also called Newton's second law for circular motion.

Let's write the relationship between force and acceleration (Newton's second law):

We replace force with torque , mass with moment of inertia and acceleration with angular acceleration , and we get:

The torque of a force on a rotating point body causes the body to move with an angular acceleration . The proportionality factor is the moment of inertia :

Here, the angular acceleration is the quotient of the tangential acceleration and the radius :

The moment of inertia of a point body is:

### Moment of inertia of a rigid body

Let us take any rigid body rotating about any axis as shown in Figure 3.

Figure 3: Moment of inertia of a rigid body

The body is divided into small cubes with mass . The squares are distant from the axis of rotation by different distances .

The first cube has its moment of inertia as:

second cube:

and any cube:

The moment of inertia of the whole body is equal to the sum of the moments of inertia of individual cubes:

 Let's write the sum above using the summation symbol : We note that:

The moment of inertia of a rigid body is the sum of the moments of inertia of the parts of the body at a given axis of rotation:

Example

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Let's also write down the moments of inertia of some other geometric bodies:

 Geometric body Axis of rotation Moment of inertia Hollow cylinder Geometric axis Full cylinder Geometric axis Full sphere Any axis passing through the center of the sphere

## Angular momentum

We already mentioned in the introduction that torque lasting a certain time changes the angular momentum of the body. But we didn't say what the angular momentum is.

Let's write again the relationship between torque , moment of inertia , and angular acceleration :

 Let's multiply both sides of the equation by : We note that: Let's now introduce the angular momentum which is given as:

We therefore define the angular momentum as the product of the moment of inertia and the angular velocity :

The angular momentum is a vector. We will write it as a scalar, noting the sign:

• Counter-clockwise rotation gives a positive torque.

• Clockwise rotation gives a negative torque.

The angular momentum is the product of the moment of inertia and the angular velocity :

The angular momentum is positive if the body rotates counterclockwise and negative if it rotates clockwise.

Angular momentum and linear momentum are related concepts and complement each other. If the linear momentum is calculated for bodies that move translationally (e.g. along a straight line), the angular momentum is calculated for bodies that rotate around a given axis.

### Conservation of angular momentum

Let's write again the equation:

The term on the left is called the angular impulse, and on the right is the change in momentum:

The written equation above is called the conservation of angular momentum.

If we act on a body that can rotate around a selected axis with an angular impulse, we change its angular momentum.

Example

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### Law of conservation of angular momentum

If there is no external angular impulse, the angular momentum is conserved:

 We note that:

If there is no external angular impulse, the angular momentum is conserved:

Example

The example is available to registered users free of charge.

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