Newtonian Mechanics With Examples

Week 1: Scalars, vectors, tensors:
Review of these mathematical objects from several point of views. For example (i) vectors as ordered n-tuple and (ii) behaviour under coordinate transformation; elementary vector operations.
The aim is to create a strong conceptual foundation of the students to take on these objects at any level of sophistication. Connections will be made to other courses e.g. electrodynamics and quantum mechanics where one frequently encounters scalars, vectors (and tensors).

Week 2: Force, torque, momentum, Newton’s laws of motion:
Review of the laws of motion emphasizing on experimental evidences (e.g. linear air track to demonstrate first law of motion); examples of common forces (including impulses) in daily life; vectorial representation of forces and moments; two basic mechanical models: point particle and rigid body.
The aim here is that students should be able to draw accurate free body diagrams to analyze mechanics problems.

Week 3-4: Statics: condition of mechanical equilibrium
(i) Using force and torque balance:
The discussion will be motivated by analyzing several examples – e.g. mechanisms like gear, and (mechanical) arms, ropes under tension, engineering structures like beam, truss, massive ropes/flexible cables/ suspension bridges.

(ii) Using principle of virtual work:
Constrained motion; degrees of freedom; generalized coordinates; explanation of virtual work principle through example problems.

(iii) Classification of equilibrium using potential energy diagram: Identifying whether a mechanical equilibrium is stable or unstable by analyzing potential energy landscape (system potential energy as a function of all possible configurations of the (conservative) system).

Week 5: Friction
The emphasis will be on understanding the nature of the friction force, and how to model it correctly at the macroscopic level (e.g. friction force can be less than mu*normal force where mu is the coefficient of friction, mu can be greater than 1); The discussion will be guided by several examples with practical applications, e.g. block on an inclined plane with friction and example of practical applications; effect of drag force on real-life projectile motion, connection will be made to Stokes’s law experiment to measure viscosity of a fluid (a standard B Tech 1st Physics experiment) etc.

Week 6: Work-energy theorem and conservation laws for energy and momentum
Work-energy theorem; conservation of energy and momentum; application of energy and momentum conservation laws to solve various collisions and scattering problems – e.g. collision in 2D, drag force on a body from collision picture; application of conservation laws to rocket motion.

Week 7-8: Translation and rotation of rigid bodies
(i) Review of basic concepts: angular velocity, angular momentum, torque, conservation of angular momentum, rotational kinetic energy.
The aim is to highlight that finite angle rotation is not a vector, angular velocity and angular momentum are not parallel in general (unlike the case for velocity and momentum); to explain under what condition torque = r X F holds for an extended object; explain the relation between torque and angular momentum;

(ii) Center of mass (CM); moment of inertia (MI); Principal axes of rotation
The aim is to emphasize on the tensor nature of MI; computation of CM, MI through examples with focus on composite objects; finding principal axes of rotation of an object.

(iii) Rigid body translation and rotation through examples
Fixed axis rotation; combination of translation and fixed axis rotation; variable axis rotation and translation.
The emphasis will be on solving several example problems (e.g. rolling motion, collision where rotation occur, mechanisms with rotating parts, example of precision of axis etc.) to explain the underlying theory.

1. Introduction to Classical Mechanics with Problems and Solutions, by David Morin, Cambridge University, 2007.
2. An Introduction to Mechanics, by Daniel Kleppner and Robert Kolenkow, 2nd Ed, Cambridge University Press, 2014.
3. Engineering Mechanics 7th Ed. Vol 1 Statics and vol 2 Dynamics, by J L Meriam and L G Kraige, John Wiley and Sons, 2006.
4. Matter and Interactions 4th Ed., Ruth Chabay and Bruce Sherwood, John Wiley and Sons, 2015.
5. The Art of Insight in Science and Engineering, by Sanjoy Mahajan, MIT Press, 2014.