System of Particles and Rotational Motion

Learning Objectives

• Understand the mathematical determination of the center of mass for systems of particles and rigid bodies, including derivations for linear and spatial distributions.
• Calculate angular momentum for particles and systems, and comprehend its conservation and relation to torque.
• Derive the relationship between torque and the rate of change of angular momentum, analogous to force and linear momentum.
• Define and explore the properties of the vector (cross) product, including its application in defining torque and angular momentum.
• Analyze the kinematics of rotational motion, including angular velocity and its relation to linear velocity, and angular acceleration.
• Formulate the conditions for mechanical equilibrium involving both translational and rotational components, with vector and scalar equation formulations.
• Apply Newton's laws to systems of particles, emphasizing the conservation of linear momentum in isolated systems.
• Differentiate between types of motion of rigid bodies, including pure translation, pure rotation about a fixed axis, and combined translation plus rotation such as rolling motion.
• Examine the motion of the center of mass under external forces, using the equation $MA = F_{\text{ext}}$, with examples like projectile explosion/fragments and binary systems.
• Understand the concept of moment of inertia as the rotational analogue of mass, its dependence on mass distribution and axis, and standard results for simple bodies.
• Utilize the perpendicular-axis theorem and parallel-axis theorem with applications to lamina, rod, disc, ring, and composite bodies.
• Explore the dynamics of rotation about a fixed axis, including the relation $\tau = I\alpha$, rotational kinetic energy, work-energy relation for rotation, and comparison with linear dynamics.
• Investigate rolling motion and combined rotation-translation, including rolling without slipping, the relation $v = R\omega$, kinetic energy of a rolling body, and rolling on an inclined plane.

Chapter Notes

Centre of Mass

• Definition: The center of mass (COM) of a system of particles is the point where the total mass of the system can be considered to be concentrated.
• Mathematical Determination:
  • For a system of particles, the position of the COM is given by: $X = \frac{\sum m_i x_i}{\sum m_i}, \quad Y = \frac{\sum m_i y_i}{\sum m_i}, \quad Z = \frac{\sum m_i z_i}{\sum m_i}$
  • For continuous mass distributions, integrals replace summations.
• Properties: The COM moves as if all external forces act at this point.

Angular Momentum

• Definition: Angular momentum ($L$) of a particle is defined as the vector product of the position vector ($r$) and linear momentum ($p$): $L = r \times p$
• Conservation: If the net external torque on a system is zero, the total angular momentum is conserved.

Torque and Angular Momentum Relation

• Relation: The rate of change of angular momentum of a particle is equal to the torque ($\tau$) acting on it: $\frac{dL}{dt} = \tau$
• Torque: Defined as the vector product of the position vector and force ($F$): $\tau = r \times F$

Vector Product of Two Vectors

• Definition: The vector product (cross product) of two vectors $\mathbf{a}$ and $\mathbf{b}$ is a vector $\mathbf{c}$ given by: $\mathbf{c} = \mathbf{a} \times \mathbf{b} = ab \sin \theta \mathbf{n}$ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$, and $\mathbf{n}$ is a unit vector perpendicular to the plane containing $\mathbf{a}$ and $\mathbf{b}$.

Rotational Motion Kinematics

• Angular Velocity ($\omega$): The rate of change of angular displacement.
• Relation to Linear Velocity: For a particle moving in a circle of radius $r$, the linear velocity $v$ is related to angular velocity by $v = \omega r$.

Equilibrium of Rigid Bodies

• Conditions for Equilibrium:
  • Translational Equilibrium: The vector sum of all forces is zero.
  • Rotational Equilibrium: The vector sum of all torques is zero.

Conservation of Linear Momentum

• Principle: In an isolated system, the total linear momentum remains constant if no external forces act on it.

Types of Motion of Rigid Bodies

• Pure Translation: All points of the body move with the same velocity.
• Pure Rotation: All points move in circles about a common axis.
• Combined Motion: Includes rolling motion, where both translation and rotation occur.

Motion of Centre of Mass

• Equation: $M \mathbf{A} = \mathbf{F}_{\text{ext}}$, where $M$ is the total mass and $\mathbf{A}$ is the acceleration of the COM.

Moment of Inertia

• Definition: A measure of an object's resistance to changes in its rotation rate.
• Dependence: Depends on the mass distribution relative to the axis of rotation.

Theorems of Moment of Inertia

• Parallel-Axis Theorem: $I = I_{\text{cm}} + Md^2$
• Perpendicular-Axis Theorem: For a planar body, $I_z = I_x + I_y$

Dynamics of Rotation About Fixed Axis

• Equation: $\tau = I\alpha$, where $\alpha$ is the angular acceleration.
• Rotational Kinetic Energy: $K = \frac{1}{2} I \omega^2$

Rolling Motion and Combined Rotation-Translation

• Rolling Without Slipping: $v = R\omega$
• Kinetic Energy: Combination of translational and rotational kinetic energy.