Question

he dimension of the ratio of angular momentum to linear momentum is
A. L⁰
B. L¹
C. L²
D. MLT

Answer 1

Hint: Linear Momentum is mathematically the product of mass and velocity.
Linear momentum of an object is given by the formula:
$p = mv$
Where,
p is the linear momentum of the object
m is the mass of the object
v is the velocity of the object

Angular momentum, (also known as moment of momentum) is mathematically the product of moment of inertia and angular velocity. It is also known as the inertia of rotational motion.
Angular momentum of an object is given by the formula:
$L = r \times mv$
Where,
L is the angular momentum of the object
r is the distance of point of application of Force from its centre of mass

Complete step by step solution:
Linear momentum of an object is given by the formula:
$p = mv$ ……….Equation 1
Where,
p is the linear momentum of the object
m is the mass of the object
v is the velocity of the object

Angular momentum of an object is given by the formula:
$L = r \times mv$ …………..Equation 2
Where,
L is the angular momentum of the object
r is the distance of point of application of Force from its centre of mass

Now, we are required to compute the ratio of angular momentum to linear momentum.
Dividing Equation 2 by Equation 1,
We get,
$ = > \dfrac{L}{p} = \dfrac{{r \times mv}}{{mv}}$
$ = > \dfrac{L}{p} = r$
We know that the dimensional formula of r is [L1]
Hence dimensional formula of $\dfrac{L}{p}$ is,
\[ = > \dfrac{L}{p} = [{L^1}]\]

Hence Option (B) is correct.

Note: Linear momentum and angular momentum are two different quantities. But in dimensions, angular momentum has one L less than linear momentum. Same is the case with other quantities like linear displacement and angular displacement, linear acceleration and angular acceleration etc…