91Ó°ÊÓ

Angular displacement is the measure of the angle through which a point or line has been rotated in a specified sense about a specified axis. Think of it like the distance a runner travels on a track; similarly, angular displacement tells us how far an angular object, like our solid cylinder from the exercise, has rotated. The unit for angular displacement is the radian (rad). In our example, the data plotted against time squared showed a straight line, indicating a constant angular acceleration, and this behavior can be described with a formula: \(\theta - \theta_{0} = \frac{\alpha}{2} t^{2}\). Here \(\theta_{0}\) is the initial angular displacement, \(\theta\) is the final angular displacement, \(\alpha\) is the angular acceleration, and \(t\) is the time.
Using the slope from a \(\theta-\theta_{0}\) versus \(t^2\) plot gives a direct relation to find the angular acceleration, which is a fundamental aspect to compute the moment of inertia.

Angular acceleration is the rate of change of angular velocity over time, defined in radians per second squared (\(\mathrm{rad/s^{2}}\)). In our cylinder scenario, once the constant force was applied, it caused the cylinder to accelerate rotationally around its stationary axle. The slope of the line on our graph is essentially the angular acceleration doubled. Hence, angular acceleration can be extracted by doubling the slope: \(\alpha = 2 * slope\).
It's pivotal for our calculations because it connects the linear force applied to the cylinder to its rotational motion. Since the force produces a torque (\(\tau\)), which in turn creates angular acceleration (\(\alpha\)), we can employ \(\alpha\) to relate this torque back to the moment of inertia (\(I\)).

Newton's second law for rotation is the rotational counterpart to the more familiar version used for linear motion. The linear form \(F = ma\) states that force is equal to mass times acceleration. Similarly, for rotation, it tells us that the net torque (\(\tau\)) acting on a body is equal to the product of its moment of inertia (\(I\)) and its angular acceleration (\(\alpha\)): \(\tau = I \cdot \alpha\).
In our cylinder problem, we applied this very principle. The horizontal force generated a torque due to the force being applied at a distance (\(r\)) from the axis. The equation links the linear dynamics of force to the rotational dynamics of torque and angular acceleration, leading us to find the moment of inertia, a property that quantifies an object's resistance to changes in its angular velocity.

Torque is essentially rotational force. It is the measure of the tendency of a force to rotate an object about an axis or pivot. You need to know two things to calculate it: the magnitude of the force applied and the distance from the axis at which it's applied, which in our case is the radius \(r\) of the cylinder. The torque is calculated by the product of the force and the radius or lever arm: \(\tau = F \cdot r\).
Our cylinder was subjected to a constant tangential force, which created a constant torque. When you apply torque to an object, it accelerates proportionally to that torque in accordance with Newton's second law for rotation. The bigger the torque, the greater the object's angular acceleration, provided its moment of inertia remains constant. By understanding this relationship, we were able to deduce the cylinder's moment of inertia.