91Ó°ÊÓ

Rotational inertia, often referred to as the moment of inertia, is a fundamental concept when studying rotational motion. It represents how much an object resists changes to its rotational motion. Just as mass determines how much a linear motion resists acceleration, rotational inertia quantifies this for rotational systems.

Imagine a spinning vinyl record. It has rotational inertia because it requires effort to change its spinning speed. The rotational inertia depends on two factors:

• The mass of the object
• The distribution of that mass relative to the axis of rotation

For the vinyl record, its mass is distributed evenly along its radius. In our problem, the record's rotational inertia is initially given as \(5.0 \times 10^{-4} \; \mathrm{kg \cdot m^2}\). When additional mass is added to the system, like the wet putty sticking to the edge, the rotational inertia increases. More mass further from the axis intensifies resistance to angular acceleration, affecting the system's rotational characteristics.

The moment of inertia is a key quantity in rotational dynamics. It acts as the rotational analog of mass in linear motion. The moment of inertia \(I\) can be calculated for various shapes depending on their mass distribution.

For instance, the formula we use when a point mass \(m\) is at a distance \(r\) from the axis is:\[ I = m r^2 \]In the exercise, the putty is treated as a point mass since it sticks at the edge of the record and affects the moment of inertia. Initially, the record had a moment of inertia \(I_i = 5.0 \times 10^{-4} \; \mathrm{kg \cdot m^2}\). After the putty sticks, it contributes additional inertia calculated by \(0.020 \; \mathrm{kg} \times (0.10 \; \mathrm{m})^2 = 2.0 \times 10^{-4} \; \mathrm{kg \cdot m^2}\).

The new total moment of inertia becomes \(I_f = I_i + I_{\text{putty}} = 7.0 \times 10^{-4} \; \mathrm{kg \cdot m^2}\). Understanding the moment of inertia helps us predict how the system's rotational behavior will change.

Angular speed measures how fast an object rotates or revolves relative to another point. Usually denoted by \(\omega\), angular speed is similar to linear speed but in a rotational context. In our scenario, it's crucial for calculating the spinning rate of the vinyl record.

Initially, the record spins at an angular speed \(\omega_i = 4.7 \; \mathrm{rad/s}\). However, when the mass (the putty) is added, this speed changes due to the conservation of angular momentum. The principle states that if no external torque acts on the system, the total angular momentum remains constant. Hence,\[I_i \omega_i = I_f \omega_f\]Substituting the known initial values, we solve for \(\omega_f\) and find that:\[\omega_f = \frac{(5.0 \times 10^{-4}) \times 4.7}{7.0 \times 10^{-4}} \approx 3.36 \; \mathrm{rad/s}\].

This end result tells us how much slower the record spins after the putty sticks due to the increased moment of inertia. Understanding this shift in angular speed is essential in applying conservation laws to solve rotational dynamics problems.