Concept Card: Rotational Kinematics – Angular Displacement, Velocity & Acceleration 1. Physical Framework & Definitions: A rigid body executes pure rotational motion about a fixed axis if every particle moves in a circular path lying in a plane perpendicular to the axis, with its center situated on the axis of rotation. Angular Position ($\theta$): The … Read more
Concept Card: Motion of the Centre of Mass 1. Kinematics & Dynamics of Centre of Mass: The velocity and acceleration of the centre of mass of a system of particles of total mass $M = \Sigma m_i$ are given by the first and second time derivatives of its position vector: $$\mathbf{\vec{v}_{\text{cm}} = \frac{d\vec{r}_{\text{cm}}}{dt} = \frac{\Sigma … Read more
Concept Card: Centre of Mass of a Rigid Body & Continuous Mass Distribution 1. Physical Framework & Integral Formulations: For extended rigid bodies comprising a continuous distribution of matter, the discrete sum $\Sigma m_i \vec{r}_i$ transforms into a definite integral over infinitesimal mass elements $dm$: $$\mathbf{\vec{r}_{\text{cm}} = \frac{\int \vec{r}\,dm}{\int dm} = \frac{1}{M}\int \vec{r}\,dm}$$ where $M … Read more
Concept Card: Centre of Mass of a Two-Particle System 1. Physical Framework & Core Meaning: The centre of mass (CM) of a system of particles is the unique spatial point that moves as though all the mass of the system were concentrated at that point and all external forces were applied directly to it. Internal … Read more
Concept Card: Oblique Collisions in Two Dimensions (2D) 1. Physical Framework & Geometric Foundations: An oblique collision occurs when the initial velocities of colliding bodies are not aligned with the line of impact. To analyze any two-dimensional collision between smooth bodies, we decompose the motion into two mutually perpendicular axes: Line of Impact (Common Normal … Read more
Concept Card: Inelastic Collisions in One Dimension & Coefficient of Restitution 1. Physical Framework & Definitions: An inelastic collision is an impact event in which the total linear momentum of the isolated system is strictly conserved, but the total macroscopic mechanical kinetic energy is NOT conserved ($K_f < K_i$). The lost kinetic energy is converted … Read more
Concept Card: Elastic Collisions in One Dimension 1. Physical Framework & Definitions: A collision is an isolated event in which two or more interacting bodies exert relatively intense mutual forces over a brief time interval $\Delta t$, such that the effect of external forces during the impact can be neglected (the impulsive approximation). A collision … Read more
Concept Card: Power in Mechanics – Average, Instantaneous & Constant Power Dynamics 1. Physical Definition of Power: While mechanical work measures how much energy is transferred or transformed, power quantifies how rapidly that work is performed or energy is delivered. Power is the time rate of doing work or transferring energy. Physical Nature: Scalar quantity. … Read more
Concept Card: Motion in a Vertical Circle & Loop-the-Loop Dynamics 1. Physical Framework & Non-Uniform Circular Motion: When a particle of mass $m$ is constrained to move along a vertical circular path of radius $R$ under uniform gravity $\vec{g}$, its speed varies continuously with height. Unlike uniform horizontal circular motion, vertical circular motion is fundamentally … Read more
Concept Card: Conservative vs. Non-Conservative Forces & Mechanical Energy Conservation 1. Conservative Forces – Definitions & Criteria: A force field $\vec{F}_c$ is conservative if the work done by it in moving a particle between two points depends solely on the initial and final position coordinates, being completely independent of the path taken. Closed-Loop Criterion: The … Read more