Angular Momentum - Rotation about Fixed Axis

If we consider an extended object as a system of small particles constituting its body, the rate of change of angular momentum with respect to time of the system of particles at a fixed point is equal to the total external torque acting on the system about that point. Can we calculate the net rate of change of angular momentum of a rotating object about a fixed axis? In this section, we will learn about the angular momentum of a particle undergoing rotational motion about a fixed axis.

Angular Momentum for an Object Rotating about a Fixed Axis

Angular momentum

Consider an object rotating about a fixed axis, as shown in the figure. Let us consider a particle P in the body that rotates about the axis as shown above. The expression of total angular momentum for this system can be given by,

\(\begin{array}{l} L = \sum_{i = 1}^{N} r _ { i } \times P _ { i } \end{array} \)

Where Pi is the momentum of the particle (which is equal to mv) and ri is the distance of the particle from the axis of rotation.

The contribution of an individual particle to the total angular momentum can be given as,

\(\begin{array}{l}l = \vec{r} \times \vec{p}\end{array} \)

Here the vector r is equal to OP, and we can write OP = OC + CP.

So we can write, 

\(\begin{array}{l}\vec{l}= \overrightarrow{OC} \times m\vec{v} + \overrightarrow{CP} \times m\vec{v} \end{array} \)

Here,

\(\begin{array}{l} v = \omega r _ { p } \end{array} \)

where rp is the perpendicular distance of the point P from the axis of rotation.

As we can see the tangential velocity v at the point P is perpendicular to the vector rp. Using the right-hand thumb rule, we can say that the direction of product CP×v is parallel to the axis of rotation.

Similarly, we can see that the product of the vectors OC×V is perpendicular to the axis of rotation.

So, we can write,

\(\begin{array}{l} l = \overline{OC} \times m \overline{v} + l_{z} \end{array} \)

Where Iz is the component of the angular momentum parallel to the fixed axis of rotation, which is along the z-axis.

\(\begin{array}{l} L = \sum l = \sum ( l_p + l_z ) \end{array} \)

The perpendicular component of momentum Lp can be given as,

\(\begin{array}{l} L _ { p } = \sum OC_{ i } \times m_{i}v_{i} \end{array} \)

And the parallel component of the momentum can be given as,

\(\begin{array}{l} L_z = ( \sum m_{i} r_{i}^{2} ) \omega \hat{k} \end{array} \)
\(\begin{array}{l} L_z = I \omega \hat{k} \end{array} \)

The object under consideration is generally symmetric about the axis of rotation; hence at a particular perpendicular distance rp, each and every particle possessing a velocity vi has a corresponding particle possessing velocity –vi located diametrically opposite on the circle. Hence, the total angular momentum due to these particles sums up to zero, as they cancel each other.

For such symmetrical objects, the total momentum of the object is given by,

\(\begin{array}{l} L = L_z = l \omega \hat{k} \end{array} \)

Where, I is the moment of inertia of the body, ω is the angular velocity of the body and gives the direction of the total angular momentum.

Watch the video and understand the angular momentum of a rigid body or a system of particles

Frequently Asked Questions – FAQs

Q1

Define angular velocity.

Angular velocity is the vector measure of the rotation rate, which refers to how fast an object rotates or revolves relative to another point.
Q2

What is angular momentum?

Angular momentum is a property of a rotating body given by the product of the moment of inertia and the angular velocity of the rotating object. It is a vector quantity, which implies that here along with magnitude, the direction is also considered.
Q3

What is angular momentum quantum number?

Angular momentum quantum number is synonymous with Azimuthal quantum number or secondary quantum number. It is a quantum number of an atomic orbital that decides the angular momentum and describes the size and shape of the orbital. The typical value ranges from 0 to 1.
Q4

What is angular acceleration?

Angular Acceleration is defined as the time rate of change of angular velocity. It is usually expressed in radians per second per second.
Q5

Can angular acceleration be negative?

The sign of angular acceleration is considered positive if the angular speed increases counterclockwise, and is taken to be negative if the angular speed increases clockwise.

To learn more about the angular momentum of an object moving around a fixed axis and other related topics, download BYJU’S – The Learning App.

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