Study Notes on Systems of Particles and Rotational Motion

Motion of a Single Particle vs. Extended Bodies
- Earlier studies emphasized the motion of a single particle, treated as a point mass without size.
- Real bodies possess finite size, necessitating an understanding of motion beyond the particle model.
- Extended bodies, defined as systems of particles, include complexities in motion that the particle model cannot adequately address.
Importance of the Centre of Mass
- The centre of mass will be a central theme in describing the motion of extended bodies and plays a critical role in simplifying problems involving rigid bodies.
- A rigid body is an idealized model where distances between particles remain constant despite forces acting on it.
- Deformations in real bodies (wheels, tops, etc.) are often negligible, allowing them to be treated as rigid bodies in many scenarios.

Pure Translational Motion
- Example: A block sliding down an inclined plane, where all particles move together with the same velocity (Fig. 6.1).
Rolling Motion
- Example: A cylinder rolling down an inclined plane moves with translational motion as well as rotational motion (Fig. 6.2).
- Particles of the cylinder exhibit different velocities at different points, especially observed at the point of contact.
Constrained Rigid Bodies
- If a rigid body is constrained (e.g., fixed along an axis), its motion is purely rotational about that axis (Fig. 6.3).
Characteristics of Rotation
- Rotation involves each particle moving in a circle about the fixed axis, characterized by a radius and an angle of rotation.
- Fixed points along the axis of rotation remain stationary during rotation.

Definition
- For two particles with masses m1 and m2, at positions x1 and x2, the centre of mass (C) is given by the formula:
  $X = rac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$
- For n particles:
  $X = rac{ ext{Sum of }(m_ix_i)}{ ext{Total mass }(M)}$ where $M = ext{Sum of } m_i$ .
- If all particles have equal mass (e.g., m), the centre of mass is midway between them.
Generalization for Systems
- For three particles with positions (x1,y1), (x2,y2), (x3,y3):
  $X = rac{1}{M} \big(m_1 x_1 + m_2 x_2 + m_3 x_3\big)$
  $Y = rac{1}{M} \big(m_1 y_1 + m_2 y_2 + m_3 y_3\big)$
- For systems distributed in space:
  egin{pmatrix} X \ Y \ Z \ ext{where} \ X = rac{ ext{Sum of }(m_ix_i)}{M}, Y = rac{ ext{Sum of }(m_iy_i)}{M}, Z = rac{ ext{Sum of }(m_iz_i)}{M} \ (M = ext{Total mass}) \ ext{also defined as:} \ R = rac{ ext{Sum of }(m_i extbf{r}_i)}{M}
Importance in Rigid Bodies
- For hefty bodies like rods or discs, methods to determine the centre of mass include using symmetrical properties of shapes.
- The centre of mass is crucial for analyzing mechanics of rigid bodies since it allows for treating the entire body as a single point for motion analysis.

Velocity of Centre of Mass
- Using the relation:
  $V = rac{P}{M}$
- P refers to the total linear momentum of the system.
- The center of mass moves as if all mass were concentrated at that point.
Acceleration of Centre of Mass
- The acceleration of a system’s centre of mass is derived from external forces only so internal forces do not affect it:
  $MA = F_{ext}$
Illustration
- In an explosion scenario, the centre of mass follows the same trajectory despite internal motion caused by the explosion due to absence of external forces changing total momentum.

Definition
- The linear momentum (p) of a particle is given by:
  $p = mv$
- For a system of n particles:
  $P = m_1v_1 + m_2v_2 + … + m_nv_n$
Relation to Centre of Mass
- From the momentum equation and using $V = rac{P}{M}$ , we have:
  $P = MV$ .
- Differentiating gives:
  $rac{dP}{dt} = F_{ext}$
- Thus, when external forces sum to zero, momentum remains constant over time.

Definition and Properties
- The vector product (cross product) of two vectors a and b yields a third vector c such that:
- Magnitude: $|c| = |a||b| ext{sin} heta$
- Direction: determined by the right-hand rule.
- Properties include:
- Non-commutative: $a imes b <br>eq b imes a$
- Distributive over addition.

Angular vs. Linear Velocity
- Relation: $v = <br>ho imes ext{angular velocity }$
- Where $<br>ho$ is the radius from the axis of rotation.

Definitions
- Torque (τ) for a particle: $au = r imes F$
- Angular momentum (L) of a particle about an axis: $L = r imes p$
Relation Between Torque and Angular Momentum
- The change in angular momentum over time equals the total torque on a system:
  $rac{dL}{dt} = au_{ext}$

Conditions for Equilibrium
- Translational Equilibrium: The net force on the body must sum to zero:
  $ext{F}_{net} = 0$
- Rotational Equilibrium: The net torque about any axis must sum to zero:
  $au_{net} = 0$
Example of a Couple
- Two forces create rotation without translation, demonstrating the principle of moments.

Introduction
- Defined as: $I = ext{Sum of } (m_{i} r_i^2)$ where Rs are the perpendicular distances from the axis.
Significance of Moment of Inertia
- Similar to mass in linear motion, it resists changes in state of motion; varies with distribution of mass and the axis of rotation.

Kinematic Relationships
- For Constant Angular Acceleration:
- Angular Displacement Equation: $heta = heta_0 + rac{1}{2} ( ext{ω} + ext{ω}_0)t$
- Angular Velocity Equation: $ext{ω} = ext{ω}_0 + ext{α}t$

Torque and Work Done
- Work by torque leads to an increase in kinetic energy, defined similarly to linear motion. The equation is analogous to linear work:
  $ext{dW} = τdθ$

Angular Momentum Conservation
- If no external torque acts on a system of particles, angular momentum remains constant:
- $L = Iω$

Rigid Body Motion: Distance between particles remains unchanged under force.
Types of Motion: Rigid bodies can exhibit translational/rotational motions.
Angular Measurements: Defined similarly to linear with analogues like torque for force.
Equilibrium Conditions: Both translational/rotational equilibrium must hold.
Work and Energy: Similar relations for rotational as linear motion, crucial for analysis.
Conservation Laws: Angular momentum conserved in absence of external torque, underpinning dynamics.