Question

Dimensional formula for Angular Momentum
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: Angular momentum is defined for a rotating body or system. The position of the rotating body changes when it is under rotational motion. The rate of change of angular position of a rotating body is defined as angular velocity. It is a vector quantity. A body always tries to resist the angular acceleration, which is defined as the moment of inertia.

Complete step-by-step answer:
Angular velocity is the rate of change of position of a rotating body. It is a vector quantity, and the rate of change or angular velocity is known as angular acceleration. The SI unit of Angular velocity is rad /sec (radian / second) and a SI unit of angular acceleration is \[radian/{{\sec }^{2}}\].
Angular momentum is the measure of a rotating body or system is product of the angular velocity of the body and moment of Inertia with respect to the rotation axis.
The Dimension of a physical defined as the power to which the fundamental quantities are raised in order to represent are enclosed in square brackets.
SI unit for angular momentum is \[kg-{{M}^{2}}/\sec \]
\[\text{Angular momentum}=\text{Angular velocity}\times \text{Moment of Inertia }.......\text{ 1}\]
\[\text{Angular momentum}=\dfrac{\text{Angular Displacement }}{\text{Time}}\text{ }.......\text{ 2}\]
\[\text{Moment of Inertia}=\text{Mass}\times {{\left( \text{Radius of gyration} \right)}^{2}}\text{ }............\text{ 3}\].
By substituting equation 2 and 3 in equation 1
\[\text{Angular momentum}=\text{Angular Displacement mass}\times {{\left( \text{Radius of gyration} \right)}^{\text{2}}}\]
\[=\dfrac{\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]}{\left[ {{T}^{1}} \right]}\times \left[ {{M}^{1}} \right]{{\left[ L \right]}^{2}}\]
\[=\left[ {{M}^{1}}{{L}^{2}}{{T}^{-1}} \right]\]
Therefore, correct choice is: (A) \[\left[ {{M}^{1}}{{L}^{2}}{{T}^{-1}} \right]\]

Note: Angular momentum is also defined as a product of the distance of the object from a rotational axis multiplied by the linear momentum. Both angular momentum and linear momentum move both are vector quantities.