91Ó°ÊÓ

Translational speed refers to the speed at which the center of mass of an object moves along a path. In the scenario with the marble and the block of ice, their translational speeds at the bottom are determined by how their gravitational potential energy is converted into kinetic energy.
For the block of ice, as it slides down the hill without friction, its gravitational potential energy is fully transformed into translational kinetic energy. The formula for translational speed is derived from the conservation of energy principle:

• Potential Energy: \(mgh\)
• Translational Kinetic Energy: \(\frac{1}{2} mv^2\)

Setting these equal gives us \(mgh = \frac{1}{2} mv^2\), and solving for \(v\), the translational speed of the ice block becomes \(v = \sqrt{2gh}\). Every bit of energy turns into translational motion, making it straightforward to determine how fast something like the ice block moves at the hill's bottom.
For the marble, both translational and rotational kinetic energy need to be taken into account, which affects how we calculate its translational speed.

Rotational kinetic energy is the energy an object possesses due to its rotation. For any rolling object, such as our marble in this problem, it contributes significantly alongside translational kinetic energy. Here, the marble rolls without slipping, meaning it directly converts part of its initial gravitational potential energy into rotational kinetic energy.
The formula that represents this form of energy for a rotating sphere like a marble is:

• Rotational Kinetic Energy: \(\frac{1}{2} I \omega^2\)

Where \(I\) is the moment of inertia for a solid sphere \(\left(\frac{2}{5}mr^2\right)\), and \(\omega\) is the angular velocity. In rolling scenarios, there is a relationship between the translational velocity \(v\) and angular velocity \(\omega\), which is \(v = r\omega\). Substituting the values and simplifying:

• Moment of Inertia: \(\frac{2}{5}mr^2\)
• \(mgh = \frac{1}{2} mv^2 + \frac{1}{2} \left(\frac{2}{5}mr^2\right)\left(\frac{v^2}{r^2}\right)\)

This leads to calculating the marble's speed as \(v = \sqrt{\frac{10}{7}gh}\). The presence of rotational kinetic energy reduces the translational speed slightly compared to the block of ice.

Gravitational potential energy is the energy an object holds due to its position in a gravitational field. It's particularly vital in scenarios like the rolling marble and sliding ice block as it represents their starting energy at height \(H\). This potential energy is given by:

• Gravitational Potential Energy: \(mgh\)

As both the marble and ice slide from rest at height \(H\), their potential energy at the top is converted entirely to kinetic energy at the bottom. What makes this interesting is how each object utilizes this energy.

• The ice block converts all of its potential energy into translational kinetic energy, resulting in \(v = \sqrt{2gh}\).
• The marble splits its energy into translational and rotational forms, thus the translational speed \(v = \sqrt{\frac{10}{7}gh}\) reflects its dual energy usage.

Even though both objects start with the same gravitational potential energy, they experience different speeds and energy types at the bottom due to the rolling motion of the marble.