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Classical Mechanics 1 Classical mechanics 1 Thisdocumentis based on Chapter 1 of the Foundations of Physics for Chemists, by G.A.D. Ritchie and D.S. Sivia, published by the Oxford University Press (2000) in the Oxford Chemistry Primers series (vol. 93). 1.1 Introduction The Oxford English Dictionary describes Physics as ‘the scientific study of the properties and interactions of matter and energy’. An important aspect of this study is a desire to understand a wide range of disparate phenomena in terms Newton (1642–1727) became Lucasian Pro- of a few basic rules or ‘laws of nature’. One of the first people to achieve a fessor at Cambridge at the age of 27, and major success in this endeavour was Sir Isaac Newton, who was able to explain stayed at the university for 30 years before why apples fell to the ground and how planets orbited the sun within a single becoming Master of the Royal Mint. While his contributions to Mathematics and Physics theory of the universal law of gravitation. We begin our discussion of Physics, were far-reaching, his greatest interests were therefore, with Newton’s work on forces and motion; this topics is usually alchemy and the occult. called classical or Newtonian mechanics. 1.2 Newton's laws of motion 1.2.1 The first law A body remains at rest, or moves at a constant speed in a straight line, unless acted upon by a force. The first law provides a definition of force: that which causes an object to change its speed or direction of motion. In lay terms, a push or a pull. It’s not too difficult to imagine that a force exerted along the line of motion will result in a change of speed, whereas a correspondingnudge in a perpendic- ular direction will alter the course of travel. This dependence on the orientation of the force tells us that it is a vector quantity, being specified by both a mag- nitude and a direction; by contrast, mass is a scalar quantity, having only the former attribute of ‘size’ (or how heavy or light something is). The statement of the first law can be made more succinct by rephrasing it in terms of a uni- form velocity, where velocity is a vector pertaining to speed (a scalar) in a given direction. Explicitly, 10 metres per second (m/s or ms−1) is a speed but 10ms−1 due east is a velocity. We will not go into a detailed discussion of vectors, or any other topic in applied mathematics, in this text (referring the reader instead to volumes 77 and 82, by Sivia and Rawlings, in the Oxford Chemistry Primers series), but will try to give a brief reminder at opportune times. 2 Classical mechanics Magnitudes, directions and basic vector algebra A vector, F say, is represented graphically by an arrow where the length of the line corresponds to its magnitude, |F|, and the pointed-end indicates the direction. Alge- braically, vector quantities are often differentiated from scalar ones through the use of a bold font. If the vector lies in the x–y plane (so that it can easily be drawn on a piece of paper), then it can be decomposed, or resolved, into two components that are perpendicular to each other. Using the x (or horizontal) and y (or vertical) direc- tions as our reference, we can express F as (Fx , Fy) where Fx and Fy are scalars; specifically, if F points at an angle θ (anti-clockwise) with respect to the x-axis, then elementary trigonometry shows that Fx = |F| cos θ and Fy = |F| sin θ . Using the identity sin2θ + cos2θ ≡ 1, or Pythagoras’ theorem, it follows that the square of the magnitude of F is given by 2 2 2 |F| = Fx + Fy . A=B ⇔ Ax =Bx , Ay =By , ... Two vectors are said to be equal if they have the same magnitude and direction; this is equivalent to requiring each of the corresponding components to be equal. Vectors are added together graphically by sequentially placing the flat-end of one arrow representation at the location of another’s apex, so that the sum is the resultant displacement from start to finish; algebraically, this amounts to separately adding- up the various components of the contributing vectors. Subtraction is just a special case of addition, with the negative vector having the same magnitude but opposite direction as its positive counterpart. Algebraically, then, C = A ± B ⇒ Cx = Ax ± Bx , Cy = Ay ± By , ... The multiplication and division of a vector by a scalar is straightforward, resulting in only a change of magnitude but not direction (to within a sign reversal, if the scalar is negative). In terms of components, C = µA ⇒ Cx = µAx , Cy = µAy , ... The product of two vector is more involved, and we defer its discussion to appropriate points in this chapter. Division by a vector is not defined, however, and this operation must never appear in our calculations. Newton’s first law of motion can be paraphrased by saying that ‘a force is required to change an object’s velocity’. A stationary body is a special case when the magnitude of the velocity, or speed, happens to be zero. Indeed, its study forms a separate branch of mechanics called statics (as opposed to dynamics); it plays a central role in working out the strength of the materials required to prevent a bridge from collapsing, or the frictional roughness of a surfaced required to stop a ladder from slipping, and so on. According to the first law, of course, the net force on an object must be zero if it is to remain stationary; this forms the basis of the topic of statics. 1.2 Newton’s laws of motion 3 1.2.2 The second law The force acting on a body is directly proportional to its rate of change of momentum. The second law is a quantitative statement about the strength, and influence, of a force. To appreciate it, however, we must first understand what is meant by momentum. In everyday parlance, we sometimes talk about a movement developing a momentum of its own; the implication being that it is a trend that is becoming increasingly difficult to argue with or stop. A simple physical picture that Exercise 1.1 Given that the quantum mechan- conjures-up the same sense is that of a car, or a truck, careering down a hill ical wavelength, λ , associated with a parti- due to brake failure. It’s not hard to imagine that the effort required to control cle having a momentum of magnitude p is the run-away vehicle will be greater the faster it’s going or the heavier it is; λ = h/p, where h is Planck’s constant, what is the speed of a neutron used in diffraction consequently, the definition of the momentum of an object as the product of experiments to study structure on an atomic its mass and velocity does not seem unreasonable. Like the velocity v, and in length scale (i.e. λ ≈ 0.1 nm)? contrast to the mass m, the momentum p is also a vector: p = mv , (1.1) so that the direction of the (linear) momentum is along the (instantaneous) line of motion. Returning to Newton’s second law of motion, we can now state this mathe- matically as d F (mv) , (1.2) ∝ dt where F is the applied force and d/d t is the differential operator meaning d dv dm ‘the rate of change, with respect to time (t), of’. If the mass of the object re- (mv)= m + v dt dt dt mains constant during the motion, so that dm/dt =0, then eqn (1.2) simplifies slightly to the more familiar form F = ma , (1.3) where dv dvx dvy a = , (1.4) ax = , ay = , ... dt dt dt the rate of change of velocity with time, is the acceleration. We should note that the proportionality of eqn (1.2) has been replaced by an equality in going to eqn (1.3) by assuming that the force, mass and velocity are given in suit- Exercise 1.2 Is an electron moving in a cir- able (SI) units: newtons (N), kilograms (kg) and metres per second (ms−1) cular orbit at a constant speed accelerating? respectively. Early on in our school life, we usually learn Newton’s second law of motion as ‘force = mass acceleration’. The discussion above shows that while this is generally true, we× must remember that force and acceleration are vectors. As such, eqn (1.3) yields separate relationships for each of its components: dv dv F = m x , F = m y , ... (1.5) x dt y dt We should also bear in mind the proviso of an invariant mass, and add a term v dm/dt to the right-hand side of eqn (1.3) if dm/dt =0 (as for the case of a rocket where fuel burning is significant). 6 4 Classical mechanics Newton’s second law of motion tells us that, for the same applied force, the magnitude of the acceleration experienced by a body is inversely proportional to its mass. That’s why motorcycleswith modestly powerful engines can easily overtake cars, or why racing cars are made from the lightest (but structurally strong) materials available. Incidentally, the change in the momentum of an object is called the impulse and, by integration of eqn (1.2), is readily shown to be equal to Fdt : R t2 t2 v1 = v(t1) Fdt = [mv ] = m(v2 v1) , (1.6) t1 − tZ1 where we have assumed that the mass does not change in the writing the term on the far right-hand side.
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