91Ó°ÊÓ

Kinetic energy is the energy an object possesses due to its motion. For rotational systems, like the pulley in this problem, kinetic energy can be split into two types: translational kinetic energy of the stone and rotational kinetic energy of the pulley. The formula for rotational kinetic energy is expressed as:

• \( KE_{rot} = \frac{1}{2} I \omega^2 \)

where \(I\) is the moment of inertia, and \(\omega\) is the angular velocity. Also, for linear motion involving the stone, kinetic energy is given by:

• \( KE_{linear} = \frac{1}{2} m v^2 \)

Here, \(m\) is the mass of the object, and \(v\) is the linear velocity. In this problem, the total kinetic energy is shared between the stone and the pulley, as the stone falls and the pulley rotates. Understanding this split helps solve for how far the stone falls and the kinetic energy distribution.

The moment of inertia is a property of a rotating object that determines how much torque is needed for a desired angular acceleration. It plays a crucial role in rotational dynamics, just like mass in linear dynamics. For the uniform disk-shaped pulley in this exercise, the moment of inertia \(I\) can be calculated with the formula:

• \( I = \frac{1}{2} M R^2 \)

where \(M\) is the mass of the disk, and \(R\) is its radius. In the given case, the disk has a mass of 2.50 kg and a radius of 0.20 m. Plugging these values into the formula gives \(I = 0.05\) kg·m². This value defines how resistant the pulley is to changes in its rotational motion and is essential for calculating the rotational kinetic energy and subsequent physics calculations.

Gravitational Potential Energy (GPE) is the energy possessed by an object due to its position in a gravitational field. In this exercise, the stone's fall transforms GPE into kinetic energy. The formula to calculate GPE is:

• \( U = mgh \)

where \(m\) is the mass of the stone, \(g\) is the acceleration due to gravity (approximately 9.8 m/s² on Earth), and \(h\) is the height from which the stone falls. As the stone descends, its potential energy decreases while its kinetic energy increases, behaving according to the conservation of mechanical energy principle. Calculate how far it should fall by equating the lost GPE to the system's total kinetic energy gain.

Angular velocity, denoted by \(\omega\), describes how fast an object rotates. It's a vector quantity that represents the rotational speed and the axis about which an object rotates. The angular velocity is linked to the linear velocity \(v\) of an object like the stone in this problem through the relationship:

• \( \omega = \frac{v}{R} \)

where \(R\) is the radius of the circular path. In the exercise, the rotational kinetic energy given helps us find the angular velocity. Solving:

• \( 4.50 = \frac{1}{2} \times 0.05 \times \omega^2 \)

we find that \(\omega \approx 13.42\) rad/s, illustrating the speed at which the pulley turns as the stone falls.