Systems of Particles and Rotational Motion

Introduction to Extended Bodies and Rigid Bodies

• Limitations of the Particle Model: Previous studies focused on point masses (particles), which have no size. Real-world bodies have finite size (extended bodies), rendering the idealized particle model inadequate for complex motions.
• Definition of an Extended Body: An extended body is regarded as a system of particles.
• Definition of a Rigid Body:   - Ideally, a rigid body has a perfectly definite and unchanging shape.   - The distances between all pairs of constituent particles remain constant regardless of external forces.   - Practical Reality: No real body is perfectly rigid as they deform under stress. However, in many contexts (wheels, steel beams, planets), deformations are negligible and the rigid body approximation is valid.

Types of Motion: Translation, Rotation, and Combined

• Pure Translational Motion:   - Characterized by all particles of the body moving with the same velocity at any given instant.   - Example: A rectangular block sliding down an inclined plane without sidewise movement. Points $P_1$ and $P_2$ on the block have identical velocities.   - Orientation: The angle a line segment within the body makes with a fixed direction (e.g., the horizontal) remains constant ($\alpha_1 = \alpha_2 = \alpha_3$).
• Pure Rotational Motion (Fixed Axis):   - The body is constrained such that it cannot move translationally, typically by fixing it along a straight line (the axis of rotation).   - Every particle moves in a circle. Each circle lies in a plane perpendicular to the axis, with the center of the circle residing on the axis.   - Particles on the axis of rotation have a radius $r = 0$ and remain stationary.   - Examples: Ceiling fan, potter’s wheel, giant wheel, merry-go-round.
• General Rotation (Fixed Point):   - In some cases, a single point is fixed rather than a line. The axis of rotation may move around this point.   - Precession: A spinning top's axis moves around the vertical through its point of contact, sweeping out a cone.   - Oscillation: An oscillating table fan where the blades rotate about an axis that itself moves sidewise in a horizontal plane about a pivot (Point $O$).
• Combined Motion (Rolling):   - Example: A cylinder rolling down an inclined plane. It undergoes both translation (moving from top to bottom) and rotation.   - Velocity variation: Unlike pure translation, different points ($P_1, P_2, P_3, P_4$) have different velocities. For a cylinder rolling without slipping, the velocity of the point of contact ($P_3$) with the ground is zero at any instant.

The Centre of Mass (CM)

• Definition for Two Particles: For particles of masses $m_1$ and $m_2$ at distances $x_1$ and $x_2$ from an origin $O$ on the x-axis, the CM is at distance $X$:   - $X = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$   - If masses are equal ($m_1 = m_2$), $X = \frac{x_1 + x_2}{2}$; the CM is exactly midway.
• Definition for $n$ Particles (Linear):   - $X = \frac{\sum m_i x_i}{\sum m_i} = \frac{\sum m_i x_i}{M}$, where $M$ is the total mass.
• Definition for Distributions in Space:   - The coordinates $(X, Y, Z)$ of the CM are defined by:     - $X = \frac{1}{M}\sum m_i x_i$     - $Y = \frac{1}{M}\sum m_i y_i$     - $Z = \frac{1}{M}\sum m_i z_i$
• Vector Representation:   - Letting $\mathbf{r}_i$ be the position vector of the $i^{th}$ particle and $\mathbf{R}$ be the position vector of the CM:   - $\mathbf{R} = \frac{\sum m_i \mathbf{r}_i}{M}$   - If the origin is the CM, then $\sum m_i \mathbf{r}_i = 0$.
• Continuous Mass Distributions: For rigid bodies with closely packed particles, the summation becomes an integral:   - $X = \frac{1}{M} \int x \,dm$, $Y = \frac{1}{M} \int y \,dm$, $Z = \frac{1}{M} \int z \,dm$   - Vector form: $\mathbf{R} = \frac{1}{M} \int \mathbf{r} \,dm$
• Symmetry and CM:   - For homogeneous bodies (uniform mass density) of regular shapes, the CM coincides with the geometric center.   - Examples include thin rods, rings, discs, spheres, and rectangular/circular blocks.   - Reflection Symmetry: For every mass element $dm$ at $(x, y, z)$, there exists an equal mass element at $(-x, -y, -z)$, causing the integrals for CM coordinates to sum to zero relative to the geometric center.

Motion of the Centre of Mass and Linear Momentum

• Velocity of the Centre of Mass (V):   - Differentiating the CM position vector with respect to time ($t$):   - $M\mathbf{V} = m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2 + ... + m_n \mathbf{v}_n$
• Acceleration of the Centre of Mass (A):   - Differentiating velocity with respect to time:   - $M\mathbf{A} = m_1 \mathbf{a}_1 + m_2 \mathbf{a}_2 + ... + m_n \mathbf{a}_n$
• Newton's Second Law for a System of Particles:   - From $F = ma$, the equation becomes $M\mathbf{A} = \mathbf{F}_1 + \mathbf{F}_2 + ... + \mathbf{F}_n$.   - Internal vs. External Forces: Summing all forces, the internal forces (action-reaction pairs) cancel out ($\sum \mathbf{F}_{int} = 0$).   - The Principle Equation: $M\mathbf{A} = \mathbf{F}_{ext}$   - Interpretation: The CM moves as if all the system's mass was concentrated at that point and all external forces were applied directly to it.
• Linear Momentum of the System (P):   - $\mathbf{P} = m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2 + ... + m_n \mathbf{v}_n = M\mathbf{V}$   - The time rate of change of momentum is $\frac{d\mathbf{P}}{dt} = \mathbf{F}_{ext}$.
• Law of Conservation of Linear Momentum:   - If $\mathbf{F}_{ext} = 0$, then $\mathbf{P} = \text{Constant}$ and $\mathbf{V} = \text{Constant}$.   - The CM of a system moves uniformly in a straight line unless acted upon by an external force, regardless of internal complexities.   - Example: Projectile Explosion: The fragments of an exploding projectile continue such that their CM follows the original parabolic path determined by gravity.   - Example: Radioactive Decay: In the decay of a Radium ($Ra$) nucleus into Radon ($Rn$) and an Alpha ($\alpha$) particle, the CM follows the parent's original trajectory. In the CM frame, the products move back-to-back while the CM remains at rest.   - Example: Binary Stars: Stars move in circular orbits about their common CM. If no external force acts, the CM moves with uniform velocity.

Vector Product (Cross Product) of Two Vectors

• Definition: The vector product of $\mathbf{a}$ and $\mathbf{b}$ is a vector $\mathbf{c} = \mathbf{a} \times \mathbf{b}$.   - Magnitude: $|\mathbf{c}| = ab \sin(\theta)$, where $\theta$ is the smaller angle ($< 180^{\circ}$) between $\mathbf{a}$ and $\mathbf{b}$.   - Direction: Perpendicular to the plane of $\mathbf{a}$ and $\mathbf{b}$. Determined by the right-hand screw rule or right-hand thumb rule (fingers curl from $\mathbf{a}$ to $\mathbf{b}$, thumb points to $\mathbf{c}$).
• Properties:   - Non-commutative: $\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a})$.   - Distributive: $\mathbf{a} \times (\mathbf{b} + \mathbf{c}) = \mathbf{a} \times \mathbf{b} + \mathbf{a} \times \mathbf{c}$.   - Self-Product: $\mathbf{a} \times \mathbf{a} = 0$ (null vector) because $\sin(0^{\circ}) = 0$.   - Reflection Behavior: Under reflection ($\mathbf{x} \rightarrow -\mathbf{x}$), $\mathbf{a} \times \mathbf{b}$ does not change sign, whereas individual vectors reach the mirror image.
• Unit Vectors:   - $\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$   - $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$   - $\hat{j} \times \hat{i} = -\hat{k}$, $\hat{k} \times \hat{j} = -\hat{i}$, $\hat{i} \times \hat{k} = -\hat{j}$
• Determinant Form:   - $\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_x & a_y & a_z \\ b_x & b_y & b_z \end{vmatrix} = \hat{i}(a_y b_z - a_z b_y) - \hat{j}(a_x b_z - a_z b_x) + \hat{k}(a_x b_y - a_y b_x)$

Angular Velocity and its Relation with Linear Velocity

• Angular Displacement (\theta): The angle swept by a particle in time $\Delta t$.
• Angular Velocity (\omega): The time rate of change of angular displacement; $\omega = \frac{d\theta}{dt}$.   - Rigid Body Property: All particles of a rigid body rotating about a fixed axis have the same instantaneous angular velocity $\omega$.
• Vector Nature of \omega:   - For fixed-axis rotation, the $\boldsymbol{\omega}$ vector is directed along the axis.   - Direction is given by the right-hand screw rule: if the body rotates as shown, the screw advances in the direction of $\boldsymbol{\omega}$.
• Relation to Linear Velocity (v):   - The magnitude of linear velocity is $v = \omega r_{\perp}$, where $r_{\perp}$ is the perpendicular distance from the axis.   - Vector form: $\mathbf{v} = \boldsymbol{\omega} \times \mathbf{r}$, where $\mathbf{r}$ is the position vector from an origin on the axis.   - $\mathbf{v}$ is perpendicular to both $\boldsymbol{\omega}$ and $\mathbf{r}$, pointing along the tangent to the circle of motion.
• Angular Acceleration (\alpha): The time rate of change of angular velocity.   - $\boldsymbol{\alpha} = \frac{d\boldsymbol{\omega}}{dt}$   - For a fixed axis, $\alpha = \frac{d\omega}{dt}$.

Torque and Angular Momentum

• Torque (Moment of Force, \tau):   - The rotational analogue of force. It depends on the magnitude of the force, its point of application, and its direction.   - Definition for a particle: $\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$.   - Magnitude: $\tau = r F \sin(\theta) = r_{\perp} F = r F_{\perp}$.   - Dimensions: $[M L^2 T^{-2}]$ (Same as work/energy, but torque is a vector).   - SI Unit: Newton metre ($N\,m$).   - Torque vanishes if $F = 0$, $r = 0$, or if the line of action of the force passes through the origin ($\theta = 0^{\circ}$ or $180^{\circ}$).
• Angular Momentum (Moment of Momentum, l):   - The rotational analogue of linear momentum.   - Definition for a particle: $\mathbf{l} = \mathbf{r} \times \mathbf{p}$.   - Magnitude: $l = r p \sin(\theta) = r p_{\perp} = r_{\perp} p$.
• Relation between Torque and Angular Momentum:   - Differentiating $\mathbf{l}$ with respect to time: $\frac{d\mathbf{l}}{dt} = \frac{d}{dt}(\mathbf{r} \times \mathbf{p}) = \mathbf{v} \times \mathbf{p} + \mathbf{r} \times \frac{d\mathbf{p}}{dt}$.   - Since $\mathbf{v} \parallel \mathbf{p}$, $\mathbf{v} \times \mathbf{p} = 0$. Since $\frac{d\mathbf{p}}{dt} = \mathbf{F}$, we get:   - $\boldsymbol{\tau} = \frac{d\mathbf{l}}{dt}$
• Systems of Particles:   - Total Angular Momentum: $\mathbf{L} = \sum \mathbf{l}_i = \sum (\mathbf{r}_i \times \mathbf{p}_i)$.   - Total Torque: $\boldsymbol{\tau} = \sum (\mathbf{r}_i \times \mathbf{F}_i)$. External forces and internal forces are considered.   - Assuming internal forces act along the line joining particles (Newton’s 3rd Law variant), internal torques cancel out ($\boldsymbol{\tau}_{int} = 0$).   - Fundamental Equation: $\frac{d\mathbf{L}}{dt} = \boldsymbol{\tau}_{ext}$ (Total external torque equals the rate of change of total angular momentum).
• Conservation of Angular Momentum:   - If $\boldsymbol{\tau}_{ext} = 0$, then $\mathbf{L} = \text{Constant}$.   - This applies even if particles within the system have complex internal motions.

Equilibrium of a Rigid Body

• Mechanical Equilibrium: A body is in equilibrium if both its linear momentum and angular momentum are constant in time.   - Condition 1 (Translational): The vector sum of all external forces is zero: $\sum \mathbf{F}_i = 0$.   - Condition 2 (Rotational): The vector sum of all external torques about any point is zero: $\sum \boldsymbol{\tau}_i = 0$.
• Independence from Origin: If translational equilibrium is satisfied, rotational equilibrium is independent of the choice of origin.
• Concurrent Forces: For a single particle, equilibrium only requires the force sum to be zero; rotational considerations do not apply.
• Partial Equilibrium:   - A body can be in translational equilibrium but not rotational (e.g., a rod with equal but opposite parallel forces at different points).   - A body can be in rotational equilibrium but not translational.
• Couples:   - A pair of forces of equal magnitude but acting in opposite directions with different lines of action.   - A couple produces pure rotation without translation (e.g., turning a lid, a compass needle in a magnetic field).   - The moment of a couple ($\mathbf{AB} \times \mathbf{F}$) is independent of the choice of origin.

The Principle of Moments and Center of Gravity

• Principle of Moments (Lever):   - An ideal lever is a light rod pivoted at a fulcrum ($F$).   - Mechanical Equilibrium: $d_1 F_1 = d_2 F_2$, where $F_1$ is the load (at arm $d_1$) and $F_2$ is the effort (at arm $d_2$).   - Mechanical Advantage (M.A.) = $\frac{F_1}{F_2} = \frac{d_2}{d_1}$. If effort arm is longer, M.A. $> 1$.
• Centre of Gravity (CG):   - The point where the total gravitational torque on a body is zero.   - $\sum (\mathbf{r}_i \times m_i \mathbf{g}) = 0$.   - Relation to CM: If the gravitational field $\mathbf{g}$ is uniform across the body, the CG coincides with the CM. For very large bodies where $\mathbf{g}$ varies, they may differ.   - CG is determined experimentally by suspending a body from different points; the CG always lies on the vertical line passing through the point of suspension.

Moment of Inertia (I)

• Concept: The rotational analogue of mass. It measures a body's resistance to change in its rotational motion.
• Mathematical Definition:   - For a system of particles: $I = \sum m_i r_i^2$, where $r_i$ is the perpendicular distance of the $i^{th}$ mass element from the axis.
• Kinetic Energy of Rotation:   - $K = \frac{1}{2} I \omega^2$.
• Dependencies: Unlike mass (fixed), $I$ depends on:   - The distribution of mass around the axis.   - The orientation and position of the axis of rotation.
• Radius of Gyration (k):   - The distance at which if the entire mass of the body were concentrated, it would have the same moment of inertia.   - $I = M k^2$ or $k = \sqrt{\frac{I}{M}}$.
• Dimensions and Units: $[M L^2]$; SI Unit: $kg\,m^2$.
• Specific Moments of Inertia:   - Thin Ring (central axis): $MR^2$   - Circular Disc (central axis): $\frac{MR^2}{2}$   - Solid Cylinder (axis of cylinder): $\frac{MR^2}{2}$   - Solid Sphere (diameter): $\frac{2MR^2}{5}$   - Thin Rod (midpoint perpendicular): $\frac{ML^2}{12}$
• Practical Application: Flywheels in engines use high moments of inertia to prevent sudden changes in speed, ensuring smooth motion.

Kinematics and Dynamics of Rotational Motion

• Kinematic Equations (Uniform \alpha):   - $\omega = \omega_0 + \alpha t$   - $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$   - $\omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0)$
• Dynamics Table (Comparison):   - Displacement: $x \leftrightarrow \theta$   - Velocity: $v \leftrightarrow \omega$   - Acceleration: $a \leftrightarrow \alpha$   - Mass/Inertia: $M \leftrightarrow I$   - Force/Torque: $F \leftrightarrow \tau$   - Momentum: $p = Mv \leftrightarrow L = I\omega$   - Second Law: $F = Ma \leftrightarrow \tau = I\alpha$
• Work and Power:   - Work Done by torque: $dW = \tau d\theta$.   - Instantaneous Power: $P = \tau \omega$.
• Angular Momentum for Fixed Axis:   - Component along the axis of rotation: $L_z = I\omega$.   - For symmetric bodies, the angular momentum vector $\mathbf{L}$ points along the axis, thus $\mathbf{L} = I\boldsymbol{\omega}$.
• Conservation Experiments:   - Swivel Chair: A person pulling arms inward while rotating on a chair decreases $I$, thus increasing $\omega$ to keep $L = I\omega$ constant.   - Acrobats/Divers: Changing body configuration in mid-air allows for control over the rate of rotation using angular momentum conservation.