91Ó°ÊÓ

Translational kinetic energy is a form of energy related to the motion of an object moving along a path. It exclusively accounts for motion in a straight line.
In the case of a block of ice sliding down a hill without friction, all gravitational potential energy is converted into translational kinetic energy. This is expressed by the conservation of energy equation:

• Gravitational potential energy = translational kinetic energy
• \[ mgh = \frac{1}{2} mv^2 \]

Here, \(m\) is mass, \(g\) is gravitational acceleration, \(h\) is height, and \(v\) is velocity.
At the bottom of the hill, the ice block reaches a velocity of \(v = \sqrt{2gh}\).
It is interesting to note that since there is no friction involved, all potential energy transforms into translational kinetic energy, making calculations more straightforward.

Rotational kinetic energy arises when an object spins around an axis. For a rolling sphere like a uniform marble, both translational and rotational kinetic energies must be considered.
As the marble rolls down the hill, the energy conserved can be described as:

• Total kinetic energy = translational kinetic energy + rotational kinetic energy
• \[ mgh = \frac{1}{2} mv^2 + \frac{1}{2} I \omega^2\]
• where \( I = \frac{2}{5}mr^2 \) and \( \omega = \frac{v}{r} \) for a solid sphere

The expression simplifies to:

• \[ mgh = \frac{7}{10} mv^2 \]
• yielding a final speed of \(v = \sqrt{\frac{10gh}{7}}\)

This showcases how rolling motion involves both sliding along the ground and rotating around its center, splitting the energy between both forms.

A uniform marble is an object with identical material composition and mass distribution throughout. This uniformity is crucial as it affects how the marble rotates and how energy is distributed between translational and rotational kinetic forms.
For a solid sphere such as a marble, the moment of inertia is given by:

• \[ I = \frac{2}{5} mr^2\]

This represents the resistance to change in rotational motion.
The combination of these factors determines how much energy ends up as rotational kinetic energy versus translational kinetic energy, impacting the marble's speed compared to the sliding block of ice. This link between moment of inertia, mass distribution, and energy conversion is key to understanding how different shapes and compositions affect motion.

Frictionless motion implies no opposing force that would convert kinetic energy into other forms of energy.
The ice block sliding down the hill illustrates this concept perfectly, lacking friction to slow it down and only using gravitational force to change its state.
In an idealized scenario, all gravitational potential energy converts to translational kinetic energy:

• Gravitational potential equals kinetic energy at the hill bottom
• \[ mgh = \frac{1}{2} mv^2\]

This setup helps students understand simple energy conversion cases.
It removes complicating factors like friction, making calculations more straightforward and focusing learning on fundamental physics principles. Such frictionless setups often serve educational purposes by simplifying scenarios to their core elements.