91Ó°ÊÓ

Potential energy is a fundamental concept in physics, particularly when studying energy conservation. It is the energy stored in an object due to its position relative to a reference point. For the boulder in our example, this reference point is the top of the hill. The boulder's potential energy at the top is calculated using the formula:

• \( PE = mgh \)

where:

• \(m\) is the mass of the boulder.
• \(g\) is the acceleration due to gravity (approximately 9.8 m/s²).
• \(h\) is the height of the hill (50.0 m, in this case).

Potential energy is what gives the boulder its ability to move or fall down the hill.

As the boulder progresses downward, this stored energy transforms into kinetic energy, both translational and rotational. This transformation aligns with the law of conservation of energy, which states that energy cannot be created or destroyed, only converted from one form to another.

Kinetic energy is the energy of motion. When the boulder rolls down the hill, the potential energy at the top is gradually converted into kinetic energy. At the bottom of the hill, the total kinetic energy consists of:

• Translational kinetic energy, given by the formula \( KE_{trans} = \frac{1}{2} mv^2 \), where \(v\) is the translational velocity.
• Rotational kinetic energy, which arises because the boulder is rolling, given by \( KE_{rot} = \frac{1}{2} I \omega^2 \).

For objects like our boulder, it's essential to consider both forms.

During the boulder's descent, the rough top half of the hill ensures it rolls without slipping, converting energy to both forms of kinetic energy without loss.

At the ice-covered bottom half, the boulder slides without the influence of friction, conserving the already converted kinetic energy to calculate final velocity.

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. It plays a crucial role in determining the boulder's rotational kinetic energy as it rolls down a hill. For our boulder, which is a solid sphere, the moment of inertia is calculated using:

• \( I = \frac{2}{5}mr^2 \)

Here,

• \(m\) is the mass of the boulder.
• \(r\) is its radius.

The moment of inertia affects how the boulder accelerates when rolling.

As the boulder reaches the hill's base, its angular velocity \(\omega\) is connected to the translational velocity \(v\) by the relationship \(\omega = \frac{v}{r}\).

Combined with the moment of inertia, these equations ensure we accurately compute both translational and rotational dynamics in assessing the energy conservation in this exercise.