Systems of Particles & Rotational Motion – Friendly Notes

1. Why talk about “systems of particles” at all?

Every real object has size, so treating it as a single point sometimes breaks down. The way around that is to picture the object as a system of many particles. When we do that, two ideas dominate: centre of mass and rigid-body motion :contentReference[oaicite:0]{index=0}.

2. Rigid body – the “no-squish” model

  • A rigid body keeps its shape; distances between any two of its particles stay the same. Real things do bend a bit, but for wheels, tops, beams, planets, etc., the bend is tiny, so the rigid-body shortcut works :contentReference[oaicite:1]{index=1}.

3. Two basic motions of a rigid body

  1. Pure translation – every particle shares the same velocity at each instant. A block sliding straight down an incline is the classic picture :contentReference[oaicite:2]{index=2}.
  2. Pure rotation about a fixed axis – every particle moves in a circle whose centre lies on that axis. Think ceiling fan blades or a potter’s wheel :contentReference[oaicite:3]{index=3}.

Most everyday motions (e.g., a rolling cylinder) mix translation and rotation. Rolling = translation of the centre plus rotation about the centre :contentReference[oaicite:4]{index=4}.

4. Seeing rotation up close

Pick any particle P in the body. If its perpendicular distance from the axis is r, it whirls around a circle of radius r. Points sitting exactly on the axis (r = 0) stay put :contentReference[oaicite:5]{index=5}.

5. Centre of mass (CM) – the “balance point”

Start simple with two particles on the x-axis. If their masses are m1 and m2, sitting at x1 and x2, the CM lies at
\(x_{\text{CM}}=\dfrac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}\)

Generalising to many particles (or a continuous object) works the same way – replace the summation with an integral when masses are spread out. This single point lets us track complicated bodies as if they were one particle. The rolling-cylinder sketches even label the CM as point O to emphasise that idea :contentReference[oaicite:6]{index=6}.

6. How the CM moves

All the external forces add up to give the total mass times the CM’s acceleration: \(\mathbf{F}_{\text{ext}} = M\,\mathbf{a}_{\text{CM}}\) So for translation problems you can “collapse” the body to its CM and ignore internal forces entirely.

7. Momentum of a system

Total linear momentum equals total mass times CM velocity: \(\mathbf{P}=M\,\mathbf{v}_{\text{CM}}\) If no external force acts, the CM glides at constant velocity.

8. Rotation essentials you’ll meet again

  • Angular velocity ω ties to linear speed by \(\mathbf{v}= \boldsymbol{\omega}\times\mathbf{r}\).
  • Torque τ is the rotational cousin of force: \(\boldsymbol{\tau}= \mathbf{r}\times\mathbf{F}\).
  • Angular momentum L for a rigid body about a fixed axis is \(L = I\,\omega\), where \(I\) is the moment of inertia.
  • Newton’s 2nd law for rotation: \(\tau_{\text{net}} = I\,\alpha\), with \(\alpha\) the angular acceleration.

9. Quick word on rolling without slipping

When a cylinder or wheel rolls smoothly, the point touching the ground is momentarily at rest. That distinguishes pure rolling from sliding and gives the handy link \(v_{\text{CM}} = \omega R\) (R = radius).

High-Yield Ideas for NEET

  1. Finding the centre of mass of two-particle and multi-particle systems.
  2. Difference between pure translation, pure rotation, and rolling motion.
  3. Relation between external force and CM motion \((\mathbf{F}_{\text{ext}} = M\mathbf{a}_{\text{CM}})\).
  4. Torque–angular momentum pair and the vector (cross) product.
  5. Moment of inertia and its role in rotational dynamics \( (\tau = I\alpha) \).

Keep exploring these ideas with real-world objects—a book sliding, a coin rolling, or a fan spinning—and the physics will stick!