91Ó°ÊÓ

In physics, energy can take various forms, one of which is rotational kinetic energy. When an object rotates, such as a rolling marble, it possesses rotational kinetic energy. This energy arises from the object's rotation around its own axis. The rotational kinetic energy can be calculated using the formula:

• \[ KE_{rotational} = \frac{1}{2}I\omega^2 \]

where \(I\) represents the moment of inertia of the object, and \(\omega\) stands for the angular velocity. For a uniform sphere like a marble, the moment of inertia \(I\) is equal to \(\frac{2}{5}mr^2\), where \(m\) is the mass of the marble and \(r\) is its radius.When the marble rolls down the rough side of the bowl, rotational kinetic energy plays a significant role. As it descends, its gravitational potential energy converts into both translational and rotational kinetic energy, ensuring the marble rolls effectively without slipping.

Translational kinetic energy relates to the movement of an object from one place to another and is given by the formula:

• \[ KE_{translational} = \frac{1}{2}mv^2 \]

where \(m\) is the mass and \(v\) is the velocity of the object. In this scenario, as the marble rolls down, the gravitational potential energy it originally possessed at the top of the bowl is converted into translational kinetic energy.On the frictionless side of the bowl, the marble can no longer roll, and hence, it stops spinning, meaning only the translational kinetic energy is present. This situation emphasizes the importance of friction for rolling motion, as it helps maintain the marble's rotation and distributes energy between translation and rotation.

When the term "rolling without slipping" is used, it hints at a scenario where an object rolls in such a way that its point of contact does not slide across the surface. For a marble to roll without slipping, there must be a relationship between its translational velocity \(v\) and angular velocity \(\omega\), given by:

• \[ v = r\omega \]

This equation means that each point on the marble's surface is momentarily at rest with respect to the surface it rolls on, which is crucial for the full conversion of potential energy into both translational and rotational kinetic energy.On the rough side of the bowl, the friction ensures the marble rolls without slipping, making it possible for both types of kinetic energy to be conserved. However, on the frictionless side, this perfect balance is disrupted as the marble slides instead of rolling, showing how friction aids in maintaining this balance during motion.