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Angular velocity is a measure of how fast an object rotates around a particular axis. It is the rate of change of angular displacement and is often represented by the Greek letter \( \omega \). If you imagine a wheel spinning, angular velocity describes how quickly it completes each circle. It is measured in radians per second (rad/s), which might sound strange but remember, one full circle creates an angle of \(2\pi\) radians.

When an object starts from rest, like in our exercise, its initial angular velocity \( \omega_0 \) is zero. Over time, as it is subjected to various forces, its angular velocity increases. It is helpful to remember:

• \( \omega = \frac{d\theta}{dt} \) – Angular velocity is the derivative of angular displacement \( \theta \) with respect to time \( t \).
• The faster the rotation speed, the higher the angular velocity.
• If angular velocity increases, angular acceleration is positive; if it decreases, angular acceleration is negative.

Torque is like a rotational force. It is what causes objects to start rotating, similar to how force causes linear motion. Imagine trying to push a door open. Torque depends on two main things: how much force you apply and how far from the pivot you apply it. The equation for torque \( M \) is given by the formula: \[ M = F \times r \sin(\theta) \]where:

• \( F \) is the force applied,
• \( r \) is the distance from the pivot point to where the force is applied,
• \( \theta \) is the angle between the force and the arm.

The moment of inertia \( I \) resists these changes in motion, much like mass resists linear acceleration. It's a property that depends on the object's mass and shape. For a disk, the moment of inertia is usually calculated as \( I = 0.5 \times m \times r^2 \). This property determines how much torque is needed to induce a given angular acceleration \( \alpha \). The relationship between torque, moment of inertia, and angular acceleration is quite simple:

• \( I \alpha = M \)
• More moment of inertia means more torque is needed to rotate the object.

Just like there are equations for linear motion, rotational motion has its set of equations. These equations help describe how an object moves through an angle over time. In our problem, understanding rotational motion is key. If you know the torque acting on an object and its moment of inertia, you can find out how its rotational speed changes.

There are several important equations:

• \( \alpha = \frac{d\omega}{dt} \) – Angular acceleration is the rate of change of angular velocity.
• \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \) – This is the rotational analog of the linear motion equation \( s = ut + \frac{1}{2}at^2 \).
• Torque plays a crucial role as it links angular acceleration and moment of inertia: \( I \alpha = M \).
• Combining these, \( \omega \) changes are computed as in: \( \omega^2 = \omega_0^2 + 2\alpha \theta \)

In our original problem, computing the angular velocity involves integrating the angular acceleration over the path of motion.