Angular Momentum

For understanding Angular Momentum you should know that there are basically two levels of studying physics. In the first level we understand the behaviour of a single particle under the given environment. The second level of physics is understanding the behaviour of a system of particles under the given environment.

The single particle study involves the object is treated as singe article and the physical laws are applied to it. In case of system of particles, there will be a combination of various particles embedded together forming a particular shape and the system of particles behavior is not arrived at by a single particle but by a group of particles. The study involves arriving at the Equations of motion under each circumstances.

Here in this section of discussing the practice problems in Physics class 11 on angular momentum, we define the angular momentum for single particle and for a system of particles.

Angular velocity of a single particle – Physics Class 11

From the basics of physics of linear velocity and angular velocity, the relationship between linear velocity and angular velocity is given by the equation , $\overrightarrow{v}=\overrightarrow{\mathit{\omega }}\times \overrightarrow{r}$ , where $\overrightarrow{v}$ is the linear velocity , $\overrightarrow{\mathit{\omega }}$ is the angular velocity and $\overrightarrow{r}$ is the position vector the particle about a reference point.. All these are vectors.

The above can be easily understood from the figure below.

Let the axes x,y and z meet at a reference point called as origin . The position vector of particle from the meeting point of x,y and z axes is (origin) is $\overrightarrow{r}$ , $\overrightarrow{\mathit{\omega }}$ is the angular velocity and $\overrightarrow{v}$ is the linear velocity of a particle as shown in above diagram.

$\overrightarrow{v}=\overrightarrow{\mathit{\omega }}\times \overrightarrow{r}$

The angular velocity of the particle is the cross product of $\overrightarrow{\mathit{\omega }}$ and $\overrightarrow{r}$ in the same order . It is important to note here that the order of cross product should not be changed. The directions to be preserved in the case of vector operations.

Angular momentum of a single particle – Physics Class 11

To understand the angular momentum, we are required to understand the concept of angular acceleration. The angular acceleration is the rate of cahnge of angular velocity similar to linear acceleration is the rate of cahnge of linear velocity. Both angular acceleration and linear acceleration are vectors. The angular momentum is defined as the cross product of position vector and the linear momentum. The angular momentum of a particle can be expressed as,

$\overrightarrow{L}=\overrightarrow{r}\times \overrightarrow{p}$

In the above equation , $\overrightarrow{L}$ is angular momentum, $\overrightarrow{r}$ is the position vector of the particle , and $\overrightarrow{p}$ is the linear momentum of the particle.

The angular momentum of the particle is the cross product of $\overrightarrow{r}$ and $\overrightarrow{p}$ in the same order . It is important to note here that the order of cross product should not be changed. The directions to be preserved in the case of vector operations here also as in definition of angular velocity.

A diagrammatic representation of angular momentum can be made as below.

Torque on Single particle

The torque is defined as the cross product of the position vector of particle and the Force acting on the particle. Here both Force and Position vectors are vectors and their cross product is a vector. Hence Torque is a vector. Hence torque on a particle is given by the equation ,

$\overrightarrow{\mathit{\tau }}=\overrightarrow{r}\times \overrightarrow{F}$

In the above equation, $\overrightarrow{\mathit{\tau }}$is called the Torque, $\overrightarrow{r}$is the position vector of the particle, $\overrightarrow{F}$is the Force acting on the particle.

For more on the vectors please refer to videos from YouTube Channel maths&science by iitian at :

https://www.youtube.com/@mathsciencebyiitian

For further study on Angular Momentum , Vectors , we will proceed to next article with a numerical solved example.

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Best of luck to every one.