Question

The position of a particle is given by $ {\mathbf{\vec r}} = \left( {{\mathbf{\hat i}} + {\mathbf{2\hat j}} - {\mathbf{\hat k}}} \right) $ and momentum $ {\mathbf{\vec P}} = \left( {3{\mathbf{\hat i}} + {\mathbf{4\hat j}} - {\mathbf{2\hat k}}} \right) $ . The angular momentum is perpendicular to
(A) X-axis
(B) Y-axis
(C) Z-axis
(D) Line at equal angles to all the three axes.

Answer 1

Hint: Momentum can be defined as the tendency of an object to be in motion. It is a product of mass and velocity. Angular momentum is the property of any rotating object given by the moment of inertia times the angular velocity.
Angular velocity can be defined as the rate of change of change of angular displacement of the body.

Complete step by step solution:
Angular momentum is represented as
 $ {\mathbf{\vec L}} = {\mathbf{\vec r}} \times {\mathbf{\vec P}} $
Where L is the angular momentum
R is the radius or position of the particle.
P is the momentum.
 $ \Rightarrow {\mathbf{\vec L}} = \left( {{\mathbf{\hat i}} + {\mathbf{2\hat j}} - {\mathbf{\hat k}}} \right) \times \left( {{\mathbf{3\hat i}} + {\mathbf{4\hat j}} - {\mathbf{2\hat k}}} \right) $
 $ \Rightarrow {\mathbf{\vec L}} = {\mathbf{0\hat i}} - {\mathbf{\hat j}} - {\mathbf{2\hat k}} $
As you can see from the equation the angular momentum is perpendicular to the X axis.
So the correct answer is Option A.

Additional Information:
Angular momentum is similar to the Azimuthal quantum number. It decides the angular momentum and shape of the orbital. The value range is from zero to one.
When an ice skater does a spin first they start off with their hands and legs distant from the body. When they get their hands and legs close to the body the angular velocity increases. Hence the angular momentum is conserved here.

Notes
We can find the direction of angular momentum using the right-hand rule. According to the right hand rule we have to position our fingers in the direction of “r”. Then the palms are curled in such a way that they point towards the direction of linear momentum. The outstretched thumb gives the direction of angular momentum.
One of the factors affecting angular momentum is rotational inertia along with mass and velocity of rotation.
Rotational Inertia can be defined as the value(scalar value) that can give an idea about how hard it is to change the rotational velocity of the object revolving around a rotational axis.