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Question: Dimensional formula for Angular Momentum A. \[\left[ M{{L}^{2}}{{T}^{-1}} \right]\] B. \[\left[...

Dimensional formula for Angular Momentum
A. [ML2T−1]\left[ M{{L}^{2}}{{T}^{-1}} \right]
B. [ML2T]\left[ M{{L}^{2}}T \right]
C. [M0LT2]\left[ {{M}^{0}}L{{T}^{2}} \right]
D. [M0L0T0]\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]

Explanation

Solution

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⁡2radian/{{\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−M2/sec⁡kg-{{M}^{2}}/\sec
Angular momentum=Angular velocity×Moment of Inertia ....... 1\text{Angular momentum}=\text{Angular velocity}\times \text{Moment of Inertia }.......\text{ 1}
Angular momentum=Angular Displacement Time ....... 2\text{Angular momentum}=\dfrac{\text{Angular Displacement }}{\text{Time}}\text{ }.......\text{ 2}
Moment of Inertia=Mass×(Radius of gyration)2 ............ 3\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
Angular momentum=Angular Displacement mass×(Radius of gyration)2\text{Angular momentum}=\text{Angular Displacement mass}\times {{\left( \text{Radius of gyration} \right)}^{\text{2}}}
=[M0L0T0][T1]×[M1][L]2=\dfrac{\left[ {{M}^{0}}{{L}^{0}}{{T}^{0}} \right]}{\left[ {{T}^{1}} \right]}\times \left[ {{M}^{1}} \right]{{\left[ L \right]}^{2}}
=[M1L2T−1]=\left[ {{M}^{1}}{{L}^{2}}{{T}^{-1}} \right]
Therefore, correct choice is: (A) [M1L2T−1]\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.