In this blog post, I explore the work-energy principle as it applies to rigid body motion, distinguishing it from its application to particles. By focusing on the work done by external forces and moments, I show how translational and rotational kinetic energies combine to describe the total energy change in two-dimensional motion. I also explain the computation of work for constant and variable forces, as well as gravity's role in energy transfer.
In this blog post, I showcase how I built a responsive visualization tool for mermaid.js using Bootstrap and JavaScript. The app allows users to input mermaid.js code in a textarea and instantly see the rendered diagram in a preview area. I explain the HTML structure, including Bootstrap for layout, custom CSS for styling, and a JavaScript script for real-time diagram rendering. By integrating mermaid.js effectively, this app provides an interactive and professional tool for testing visualizations before embedding them in web pages.
In this blog post, I explore the calculation of kinetic energy in systems involving both translational and rotational dynamics. I examine scenarios such as a wheel rolling without slipping, an unbalanced wheel, and a rotating bar, discussing the interplay between translational motion, rotational motion, and moment of inertia. Using detailed derivations, I demonstrate how to account for the effects of offset mass centers and different reference points in the analysis.
I recently integrated mermaid.js into my work to create advanced visualizations for technical concepts. mermaid.js allows me to transform plain text into professional diagrams, such as Gantt charts for project management, sequence diagrams for process flows, and entity-relationship diagrams for system modeling. Its intuitive syntax and customization options make it a reliable tool for illustrating complex ideas effectively. By combining this with customized themes, I can ensure diagrams meet precise aesthetic and functional requirements.
In this blog post, I explore the kinetic energy of a rigid body in planar motion, deriving the equation for total energy as the sum of translational and rotational components. Using a systematic approach, I relate the velocity of any point in the body to the center of mass and angular velocity. The derivation simplifies to the well-known result that the translational kinetic energy, proportional to the square of the center of mass velocity, and the rotational energy, linked to angular velocity and the moment of inertia about the center of mass. This analysis is essential for understanding rigid body dynamics in physics and engineering.
In this blog post, I explore the equivalence principle, a core concept in general relativity. Using the classic thought experiment of an observer in an elevator, I examine how uniform acceleration becomes locally indistinguishable from a uniform gravitational field. This simple premise leads to the revolutionary conclusion that gravity must affect the trajectory of light, transforming our understanding of its path through space. The discussion then addresses the limitations of this principle on a global scale, introducing tidal forces as the true signature of a gravitational field and linking these physical phenomena to the mathematical concept of spacetime curvature.
In this blog post, I explains Euler's second law for continuous bodies, the significance of mass moments of inertia, and products of inertia. Planar rigid body kinetics examines the motion of rigid bodies in two dimensions, combining kinematics with Newton's laws of motion to analyze forces, moments, and their effects on linear and angular motion. I also detail how to compute the mass moment of inertia for a homogeneous cylinder, emphasizing practical applications. By presenting these principles, I aim to provide a clear framework for understanding problems involving linear and angular acceleration, dynamic equilibrium, and energy methods in planar motion.
In this blog post, I explore the relationship between relative and absolute velocities in the context of rotating reference frames. Starting from the position vector of a point relative to two frames, I derive the velocity equation by carefully applying differentiation rules, accounting for both translational and rotational motion. The angular velocity term introduces a cross product, highlighting the coupling between frame rotation and position vectors. This treatment is particularly relevant for planar motion, where the angular velocity vector simplifies to a scalar. I aim to clarify how these concepts combine into a single, practical velocity expression used in mechanics and dynamics.
In this post, I explore the relationship between absolute and relative acceleration using a rigorous mathematical approach. By differentiating velocity in non-inertial reference frames, I derive key components such as tangential, normal, and Coriolis accelerations. These terms highlight how rotational dynamics influence motion when frames of reference are in relative motion. I explain each term clearly, providing context for its physical interpretation. My analysis assumes planar motion, making it relevant for applications in two-dimensional dynamics.
In this blog post, I explore the relationship between relative and absolute velocities in the context of rotating reference frames. Starting from the position vector of a point relative to two frames, I derive the velocity equation by carefully applying differentiation rules, accounting for both translational and rotational motion. The angular velocity term introduces a cross product, highlighting the coupling between frame rotation and position vectors. This treatment is particularly relevant for planar motion, where the angular velocity vector simplifies to a scalar. I aim to clarify how these concepts combine into a single, practical velocity expression used in mechanics and dynamics.
In this blog post, I explore the intricacies of reference frame changes in planar kinematics, focusing on how position, velocity, and acceleration vectors transform between fixed and moving frames. By systematically linking multiple frames and utilizing trigonometric relationships, I derive the fundamental Derivative Formula that facilitates the transition of vector derivatives from a moving frame to a fixed frame.
In this blog post, I explore the dynamics of a wheel rolling without slipping on a curved surface, analyzing velocity and acceleration using normal and tangential coordinates. Starting from the relative velocity equation, I derive expressions for the center of the wheel's motion and its contact point with the surface, incorporating the effects of angular velocity, angular acceleration, and curvature radius. These results reveal how tangential and normal components contribute to the wheel's behavior, providing a clear mathematical framework to understand rolling motion.
In this blog post, I examine a wheel rolling without slipping on a fixed horizontal surface. By identifying the instantaneous center of rotation and applying the relative velocity framework, I derive expressions for both the velocity and acceleration of the wheel's geometric center. The analysis demonstrates that the center's acceleration is directed exclusively along the horizontal axis, while the point of contact with the ground experiences a centripetal acceleration. These insights enhance our comprehension of rotational kinematics and provide a solid basis for advanced applications in mechanical engineering and related fields.
In this blog post, I explore the kinematics and acceleration of a rigid body with emphasis on angular motion. By examining the relationship between angular velocity, angular acceleration, and the position vector between two points on the body, I derive the expressions for absolute velocity and acceleration. The discussion includes the decomposition of acceleration into its tangential and normal components, highlighting their dependence on angular variables.
In this blog post, I explore the kinematics and acceleration of a rigid body with emphasis on angular motion. By examining the relationship between angular velocity, angular acceleration, and the position vector between two points on the body, I derive the expressions for absolute velocity and acceleration. The discussion includes the decomposition of acceleration into its tangential and normal components, highlighting their dependence on angular variables.