Course Contents

VECTOR ANALYSIS PARTICLE DYNAMICS WORK ENERGY AND POWER ROTATIONAL DYNAMICS ANGULAR MOMENTUM CONSERVATION OF ENERGY SYSTEM OF PARTICLES COLLISIONS GRAVITATION SPECIAL THEORY OF RELATIVITY

Course Synopsis

VECTOR ANALYSIS Vector in 3 dimensions: Review of vector and operations, Cosines, Spherical polar Coordinates, applications. Vector derivatives and operation, Divergence and curl of a vector, and gradient of a scalar. Gradient, Divergence and Curl of a vector: Physical significance of each type: Divergence and flux of a vector field. Curl and line integral (mutual Relation). Vector identities. Divergence Theorem, Stokes Theorem: Derivation, physical importance and applications to specific cases.Converting from differential to integral forms PARTICLE DYNAMICS: Force laws, Frictional forces, Microscopic Basis of friction, conical pendulum, the rotor, the Banked curve, equation of motion (constant forces) Dynamic of Uniform: The rotors, circular motion the banked curve. Equations of Motion: Deriving kinematics equations x (y). V(t) using integrations. Constant and Non Constant forces and Special examples Time Dependent Force: Obtaining x (t), v(t) for this case using integration method. Effect of Drag Forces on Motion: Applying Newton’s Law to obtain V (t) for the case of motion with the time dependent (Integration Approach) drag (Viscous) forces, Terminal Velocity, Projectile Motion/ air resistance Non Inertial Frame and Pseudo Forces: Qualitative discussion to develop understanding, calculation of Pseudo forces For simple cases (linearly accelerated references frame). Centrifugal force as an example of Pseudo, Coriolis force WORK ENERGY AND POWER: Work done by a constant force, work done by a variable force (1-dimension) (Essentially a review of grade –XII concepts use of integration technique to calculate work done (e.g. in vibration of a spring obeying Hook’s Law). Work Done by a Variable: Obtaining general expression force (2-dimensional case ) and applying to simple cases e.g. pulling a mass at the end of fixed string against gravity. Work Energy Theorem, General Proof of Work Energy Theorem: Qualitative review of work energy theorem. Derivation using integral calculus. Basic formula and applications. Reference Frame: Energy changes with respect to observe in different inertial frame. ROTATIONAL DYNAMICS: Overview of Rotational Dynamics, Relationship between linear and angular variables, scalar and vector from kinetic energy of rotation, moment of inertia Parallel Axis theorem, Perpendicular Axis: Prove and illustrate, apply to simple cases. Determination of Moment of Inertia of Various Shapes. Rotational Dynamics of Rigid Bodies: Equations of rotational motion and effects of application of torques. Combined Rotational and Translational Motion Rolling without slipping ANGULAR MOMENTUM: Angular Velocity: Definition, conservation of angular momentum effects of Torque Relation. Stability of spinning Objects: Discussion with examples, the spinning Top, Effects of Torque on the Angular Momentum Processional Motion. CONSERVATION OF ENERGY Conservation and Non conservation of Energy Definition of either type of force & examples, work done in a closed path. One Dimensional Conservative system, Force as the gradient of Potential energy, Application to the case a spring and force of Gravity. One Dimensional Conservative system: Obtaining velocity in terms of U and E, Stable, Unstable and neutral Equilibrium, analytic solution for x (t). 2, 3 dimensional conservative System: Change in P.E for motion in 3-d force as the gradient of the potentials. Work done in 2, 3 dimensional motion. Conservation of Energy in a System of Particles: Law of conservation of total energy of an isolated system. SYSTEM OF PARTICLES: Two particles system and generalization of many particle systems, Centre of mass, its position velocity and equation of motion. Centre of Mass of Solid Objects: Calculations of centre of mass of solid objects using integral calculus. Calculating C. M of Uniform rod, Cylinder, Sphere, Momentum changes in a system of variable mass. Derivation of basic equation application to motion of a rocket (Determination of its mass as function of time) COLLISIONS Elastic collision, conservation of Momentum during Collision a) - one Dimensional (concept) b) - Two dimensional (Oblique Collisions) c)- (Mathematical Approach) Inelastic Collision, Collision in Centre of Mass Reference Frame: One and two dimensions, simple applications obtaining, velocities in c.m frame. GRAVITATION Review of basics concepts of gravitation, Gravitational Effect of a spherical Mass Distribution, Mathematical treatment. Gravitational Potential Energy: Develop using integration techniques, calculation of escape velocity. Gravitational Field & Potential: Develop the idea of field of force, Universal Gravitational Law, Motion of Planets and Kepller’s Laws (Derivation & Explanation), Motion of Satellite, Energy consideration in Planetary and Satellite Motion, Quantitative law to the Galaxy SPECIAL THEORY OF RELATIVITY Trouble with classical, Qualitative discussion of the Mechanics Inadequacy or paradoxes in classical; idea of time, length and Velocity. Postulates of Relativity: Statement and discussion The Lorentz Transformations: Derivation, Assumptions on which inverse transformation derived application of the same transformation of velocities. Consequences of Lorentz Transformations: Relativity of Time, Relativity of Length, Relativity of Mass. Relativistic Momentum: Derivation, Relativistic Energy, Derive E=mc2

Course Learning Outcomes

After the completion of the course the students will be able to 1. understand the concept of vector and their various properties. 2. describe the laws of motion and their applications in daily life. 3. understand the mathematical concept and expressions of various physical parameters used in mechanics. 4. finding rotational inertia, centre of mass 5. derivation and applications of Lorentz transformation equations.