Name: Radius of Gyration
Uploaded: 2019-07-19
Duration: 1 min 27 s
Description: If you don't know about the radius of gyration, its relation with a moment of inertia and the unit of the radius of gyration. Watch this video to learn more.

Question

Explain the concept of angular momentum and obtain an expression in Cartesian coordinate.

Accepted Answer

Angular momentum is a vector quantity that is a measure of the rotational momentum of a rotating body or system, that is equal in classical physics to the product of the angular velocity of the body or system and its moment of inertia with respect to the rotation axis, and that is directed along the rotation axis.

If $p_{x} = \frac{{dp}_{x}}{dt} = \frac{dp ({mv}_{x})}{dt}, p_{y} = \frac{{dp}_{y}}{dt} = \frac{d ({mv}_{y})}{dt}$ Are the components of linear momentum od in X-Y directions Then, the torque $τ = {xF}_{y} - {yF}_{x}$ , can be written as :

$τ = x m \frac{d v_{y}}{d t} - y m \frac{d v_{x}}{d t} = m [x \frac{d v_{y}}{d t} - y \frac{d v_{x}}{d t}] . . . . . . . . . . . . . (i)$

The quantity under the bracket can can be written using the differentiation products as

$\frac{d}{dt} ({xu}_{y} - {yu}_{x})$ , where $v_{x} and v_{y}$ are the components of velocity in x, Y directions

so we can write

$τ = m \frac{d}{dt} ({xv}_{y} - {yv}_{x})$ or $τ = \frac{d}{dt} ({xmv}_{y} - {ymv}_{x})$

As ${mv}_{x,} {mv}_{y}$ are the components of momentum in x, y directions we have

$τ = \frac{d}{dt} (x p_{y} - y p_{x})$

The quantity $({xp}_{y} - {yp}_{x})$ is called the angular momentum of the body $L$

so $τ = \frac{dL}{dt}$

Thus, the angular momentum of a body about a given axis is the moment of the linear momentum of the body about that axis.

Explain the concept of angular momentum and obtain an expression in Cartesian coordinate.

Important Questions on Rotational Motion of a Rigid Body : Moment of Inertia

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