91Ó°ÊÓ

Rotational kinetic energy is the energy possessed by a rotating object due to its motion. For a hollow sphere, this energy can be described by the formula: \[ K_r = \frac{1}{2} I \theta^2 \] In the case of the problem, we know the total kinetic energy is 20J and need to determine how much of this is rotational. Using the properties of a hollow sphere, we apply the parallel axis theorem which gives the moment of inertia as: \[ I_h = \frac{2}{3} M R^2 \] Given rotational inertia, I = 0.040 kg·m², we substitute to find: \[ K_r = \frac{1}{2} I_h \theta^2 \] By dividing the total kinetic energy by the percentage of rotational inertia, we determine the amount of kinetic energy attributed to rotation.

The principle of conservation of energy states that the total energy of an isolated system remains constant. For the hollow sphere rolling up the inclined plane, this principle implies that the sum of kinetic and potential energies remains unchanged. Initially, the sphere's total kinetic energy is 20J. As it moves up the incline, potential energy increases. The increase in potential energy can be calculated using: \[ PE = mgh = mgd \times \text{sin}(\theta) \] This creates a direct relationship between the height gain and the decrease in kinetic energy resulting in less speed. By subtracting potential energy from the initial kinetic energy: \[ K' = K - PE \] we obtain the remaining kinetic energy after moving up the incline.

Translational kinetic energy refers to the energy an object possesses due to linear motion. It is given by: \[ K_t = \frac{1}{2} M v^2 \] For the initial scenario, the translational kinetic energy can be found by subtracting the rotational kinetic energy from the total kinetic energy: \[ K_t = K_{total} - K_r \] This allows us to determine the specific amount of energy devoted to the translation of the sphere along the incline.

The speed of the center of mass of an object provides insight into how fast the object is moving linearly. Given the translational kinetic energy, the speed of the center of mass can be determined using: \[ K_t = \frac{1}{2} M v^2 \] Rearrange this to find the velocity: \[ v = \frac{\text{K}_t}{\frac{1}{2}M} \] This formula lets us solve for the speed of the hollow sphere at the initial position. After moving up the incline, the speed of the center of mass can be recalculated using the new kinetic energy: \[ K' = \frac{1}{2} M v'^2 \] By solving for the new velocity, we obtain the updated speed of the center of mass after the move.

An inclined plane is a flat surface tilted at an angle to the horizontal. Physics problems involving inclined planes often involve calculating the force components acting on an object moving on the plane. The angle of inclination influences the motion. For our hollow sphere on a 30° incline, key considerations include the relationship between gravitational force, friction, and normal force. When calculating potential energy changes, it's important to consider the height component of the movement: \[ PE = mgh = mgd \times \text{sin}(\theta) \] This equation allows for determining how much kinetic energy converts into potential energy, thus affecting the motion and speed of the sphere moving along the incline.