Question

Dimensional formula for angular momentum is
(A) $\left[ {M{L^2}{T^{ - 1}}} \right]$
(B) $\left[ {M{L^2}T} \right]$
(C) $\left[ {{M^0}L{T^2}} \right]$
(D) $\left[ {{M^0}{L^0}{T^0}} \right]$

Answer 1

Hint : The dimension of the angular momentum can be given by the dimension of the quantities in any of its formula, operating on them as in the formula. In one form, angular momentum is a product of mass, linear speed and radius of circulation.

Formula used: In this solution we will be using the following formula;
$\Rightarrow L = mvr$ where $L$ is the angular momentum of a circulating body, $m$ is the mass of the body, $v$ is the tangential speed at a point and $r$ is the radius of the circular path.
$\Rightarrow L = I\omega$ , where $L$ is the angular momentum of a rotating body, $I$ is the moment of inertia, and $\omega$ is the angular velocity of the body.

Complete step by step answer
Angular momentum, in general, is the rotational equivalent of linear momentum. Linear momentum can be considered the quantity that gives a measure of the tendency of an object to resist a change in direction of linear motion. Similarly, the angular momentum can be considered the tendency of an object to resist a change in direction, in a circular or curved direction. This to say, when an object is moving in a clockwise direction, it’s the resistance to a change in direction such that the object starts to move in the anti-clockwise direction.
For a circulating body, the angular momentum is given as
$\Rightarrow L = mvr$ ,where $m$ is the mass of the body, $v$ is the tangential speed at a point and $r$ is the radius of the circular path.
Hence, the dimension of angular momentum can be given as
$\Rightarrow \left[ L \right] = \left[ m \right]\left[ v \right]\left[ r \right]$ where the square brackets stand for dimension.
Hence,
$\Rightarrow \left[ L \right] = M\left( {L{T^{ - 1}}} \right)L$
$\Rightarrow \left[ L \right] = M{L^2}{T^{ - 1}}$
So the correct answer is option A.

Note
Alternatively, the angular momentum of a rotating body can be given as
$\Rightarrow L = I\omega$ , where $I$ is the moment of inertia, and $\omega$ is the angular velocity of the body.
Hence, the dimension is
$\Rightarrow \left[ L \right] = \left[ I \right]\left[ \omega \right]$ ,
which is
$\Rightarrow \left[ L \right] = \left( {M{L^2}} \right){T^{ - 1}}$
$\Rightarrow \left[ L \right] = M{L^2}{T^{ - 1}}$
Hence, the correct answer is A.