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The moment of inertia is a measure of an object's resistance to changes in its rotation rate. It is a rotational equivalent of mass for linear motion. Moment of inertia depends on the distribution of an object's mass and its axis of rotation. For example, a spinning figure skater pulls in her arms to reduce her moment of inertia, which, in turn, increases her angular velocity without applying an external torque.

Mathematically, it's represented as 'I' and calculated differently for various shapes. A solid sphere, a ring, a disk, and a rod all have different formulas for calculating their moment of inertia. A higher moment of inertia means it's harder to change the object's rotational speed. For instance, in our exercise, the object has a moment of inertia of \(2.0 \text{kg} \times \text{m}^{2}\), which signifies the rotational inertia relative to its rotation axis.

Angular acceleration is the rate of change of angular velocity over time. It corresponds to how quickly an object speeds up or slows down its rotation. The faster the angular acceleration, the quicker the change in the rotation rate. It's like how the acceleration in linear motion refers to changes in velocity, angular acceleration refers to the changes in angular velocity.

In equations, it's denoted by the Greek letter alpha (\(\alpha\)). Its units are radians per second squared (\(\text{rad/s}^2\)). A higher angular acceleration means that the net torque applied to the object is causing it to change its rotational motion more aggressively. This concept is illustrated in our solved problem, where we have an angular acceleration of \(4.0 \text{rad/s}^2\).

Torque is the measure of the force that can cause an object to rotate about an axis. Just like force causes an acceleration in linear motion, torque causes an angular acceleration in rotational motion. The torque formula is central to understanding how objects rotate and react to forces.

The formula for calculating torque (\(\tau\)) is \(\tau = I \times \alpha\). Here, 'I' is the moment of inertia and '\(\alpha\)' is the angular acceleration. This equation tells us that the torque applied to an object is directly proportional to the moment of inertia and the angular acceleration. It's as though the object's rotation responds directly to the product of its 'rotational mass' and the rate at which it's trying to change that rotation. In this example, a net torque on the object would have been calculated by multiplying its moment of inertia and the angular acceleration.

Angular velocity refers to the speed of rotation and the direction of the axis of rotation. In simpler terms, it describes how fast something spins around a certain point. Angular velocity is denoted by the symbol omega (\(\omega\)) and is measured in radians per second (\(\text{rad/s}\)).

For instance, when a figure skater pulls their arms in, their rotational speed increases, showing a higher angular velocity. It's this velocity that the angular acceleration changes - the two concepts are intimately linked. When we talk about increasing angular velocity, we often refer to a positive torque, which is exactly what is occurring in our example problem - the object's angular velocity is increasing because of the net torque applied to it.