how to explain the relationship between the mass of the rotating object and its period?

period of what?

what is relationship between mass of rotating object and its period

To explain the relationship between the mass of a rotating object and its period, let's start by understanding what period means.

The period of a rotating object is the time it takes for the object to complete one full rotation or cycle. It is usually measured in seconds.

When it comes to the relationship between the mass of a rotating object and its period, we need to consider some key principles of rotational motion. One of these principles is the conservation of angular momentum.

Angular momentum (L) of a rotating object is the product of its moment of inertia (I) and its angular velocity (ω). Mathematically, it can be expressed as L = Iω.

The moment of inertia depends on the mass distribution of the rotating object. For a simple case, such as a rotating object with a point mass at a given distance from the axis of rotation, the equation for the moment of inertia is I = mr², where m represents mass and r represents the distance from the axis of rotation.

Now, let's consider two scenarios – one with a larger mass and the other with a smaller mass:

1. Larger Mass: When the mass of the rotating object increases, the moment of inertia also increases, assuming the shape and the radius of rotation remain the same. Consequently, according to the conservation of angular momentum, to maintain the same angular momentum, the angular velocity (ω) of the rotating object must decrease.

Since period (T) is the inverse of angular velocity (T = 1/ω), this means that as the mass increases, the period of the rotating object also increases. In other words, a larger mass leads to a longer period.

2. Smaller Mass: On the other hand, when the mass of the rotating object decreases, the moment of inertia decreases. To conserve angular momentum, the angular velocity must increase. As a result, the period of the rotating object decreases. In simpler terms, a smaller mass leads to a shorter period.

In summary, there is an inverse relationship between the mass of a rotating object and its period. As the mass increases, the period increases, and as the mass decreases, the period decreases. This relationship can be explained using the principles of angular momentum and the conservation of angular momentum.