Newtonian Mechanics

Introduction

Ancient philosophers (such as the Greeks) were interested in the natural world around them, in particular how bodies moved. While the solutions to their pondering would only come much later, they contributed immensely to our understanding of the natural world by starting the initial exploring and definitions related to motion. In this set of notes, we attempt to build up our knowledge from basic axioms. While basic knowledge is assumed (in the form of basic definitions and equations), this set of notes is intended to flow through as we build up to larger concepts, to ensure that by the end of these notes, any loopholes or gaps in understanding have been covered, and the reader is brought to greater awareness about how the methods he or she has learnt may be applied to other situations. Definitions will only be repeated if the author deems them necessary (e.g. due to presence of subtleties to take note of).

1. Statics

To understand why things move, we first need to understand why things do not move. A state in which a stationary body, if left alone, remains stationary or a body in motion remains in motion with the same parameters (See Newton's First Law) is known as equilibrium. Analysis of Equilibrium involves asking four (or more) questions:
  1. What are the forces acting on the body?
  2. Where are the forces acting towards?
  3. Where are the forces acting on the body?
  4. Is it a stable, unstable, or neutral equilibrium?
Questions 1 and 2 will doubtless be too familiar to you. The third question, related to the moment of a force, should likewise be familiar to you. The ideas involved will be explored in greater depth in later chapters, when we cover the dynamics of rigid-body motion.
Let us discuss question 4 now, by considering a pencil on a table. If the pencil were to be placed vertically on the table with its centre of gravity precisely above the contact point, it would be in a state  of unstable equilibrium, as any small perturbation of the body away from its equilibrium position results in a greater motion away from (and out of) equilibrium. If the pencil were to be placed horizontally on the table, it would be in a state of stable equilibrium, as any small perturbation of the body away from its equilibrium position (i.e. upwards) results in a motion back down towards the equilibrium position. In this case, we also consider the pencil to be in neutral equilibrium, as it can roll along the table and find another equilibrium position. Note that this example allows us to understand that it is possible for an object to be in different states of equilibrium in relation to different coordinates simultaneously -- equilibrium states are linked to so-called degrees of freedom. Note also that it is possible for an object to be in stable equilibrium along one direction, and in unstable equilibrium along another direction.
Olympiad questions, while relying on the same basic concepts of equilibrium, can still be extremely challenging, e.g. by considering a complicated system, or involving an accelerating frame of reference.
In solving questions, we recommend that you analyse when it might be necessary to impose force balance on individual components in a system, versus for the system as a whole. An extreme example may be seen in the following question:

1.1 A chain of mass \(M\) hangs between two walls, with its ends at the same height. The chain makes an angle \(\theta\) with each wall. Find the tension in the chain at the lowest point. Solve it via: (1) Considering the forces on half the chain (Solving as a system), and (2) Using the fact that the height of a hanging chain is given by \(y\left(x\right)\ \ =\ \left(\frac{1}{\alpha}\right)\cosh\left(\alpha x\right)\), and considering the vertical forces on an infinitesimal piece at the bottom (Solving per components).

Extension: derive the equation for the shape of a uniform chain suspended by its ends in a uniform gravitational field.

2. Linear Kinematics and Dynamics

An overall imbalance in the forces on an object results in a change in motion (Newton's second law relates this resultant force to the rate of change of linear momentum). In solving Olympiad problems, you should be conscious of the directions in which forces are acting, and translating these directions to mathematical expressions accordingly. A negative sign usually changes the physics profoundly, and you should additionally develop the habit of quickly checking for reasonableness as you work through a problem.

2.1 "String Theory"

In dealing with problems that have a pulley and inextensible string, it is often useful to realise that we are able to obtain an equation (of kinematic constraint) by summing up the lengths of all the components of strings, and noting that it must be a constant, thus the time derivative has to be zero. 
 

2.2 Polar Coordinates

Up till now you might have been entirely comfortable working with Cartesian coordinates. However, for certain problems (e.g. central force motion), it may sometimes be easier to convert to polar coordinates. To do this effectively, we need to comfortable with the form of Newton's law expressed in polar coordinates. We begin with the basic relationships between the two coordinate systems: