1. MA4248 Weeks 1-3. Topics Coordinate Systems, Kinematics, Newton’s Laws, Inertial Mass, Force, Momentum, Energy, Harmonic Oscillations (Springs and Pendulums) Mechanics a drama authored by physical law whose stage is four dimensional space-time Space-time has an affine structure, and additional structure for either classical or relativistic mechanics 1

2. VECTOR SPACE Definition a set, whose elements are called vectors, together with two operations, called vector addition and scalar multiplication (by elements of R = reals) - the two operations must be related - each operation must satisfy certain properties Examples d-tuples of real numbers, real valued functions on a specified set S, set of functions on R having the form x  a cos (x+b) 2

3. AFFINE SPACE Definition a set A, whose elements are called points, together with a vector space V and an operation (called translation), that associates to every p in A and u in V an element in A (denoted by p+v), and that satisfies the following two properties: 3 Examples lines, planes, space without the Euclidean structure (dot product and derived angle and length)

4. EUCLIDEAN VECTOR SPACE Definition: A vector space together with an operation V x V  V, that associates to a pair (a,b) of vectors a, b in V a real number (called their dot product and denoted by ), that satisfies the following Definition: Length and Angle between nonzero vectors are then defined by 4 Euclidean Affine Space or Euclidean Space is an Affine Space whose Associated Vector Space has a Euclidean Vector Space structure

5. VECTOR ALGEBRA Vectors can be represented by their coordinates with respect to a choice of basis and then so can vector operations 5 If the basis orthonormal, then

6. KINEMATICS A trajectory in an affine space is a function Smooth trajectories in affine space define trajectories in the associated vector space, called velocities 6

7. KINEMATICS Example 1 Choose p, q, r be points in A and construct 7

8. KINEMATICS Smooth trajectories in affine space define trajectories in the associated vector space, called accelerations 8 Remark Here, in contrast to the situation for velocity, the numerator is the difference between two vectors

9. KINEMATICS Example 2 Harmonic Oscillation of a small body, “particle”, along a line is described by 9

10. KINEMATICS Example 3 Circular Motion of a small body is described with by orthogonal unit vectors u, v 10

11. NEWTON’s FIRST LAW The velocity of an isolated body is constant 11 Criticize the following versions of this law given by Halliday, Resnick and Walker: page 73 If no force acts on a body, then the body’s velocity cannot change; that is, the body cannot accelerate page 74 If no net force acts on a body, then the body’s velocity cannot change; that is, the body cannot accelerate

12. NEWTON’s FIRST LAW Inertial Frames are preferred coordinate systems for space-time for which Newtons laws hold 12 Given an inertial frame, we can obtain others by translating the original in space and time, by rotating the original through some angle about an axis, and by

13. NEWTON’s THIRD LAW 13 Criticize the following version of this law given by Halliday, Resnick and Walker: page 84 When two bodies interact, the forces on the bodies from each other are always equal in magnitude and opposite in direction Logically formulate this law by using Newton’s second law on page 84 The net force on a body is equal to the product of the bodies mass and the acceleration of the body

14. NEWTON’s SECOND and THIRD LAWS 14 Deal with the effects of interactions between bodies on their motion that cause them to accelerate When bodies i and j interact (only) with each other, their acceleration magnitudes satisfy is independent of the interaction and for bodies i, j, k (interacting pairwise), these ratios satisfy the equation (this is not an algebraic identity)

15. DEFINITION OF INERTIAL MASS 15 Choose a standard body and assign it a mass, for example the SI standard of 1 kilogram mass is that of the paltinum-iridium cylinder kept at the International Bureau of Weights and Measures near Paris Define the mass of any body i to be then the three body equation implies

16. DEFINITION OF FORCE 16 Define the force on a body to have magnitude ma and direction given by direction of its acceleration Then Newton’s second law can be expressed as Remark: this is a consequence of Newton’s laws, as discussed in Calkin’s book, together with the definitions of mass and force

17. NEWTON’s SECOND LAW 17 Deals with the interaction of three or more bodies or, equivalently Law is an empirical observation that says the net acceleration of any body is the sum of the acclerations that it experience from its interaction with each of the other bodies individually

18. BOUND COMBINATIONS 18 Suppose that particles 1, 2 and 3 interact. Then If particles 2 and 3 interact so that they are bound together as a single particle, then where

19. MOMENTUM 19 Suppose that particles 1 and 2 interact over time Therefore the momentum of the system, defined by is constant or invariant. This is the case for any system of particles.

20. ENERGY 20 Suppose that a particle accelerates under a force Further assume that the force is conservative, which by definition means that there is a real valued potential energy function U(x) on space such that Then energy is constant since

21. HARMONIC OSCILLATIONS Consider an object attached to a spring that moves horizontally near equilibrium Then 21

22. HARMONIC OSCILLATIONS Consider a pendulum - an object on a swinging lever. Then for small L 22

23. IN LINE COLLISIONS Consider the collision of two objects Since kinetic energy is conserved Since momentum is conserved 23

24. INLINE COLLISIONS From the two equations we derive 24

25. STATICS Compute the force that each string exerts on the body Hint: The direction of each strings force is along the string and away from the body 25