91Ó°ÊÓ

Kinetic energy is the energy a body possesses due to its motion. For our hoop, the kinetic energy (KE) is a sum of two types: translational and rotational. Let's break it down:

• Translational kinetic energy: This is associated with the linear motion across the surface, calculated with the formula: \( KE_{trans} = \frac{1}{2}mv^2 \). Where \( m \) is the mass of the hoop and \( v \) is its velocity.
• Rotational kinetic energy: This is due to the hoop's rotation, calculated as: \( KE_{rot} = \frac{1}{2}I\omega^2 \). Here, \( I \) represents the moment of inertia and \( \omega \) is the angular velocity.

The total kinetic energy \( K \) for a rolling hoop, like in the problem, combines these two forms, leading us to: \( K = \frac{1}{2}mv^2 + \frac{1}{2}I\left(\frac{v}{R}\right)^2 \). Plugging in the given values, our hoop's kinetic energy is 600 Joules, which tells us how much energy the hoop has while rolling at 10 m/s.

The Work-Energy Principle is a powerful concept that connects the concepts of work and kinetic energy. It states that the total work done on an object is equal to the change in its kinetic energy. In simpler terms, to stop the rolling hoop, the work done must remove all of its kinetic energy.
The work required to bring our 600 Joule hoop to a stop is exactly 600 Joules. This is because the work done by a force when stopping or moving an object is equal to the object’s initial kinetic energy. Thus, in this scenario, the work simply converts the hoop's kinetic energy into other forms such as heat or sound.
In part (b) of the exercise, as the hoop climbs the incline, its kinetic energy converts to potential energy. As the hoop comes to a stop momentarily, all its initial kinetic energy has been converted into gravitational potential energy.

Moment of inertia, often symbolized as \( I \), is the rotational analog of mass in linear motion. It tells us how difficult it is to change the rotational motion of an object. For our hoop, the moment of inertia is calculated by the formula: \( I = MR^2 \), where \( M \) is the mass of the hoop and \( R \) its radius.

• The calculated moment of inertia for our hoop is \( 6.0 \,\text{kg}\cdot\text{m}^2 \), which represents the hoop's resistance to rotational acceleration about an axis through its center.

This concept is crucial during the calculations of rotational kinetic energy, as it factors into the energy associated with how the hoop spins as it rolls across the surface. The moment of inertia allows us to incorporate rotational dynamics into our energy calculations.

Rotational motion involves objects that spin around an axis. For the hoop, this means it rolls across the floor while simultaneously spinning around its own center. The motion is characterized by parameters like angular velocity (\( \omega \)), which relates linear speed (\( v \)) and radius (\( R \)) through the relation: \( v = R\omega \).

• Rotational motion here contributes to the total kinetic energy. This is because as the hoop spins, it carries not only a translational speed but also an angular speed.
• The relationship between the two speeds is captured in how the rotational kinetic energy was calculated in previous sections.

Understanding rotational motion is key to grasping why the hoop has rotational kinetic energy in addition to translational energy. This dual motion is what allows us to apply concepts like the work-energy principle to both linear and rotational aspects of the problem.