Chapter 6: Systems of Particles and Rotational Motion Study Notes

Earlier chapters focused on the motion of a single particle, represented ideally as a point mass without size.
This approach was used to apply results to bodies of finite size, treating them as particles under motion.
Real bodies have finite sizes, making it necessary to move beyond the particle model to deal with the motion of extended bodies, which are systems of particles.
Key Concept: The centre of mass of a system of particles is crucial for understanding the motion of extended bodies.
We will explore the motion of the centre of mass and use it to solve problems involving rigid bodies.
Rigid Body Definition: An ideal rigid body maintains a definite shape where distances between all pairs of particles remain constant.
No real body is perfectly rigid due to deformation under forces, but many practical situations allow us to consider bodies as rigid, such as wheels, tops, and planets.

Types of motion that a rigid body can have:
- Consider a rectangular block sliding down an inclined plane, moving as a rigid body in pure translational motion (all particles move with the same velocity).
- Contrast this with a rolling motion of a cylinder down an inclined plane where different particles do not share the same velocity, indicating a combination of translational and rotational motion.
Fixed Rotation: If a rigid body cannot translate (e.g., it is fixed in place), it can only rotate about its axis, which is termed the axis of rotation.
Examples include ceiling fans and potter’s wheels, where only rotary motion is observed.
Points along the axis of rotation remain stationary while particles away from the axis describe circular paths.

Definition: The centre of mass (CM) of a two-particle system is determined by their respective masses and distances from an origin:
- $X = \frac{m1 x1 + m2 x2}{m1 + m2}$
If both particles have equal mass, the CM lies halfway between them.
For an n-particle system along a straight line, the CM position is:
- $X = \frac{\sum{i=1}^{n} mi xi}{\sum{i=1}^{n} m_i}$
For particles in a plane with coordinates $(x1, y1), (x2, y2), (x3, y3)$ :
- $X = \frac{m1 x1 + m2 x2 + m3 x3}{m1 + m2 + m_3}$
- $Y = \frac{m1 y1 + m2 y2 + m3 y3}{m1 + m2 + m_3}$
In three-dimensional space, the CM is located using:
- $X = \frac{\sum{i=1}^{n} mi xi}{M}, \ Y = \frac{\sum{i=1}^{n} mi yi}{M}, \ Z = \frac{\sum{i=1}^{n} mi zi}{M}$ where $M = \sum{i=1}^{n} m_i$ is the total mass.
If the reference frame's origin is the CM, then $\sum mi \mathbf{r}i = 0$ .
Rods and Symmetric Bodies: The CM of uniform objects like rods, rings, and spheres coincides with their geometric centers.

From the CM definition, if $R$ is the position vector of the CM:
- $\mathbf{R} = \frac{\sum mi \mathbf{r}i}{M}$
Velocity of the CM is given by:
- $V = \frac{P}{M}$ where $P$ is the total linear momentum of the system.
If no external forces act on the system, the CM moves in a straight line with constant velocity (conservation of momentum).
From Newton's second law, the total external force acting on the CM is:
- $MA = F_{ext}$
Application Example: In cases like a projectile exploding into pieces, the CM follows the parabolic path unless external forces change this trajectory.

The linear momentum $p$ of a particle is defined as:
- $p = mv$ .
For n particles, the momentum is the vector sum:
- $P = m1v1 + m2v2 + … + mnvn$
Thus the total momentum equals total mass times the CM velocity:
- $P = MV$
Total momentum conservation: If $F_{ext} = 0$ , then total momentum $P$ remains constant.

Vector Product Definition: The vector product of vectors a and b yields vector c:
- Magnitude: $|c| = |a||b| \sin(\theta)$ where $\theta$ is the angle between a and b.
- Direction: c is perpendicular to the plane of a and b defined by the right-hand rule.
Cross Product Notation: Represented by $a \times b$ .
Properties:
- $a \times a = 0$ ,
- $a \times b = - (b \times a)$ ,
- Distributive property applies: $a \times (b+c) = a \times b + a \times c$

Angular velocity $\omega$ : Describes the rotational speed:
- $v = \omega r$ , where r is the distance from the rotation axis to the particle.
Each particle rotates about a central axis, maintaining angular velocity which is the same for the entire rigid body.
The direction of v is tangential to the circular path at any given moment.
The relationship for linear velocity remains consistent for all particles: $v = \omega r_i$ .

Torque (Moment of Force): The torque $\tau$ resulting from a force F about a point is defined:
- $\tau = \mathbf{r} \times \mathbf{F}$ .
Torque's effectiveness depends on the distance and angle of application, described by:
- $\tau = r F \sin(\theta)$
Angular Momentum (L): Defined for a particle as:
- $L = \mathbf{r} \times \mathbf{p}$ where $p$ is the momentum of the particle.
Total angular momentum for a system of n particles expressed as:
- $L = \sum{i} Li = \sum{i} ( \mathbf{r i} \times mi \mathbf{vi} )$ .
Conservation of angular momentum states that for an isolated system,
- If $\tau_{ext} = 0$ , then $L$ remains constant.

A rigid body is in mechanical equilibrium if the total forces and the total torques acting on it are both zero:
- For translational equilibrium: $\sum F = 0$ .
- For rotational equilibrium: $\sum \tau = 0$ .
These conditions can be expressed in component forms across axes (X, Y, Z).
Special cases involve body scenarios, such as levers, where moments can demonstrate equilibrium in distinct force applications.

Moment of Inertia (I): Characterizes how mass is distributed relative to an axis. Defined as:
- $I = \sum m r_{i}^{2}$ where r is the distance from an axis.
Moment of inertia plays a central role in the dynamics of rotational motion, analogous to mass in linear motion.
It varies based on the distribution of the mass relative to the respective axis and can be derived in simple cases like rings and rods.

The angular displacement $θ$ corresponds to a linear equivalent $x$ in linear motion. Key kinematic equations include:
- $ω = ω_0 + αt$ ,
- $θ = ω_0t + \frac{1}{2}αt^{2}$ ,
- $ω^2 = ω0^{2} + 2α(θ - θ0)$ .
These equations allow for analyzing rotational motion under uniform acceleration.

The relationship between torque and angular acceleration:
- $\tau = Iα$ where I is the moment of inertia and α is angular acceleration.
Power produced through torque is given by:
- $P = \tau ω$ .

Conservation laws apply where if no external torque is applied:
- $L = Iω$ .
This leads to significant applications in physics, demonstrating dynamic balance during interactions of rotational bodies.

Rigid body dynamics intertwine translational and rotational elements deeply reliant on notions of force, torque, and mass distribution.
The interconnectivity guides various practical applications, elucidating concepts crucial for engineering and physics.

External forces predominantly define the motion of the centre of mass.
Identifying motion dynamics around the CM clarifies applications in system analysis.
The vanishing of forces and torques operates independently under certain physical laws.