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Mechanics Part One Notes

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Classical Mechanics: Part I 2014

Contents 1 Introduction

2

2 Kinematics

2.1 Projectile motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . .

2 4 5

3 Dynamics

3.1 Newton's Laws of Motion

3.2 N1, Inertial Frames . . . .

3.3 Galileian Relativity . . . .

3.4 N2, Talking about Forces

3.5 Fundamental Forces . . .

3.6 Friction and Drag . . . . .

3.7 N3, Action and Reaction .

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8 8 8 9 11 11 12 15

1

INTRODUCTION

1

2

Introduction Classical mechanics is an ambitious theory. Its purpose is to predict the future and reconstruct the past, to determine the history of every particle in the Universe. Introduction to Mechanics, D Tong

Classical Mechanics deals with the motion of bodies, ranging from billiard balls to galaxies. The study of motion constitutes one of the most ancient areas of Physics (possibly pre-dated only by Astronomy), but it took the genius of Galileo and Newton to obtain the correct set of principles and the mathematical framework that we still identify as Classical (or indeed "Newtonian") Mechanics. The beauty of this formalism is that the motion of any body is completely determined, given the initial conditions, by a single input: the applied force. In this first chapter we will be studying the machinery laid out by Newton, and "turn the handle" several times, exploring the consequences of his famous F = ma. We will not have time to study the origin of the macroscopic forces that we see in action in our world. However it is worth pointing out now that Nature only presents us with four fundamental forces, only two of which can be meaningfully described within the classical regime: Gravity and Electromagnetism 1 . A very elegant (and immensely useful!) consequence of the laws of motion are the conservation laws for momentum, energy and angular momentum. These provide not only an alternative problem-solving route, that can be pursued when the exact force law is not known, but also a more fundamental approach to Mechanics, which has proved useful in the subsequent development of Quantum Mechanics. Indeed the modern theoretical Physicist learns to forget F = ma as soon as possible, in favour of more elegant (but more abstract) objects, called Lagrangians and Hamiltonians.2 We now know that Classical Mechanics, though immensely successful, is NOT the right theory of motion. Motion on the smallest scales follows the completely counterintuitive rules of Quantum Mechanics, while particles moving at high velocities (comparable to the speed of light) follow Relativistic Mechanics3 . In the realm of Elementary Particles, where the objects of interest are both "small" and "fast", a theory containing both quantum mechanics and relativity is needed, called Quantum Field Theory.4

2

Kinematics

We will start by discussing the dynamics of particles (also called more precisely material points). A particle in mechanics denotes a body of insignificant size (ie: which 1 The

Strong and the Weak Interactions can only be treated quantum mechanically. would argue that the generality and beauty of this formulation widely surpasses that of Newton's Laws. Indeed Lagrangians and Hamiltonians can be used to describe everything from classical pendulums to the Standard Model of Particle Physics! It is only too sad that the subject matter is deemed too mathematically challenging for the first year students, and currently properly taught in Cambridge only in Part II of the Physics or Mathematics Tripos. 3 I will have the enormous pleasure of introducing you to this topic in chapter 4 Needless to say you won't hear anymore about Quantum Field Theory this year, and you might never hear about it again unless you decide to stick with Physics until fourth year!
2I

2

KINEMATICS

3

ideally only has one dimension) and has nothing to do with an elementary particle, like an electron or a quark. Indeed anything could be approximated as a particle if we are not interested in the motion of its parts. If we are, on the other hand, interested in the rotational motion of a planet or the motion of passengers inside a car, we need to add further complications to the problem, and deal with extended (often assumed rigid 5 ) bodies, whose motion is more interesting and complicated (and will be dealt with in section). To study motion, we must start by defining a reference frame, which, unless otherwise stated, will always be taken to be Cartesian (the usual right-handed set of orthogonal axes). The position of a particle as a function of time is called its trajectory and in 3D Euclidean space is given by a vector 6 : x(t)

(1)

Velocity is defined as the time derivative of the position vector: v(t) =

dx(t) dt

(2)

Mathematical Note You might have never met the derivative of a vector before. There is nothing special about it, it is just the derivative of each component! Consider the vector A = (A1 , A2 , A3 ), its derivative is given by: dA1 dA2 dA3 dA(t)
=
, , dt dt dt dt

(3)

For future reference, all the rules that apply to differentiation of scalars apply also to vectors, in particular the chain rules works with respect to both the scalar and the cross product. Ie: dA·B = dA · B + dB · A. dt dt dt Regarding notation, it is customary to denote differentiation with respect to time (and time only!) with a dot. So v(t) =

dx(t)
˙
= x(t). dt

(4)

Since we defined the first derivative of the position vector, we can certainly define the second derivative, which we call acceleration. a(t) =

dv(t) d2 x(t)
¨
=
= x(t) dt dt2

(5)

It turns out that in mechanics acceleration has a very important role to play, while any higher derivative of the trajectory is generally useless. Let us deal with Kinematics, the study of motion, independently from its causes. In other words, we seek to answer questions of the sort: How fast will the car go? How 5 The

motion of non-rigid bodies is a very complicated business, that we not will treat this year (but you can learn about it in IB Mechanics, or by becoming and Engineer!
6 Remember vectors are denoted in bold

2

KINEMATICS

4

far will the bullet reach? To answer such questions we clearly need to know something about the motion of these objects. In particular the motion of a body is fully determined, if we know its acceleration at all times, and its velocity and position at the beginning. While it sounds reasonable that we need to know where something starts, if we want to know where it gets to, it seems slightly arrogant and unjustified to ask about it acceleration at all times! However, we will soon see that the acceleration is tightly related to the force acting on a body. You can imagine a force as knowing how hard you're pushing the car. A better way of thinking about forces would be to think about fundamental interactions, like gravity, which act on all the stuff in the Universe. Since Physicists are smart, they have long figured out the Law of Gravity, and know about the force on, for example, a falling bullet. In the next chapter we will see how knowing the force, we can calculate the acceleration (Hint: F = ma). Once you have the acceleration, all that remains to do is to integrate to find velocity and position. The initial velocity and initial position are needed to fix the integration constant. Voila', we are done with Kinematics! However before leaving this subject, I would like to talk you though a simple example, which you have certainly met in your previous studies of Physics, and show you how it all fits together within the framework I have just described.

2.1

Projectile motion

Motion under constant acceleration is the Physicist's first love. Its beauty probably lies in the fact that we can observe bodies performing this type of motion any time we want, just by dropping an object and letting it fall under gravity. In this section, however, we shall not be concerned about why gravity on the surface of the Earth makes all bodies accelerate at g, but rather take this as granted and focus on the kinematics, i.e. on the integration. The equation of motion for a body under constant acceleration a is simply
¨
x = a. (6) To get the velocity we integrate the constant acceleration with respect to time
¨
˙
x = [x]v0 =

˙
adt = x − v0 t,

(7)

assuming we start measuring time from t = 0. Similarly to obtain the displacement we integrate once more: s − s0 =

(at + v0 )dt =

1 2 at + v0 t. 2

(8)

Note that the above equations are vector equations, and contain motion with a constant speed as a special case (a = 0). The example of projectile motion will illuminate the concept. In projectile motion, a body (the projectile) is falling under gravity (a = −g) in the vertical direction, but is also travelling in the horizontal direction. If we call v2 the initial velocity in the y-direction and v1 the initial velocity in the x-direction, we can write the solution to the projectile motion along the two axes using equation 8 by using the x and y components of the vectors involved: x = v1 t

(9)

2

KINEMATICS

5

and

1 y = − gt2 + v2 t. (10) 2 These two equations describe a parabola, if you don't believe me try eliminating time. The vertex of the parabola is the point where the projectile reaches maximum height (to find it try maximizing equation 10 with respect to time, you should 2 v2 get t = v1 , y = 2g , x = v1gv2 ). g The range of the projectile is defined as the distance travelled horizontally when the projectile hits the ground. To calculate this we need to solve for time the equation y = 07 , and to substitute this time into 9. The answer is x = v1gv2 . There is a cute trick to get this result knowing the previous one about the maximum height: since the parabola is symmetric (ie: the ascending part is a mirror image on the descending part) the x-coordinate of the maximum height point is exactly half of the range. And indeed, if you look at the formulae we found, it is!
Now a word of warning: Symmetries are everywhere in Physics. In the above case using symmetry might have saved you a couple of lines of algebra, but sometimes it saves you pages of work and might greatly simplify problems that otherwise appear intractable8 . In short I feel it is a good time to emphasize the concept: in Physics use symmetry whenever possible!

Figure 1: A sketch of projectile motion. The total velocity (in red) is split in the x and y components. While the x component remains constant, the y component follows the law of uniform acceleration. The range is denoted by R

2.2

Simple Harmonic Motion

A Physicist's most faithful friend. It is not an exaggeration to say that simple harmonic motion is probably the most useful tool in Physics, because of its simplicity and its 7 One of the solutions we get is t = 0, which is hardly surprising, and actually acts as a good reality check: before being shot the projectile is where we expect it to be!
8 I remember spending about half an hours solving a supervision question which I thought was quite hard. Once in the supervision, my supervisor said: "Oh this is obvious, by the symmetry of the system!" I cannot deny I felt quite cheated, and it took me a while to understand the beauty of his reasoning and the power of the method.

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