91Ó°ÊÓ

Gravitational potential energy is the energy stored due to an object's height above the ground. It is calculated using the formula:

• \( U = mgh \)

where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is the height. This form of energy depends on the position and is maximal at the top of a hill.
As the stone begins its descent, this stored energy is gradually converted into other forms of energy, such as kinetic.
In cases where two scenarios need comparison, like sliding versus rolling down a hill, understanding the initial potential energy is crucial. This potential energy is entirely based on the starting height \( H \) before the stone begins its movement.

Kinetic energy is the energy an object possesses due to its motion. There are two types: translational and rotational. For translational kinetic energy, the formula is:

• \( KE_{trans} = \frac{1}{2}mv^2 \)

In the frictionless scenario, the spherical stone converts all gravitational potential energy into translational kinetic energy as it slides down the hill.
Since all energy is used in this motion, the stone eventually reaches a height \( h = H \) again after being vertically launched.
However, when friction is present and the stone rolls, part of this potential energy becomes rotational kinetic energy, too. This means not all of it is directed into upward motion.

Rotational motion occurs when the stone rolls down the hill instead of sliding. This involves another type of kinetic energy:

• Rotational kinetic energy: \( KE_{rot} = \frac{1}{2}I\omega^2 \),

where \( I \) is the moment of inertia and \( \omega \) is the angular velocity.
For a solid sphere, \( I = \frac{2}{5}mr^2 \) and \( \omega = \frac{v}{r} \).
Rolling without slipping ensures energy is split into both translation and rotation.
As a result, less energy remains for height gain, hence the stone reaches a smaller height \( h = \frac{5}{7}H \). This division explains why the total energy is dispersed differently compared to sliding.

Friction is the force that resists motion between two surfaces in contact. In the context of the rolling stone, friction provides the necessary torque for rotation without slipping.
While it might seem like an energy drain, friction fosters rolling, converting some potential energy into rotational kinetic energy.
This interaction is described by how potential energy transitions into both types of kinetic energy in the equation:

• \( mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \)

The presence of friction ensures that energy isn't entirely funneled into translational movement. Consequently, this causes a reduced maximum height \( h = \frac{5}{7}H \) when compared to a frictionless case.
It highlights the importance of friction to initiate and sustain rolling motion, altering how high the stone travels.