Question

Dimensional formula of angular momentum is
A. $M{{L}^{2}}{{T}^{-1}}$
B. ${{M}^{2}}{{L}^{2}}{{T}^{-2}}$
C. $M{{L}^{2}}{{T}^{-3}}$
D. $ML{{T}^{-1}}$

Answer 1

Hint: To find the dimension of any physical quantity first express it in terms of the fundamental quantities. Here angular momentum can be expressed as the product of perpendicular distance and the linear momentum. Again, the linear momentum can be expressed as the product of mass and velocity. In this way express the angular momentum in terms of the fundamental quantities and then we can find the dimension of this quantity.

Complete step by step answer:
Angular momentum can be defined as the rotational analogue of linear momentum. The angular momentum of a rigid body is the product of the moment of inertia and the angular velocity of the object.
So, we can mathematically express the angular momentum as,
$L=I\times \omega =r\times p$
Where, L is the angular momentum, I is the moment of inertia and $\omega$ is the angular velocity.
And we can write,
$L=r\times p=mvr,\text{ where, }p\text{ }=\text{ }mv$
Here p is linear momentum, m is mass of the object and v is the linear velocity of the object.
Now, dimension of mas is $M{{L}^{0}}{{T}^{0}}$
Dimension of velocity is ${{M}^{0}}L{{T}^{-1}}$
And dimension of r is ${{M}^{0}}L{{T}^{0}}$
So, the dimension of angular momentum will be given as,
$M{{L}^{0}}{{T}^{0}}\times {{M}^{0}}L{{T}^{-1}}\times {{M}^{0}}L{{T}^{0}}=ML{{T}^{-1}}$
The correct option is (D).

Note: The SI unit for linear momentum is $kgm{{s}^{-1}}$. The SI unit of angular momentum is $kg{{m}^{2}}{{s}^{-1}}$.
All the derived physical quantities can be expressed in terms of the fundamental quantities. The derived units are dependent on the 7 fundamental quantities. Fundamental units are mutually independent of each other.
Dimension of a physical quantity is the power to which the fundamental quantities are raised to express that physical quantity.