91Ó°ÊÓ

Angular displacement is crucial to understanding how much an object has rotated around a circle. It's typically expressed in radians, a unit based on the radius of a circle. In this exercise, the rotational movement of the circular saw blade is analyzed through 155 revolutions. Since one full revolution is equal to \(2\pi\) radians, we need to convert revolutions into radians to make calculations possible.
For instance, converting 155 revolutions to radians involves multiplying the number of revolutions by \(2\pi\): \(\theta = 155 \times 2\pi = 310\pi\) radians. This conversion gives a precise measure of the angular displacement and sets the stage for further calculations involving rotational kinematics.
Grasping angular displacement is key for further understanding how rotational motion translates into tangential or linear movements, especially when dealing with rotating objects transitioning into different forms of motion, like our saw blade piece.

Angular velocity describes how quickly an object rotates. In the context of the saw blade, it started from rest and accelerated due to a constant angular acceleration. Calculating the angular velocity at any given point requires understanding initial conditions and the effect of angular acceleration over time.
Using the equation \(\omega_f^2 = \omega_i^2 + 2\alpha\theta\), where \(\omega_i\) is the initial angular velocity (here, zero), \(\alpha\) is the angular acceleration, and \(\theta\) is the angular displacement, we solve for the final angular velocity \(\omega_f\).
This final angular velocity can be converted to tangential velocity using the relationship \(v_t = r \times \omega_f\), where \(r\) is the radius. Making these conversions connects rotational motion with linear, setting up for analyzing when the saw blade piece breaks off and becomes a projectile, with its initial horizontal velocity being this tangential velocity.

Projectile motion is all about understanding how objects move when they are launched into the air and acted upon by gravity alone. For the saw blade piece, once it breaks off, it transitions from rotational to linear motion, traveling horizontally while falling vertically towards the ground.
Key to solving projectile motion problems is separating the horizontal and vertical components of motion. The horizontal distance traveled depends on the initial horizontal speed and the time in the air. Using the equation \(h = \frac{1}{2} g t^2\), the time required for the piece to hit the floor can be calculated. Here, \(h\) is the vertical distance (0.820 m), and \(g\) is the acceleration due to gravity (9.81 m/s²).
After finding the time the piece stays in the air, we calculate its horizontal travel using \(d = v_t \times t\). This formula uses the initial horizontal velocity (equal to the tangential velocity from the saw blade) and the time to determine how far it moves horizontally before hitting the ground. Combining these calculations reveals the path traveled by the piece once it shifts from circular to straight-line motion.