A weightlifter's barbell consists of two 25 -kg masses on the ends of a \(15-\mathrm{kg}\) rod \(1.6 \mathrm{m}\) long. The weightlifter holds the rod at its center and spins it at 10 rpm about an axis perpendicular to the rod. What's the magnitude of the barbell's angular momentum?

The moment of inertia, often represented as 'I', is a quantity that describes an object's resistance to changes in its rotational motion. It is analogous to mass in linear motion; just as an object with greater mass requires more force to change its speed, an object with greater moment of inertia requires more torque to change its rate of spin.

The formula to find the moment of inertia depends on the geometry and mass distribution of the object. For a solid rod rotating around an axis through its center, as in the weightlifter's barbell scenario, the formula is \[\begin{equation} I_{\text{rod}} = \frac{1}{12}ML^2 \, \end{equation}\] where 'M' is the mass of the rod and 'L' is its length. If we attach point masses, each with mass 'm' at a distance 'r' from the axis, then the moment of inertia for each is \[\begin{equation} I_{\text{point}} = mr^2 \, \end{equation}\] and the total moment of inertia is the sum of all individual moments. For the weightlifter's barbell, this included the rod itself and the two distributed masses on the ends.

Angular velocity, denoted by the Greek letter \( \omega \) (omega), measures the rate of rotation of an object around an axis. It's the rotational equivalent of linear velocity, which tells us how fast an object is moving in a straight line. The standard unit for angular velocity is radians per second (rad/s).

Understanding angular velocity is crucial when dealing with rotational systems, as it impacts other rotational quantities such as angular momentum and kinetic energy. With the weightlifter's barbell example, finding the angular velocity was an essential step before we could calculate the barbell's angular momentum. Knowing how quickly the barbell spins provides insight into both the amount of kinetic energy it possesses and the required effort to stop or change its rotation.

When we deal with rotating objects, it's common to encounter different units of measurement for angular velocity. While some situations call for revolutions per minute (rpm), others require radians per second (rad/s). Converting between these units allows for a consistent approach to solving rotational motion problems.

To convert from rpm to rad/s, one can use the fact that one revolution corresponds to \(2\pi\) radians and there are 60 seconds in a minute. Consequently, the conversion formula is \[\begin{equation} \omega \_{\text{rad/s\}} = rpm \times \dfrac{2\pi}{60}\, \end{equation}\] This formula ensured the correct unit for angular velocity was used to determine the barbell's angular momentum in the exercise. With \( \omega \_{\text{rad/s\}} \) calculated, it could be multiplied with the moment of inertia to find the angular momentum of the system.