System of Particles — 8-topic bundle

When we study mechanics, we often start with a single particle – a point mass with no size, just mass. This is great for simple situations, but real-world objects are rarely just points! That's where systems of particles and rigid bodies come in.

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The Centre of Mass (CM) is a fundamental concept in physics, especially when dealing with the motion of multiple particles or extended objects. Instead of calculating forces and movements for every tiny piece of an object, we can often simplify things by considering only the CM.

When we talk about a "system of particles," we mean a collection of objects (which we can treat as tiny points of mass). These objects can interact with each other (internal forces) and with the outside world (external forces).

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Its magnitude is given by:
$|\mathbf{C}| = |\mathbf{A}| |\mathbf{B}| \sin(\theta)$
where $\theta$ is the angle between $\mathbf{A}$ and $\mathbf{B}$.

Its direction is perpendicular to both $\mathbf{A}$ and $\mathbf{B}$, determined by the right-hand rule. If you point the fingers of your right hand in the direction of $\mathbf{A}$ and curl them towards $\mathbf{B}$, your thumb points in the direction of $\mathbf{C}$.

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Torque, often represented by the Greek letter tau (τ), is a twisting force that can cause an object to rotate around an axis. It's not just about the strength of the force, but also where and how you apply it.

1. Translational Equilibrium: The sum of all external forces acting on the body must be zero. This means the body isn't accelerating linearly (it's either at rest or moving at a constant velocity).
   • Mathematically: $\Sigma \vec{F} = 0$, which breaks down into $\Sigma F_x = 0$, $\Sigma F_y = 0$, and $\Sigma F_z = 0$ (for 3D problems).

Moment of inertia, often denoted by I, is a fundamental concept when studying the rotation of objects. It helps us understand how much torque is needed to achieve a certain angular acceleration.

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Just like objects can move in a straight line, they can also rotate around an axis. We use similar concepts to describe this, but with an "angular" twist.

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