Question

Linear momentum and angular momentum have the same dimensions in:

A. Mass and length
B. Length and time
C. Mass and time
D. Mass, length and time

Answer 1

Hint: As according to the definition of the linear momentum the linear momentum is the product of mass and its velocity and it is measured in $kg.m.{{s}^{-1}}$. On the other hand we can say that the angular momentum is the rotational equivalent of linear momentum and it is measured in $kg.{{m}^{2}}.{{s}^{-1}}$.

Formula used:
For linear momentum the formula: $\overrightarrow{p}=\left( m \right)\left( \overrightarrow{v} \right)$
Where $m$= mass of the object
$v$ = velocity of the object
And $p,v$ are vector quantities.
For angular momentum: $\overrightarrow{L}=I\times \overrightarrow{\omega }$ ( for extended objects)
$\overrightarrow{L}=r\times \overrightarrow{p}$ (for point objects)
Here $I$= rotational inertia
$r$ = radius (distance between the object and the fixed point about it revolves)
$\overrightarrow{\omega }$ = angular velocity
$\overrightarrow{p}$= linear momentum

Complete answer:
The dimensional formula for linear momentum is: $\left[ {{M}^{1}}{{L}^{1}}{{T}^{-1}} \right]$
And the dimensional formula for the angular momentum of any object is: $\left[ {{M}^{1}}{{L}^{2}}{{T}^{-1}} \right]$
Here we can see that the mass and time have the same dimensions.

So, the correct answer is “Option C”.

Additional Information:
The linear Momentum of an object is directly proportional to the object’s mass and also its velocity. Thus, the higher an object’s mass or the more its velocity, the greater its momentum. Momentum p may be a vector having an equivalent direction because the velocity v. Force and momentum are intimately related. Force acting over time can change momentum, and Newton’s second law of motion is often stated in its most broadly applicable form in terms of momentum. Momentum continues to be a key concept within the study of atomic and subatomic particles in quantum physics.

Note:
As the term defines itself linear momentum is used for the objects which are having the motion in linear direction while the angular momentum is a property of a rotating body having angular velocity. So don’t get confused between linear and angular momentum.