The question asks for the mathematical expression that represents angular momentum, given angular velocity (ω) and moment of inertia (I).

Angular momentum is a fundamental quantity in rotational motion, similar to linear momentum in linear motion. It describes the "amount of rotation" an object has, taking into account its mass distribution and how fast it is rotating.

Understanding Angular Momentum, Moment of Inertia, and Angular Velocity

• Angular Momentum: This is a vector quantity that represents the rotational inertia of a body in motion about an axis. It depends on the object's mass distribution and its angular velocity.
• Moment of Inertia (I): This property represents how the mass of an object is distributed relative to its axis of rotation. It is the rotational equivalent of mass in linear motion. A higher moment of inertia means it's harder to change the object's rotational speed.
• Angular Velocity (ω): This is the rate at which an object rotates or revolves about an axis. It's a vector quantity, and its magnitude is typically measured in radians per second.

Mathematical Relation for Angular Momentum

For a rigid body rotating about a fixed axis, the angular momentum (L) is directly proportional to its moment of inertia (I) about that axis and its angular velocity (ω). The mathematical relationship is given by:

$$L = I\omega$$

Where:

• L is the angular momentum
• I is the moment of inertia
• ω is the angular velocity

This formula shows that angular momentum is the product of moment of inertia and angular velocity. This relationship is analogous to the linear momentum formula, p = mv, where p is linear momentum, m is mass (analogous to I), and v is linear velocity (analogous to ω).

Let's look at the given options:

• Option 1: $I + \omega$. This is the sum of moment of inertia and angular velocity, which does not represent angular momentum.
• Option 2: $I\omega$. This is the product of moment of inertia and angular velocity, which is the correct formula for angular momentum.
• Option 3: $I - \omega$. This is the difference between moment of inertia and angular velocity, which does not represent angular momentum.
• Option 4: $I\omega^2$. This expression involves the square of angular velocity multiplied by moment of inertia. This form is related to rotational kinetic energy ($KE_{rot} = \frac{1}{2}I\omega^2$), but not angular momentum.

Based on the definition and formula for angular momentum, the correct expression is the product of moment of inertia and angular velocity.