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Introduction

The circular and rotational motion is one of the most important phenomena in physics and engineering, as it forms a fundamental basis for understanding many vital phenomena and applications in our daily and industrial lives. The significance of these motions lies in their ability to elucidate the movement of bodies under various conditions and their effects, whether in outer space or on the surface of the Earth.

In this report, we will delve into circular and rotational motion in detail, starting from analyzing the concepts of angular velocity and angular acceleration, to understanding Newton's laws that govern these motions precisely. We will also provide practical examples and real-life applications of these motions, such as the movement of planets in the solar system and their effects on their orbits and other celestial bodies.

We will also focus on the engineering and technological applications of circular and rotational motion, such as designing engines, machinery, controlling robots, and spacecraft. Additionally, we will discuss its everyday applications in fields such as gaming industry and electronics, where rotational motion plays a crucial role in the design and functionality of these devices.
Rotation
Key Ideas

To describe the rotation of a rigid body about a fixed axis, called the rotation axis, we assume a reference line is fixed in the body, perpendicular to that axis and rotating with the body. We measure the angular position θ of this line relative to a fixed direction. When θ is measured in radians,
θ =s/r (radian measure)
where s is the arc length of a circular path of radius r and angle θ
• Radian measure is related to angle measure in revolutions and degrees by
1rev =360°=2π rad
A body that rotates about a rotation axis, changing its angu- lar position from θ₁ to θ_2, undergoes an angular displacement
Δθ =θ_2-θ_1
Where Δθ is positive for counterclockwise rotation and nega- tive for clockwise rotation.
◆ If a body rotates through an angular displacement Ao in a time interval At, its average angular velocity ω_avg is
ω_avg=∆θ/∆t
The (instaneous) angular velocity ω of the body is
ω=dθ/dt
Both ω_avg and ω Both Wavg and ware vectors, with directions given by a right-hand rule. They are positive for counterclockwise rota- tion and negative for clockwise rotation. The magnitude of the body's angular velocity is the angular speed.
• If the angular velocity of a body changes from ω_1 to ω_2 in a time interval ∆t=t_2-t_1, the average angular acceleration α_avgof the body is
α_avg=ω_(2- ω_1 )/(t_2-t_1 )=∆ω/∆t
The (instantaneous) angular acceleration α of the body is
α=dω/dt
Both α_avg and α are vectors