Question

Does angular momentum change with radius?

Answer 1

We are asked to state the dependency of angular momentum on radius. We can start to answer this question by defining what angular momentum is. Then we can move onto writing down the equation or the formula to find the angular momentum and then substituting a set of values we can see if angular momentum is dependent on the radius.

Formulas used: The formula used to find the angular momentum of a system is given by $J = m\left( {\vec r \times \vec v} \right)$
Where $\vec r$ is the radius of the motion
$\vec v$ is the linear velocity of the motion
$m$ is the mass of the body undergoing the motion

Complete step by step solution:
Let us start by defining what angular momentum is. Angular momentum can be easily defined as the rotational equivalent of linear momentum but it actually is the property of a rotating body which is the product of the moment of inertia and the angular velocity of the body.
The formula to find the angular momentum of a body is given by $J = m\left( {\vec r \times \vec v} \right)$
When we change this into the scalar form, we get $J = mvr$
We know that $v = r\omega$
Substituting this in the equation above, we get $J = m{r^2}\omega$
If the angular velocity of the motion is a constant, and the mass is a constant. We get
$J \propto {r^2}$
This means that the angular momentum depends on the radius. The angular momentum is directly proportional to the square of the radius of motion.

Note:
The property of a rotating body which is the product of the moment of inertia and the angular velocity of the body. From this definition of angular momentum, we can come up with a formula for it, which will be $J = I\omega$
Where $I$ is the moment of inertia of the system and
$\omega$ is the angular momentum