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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 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) - each operation must satisfy certain properties - the two operations must be related 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 AFFINE SPACE Definition a set A, whose elements are called points, together with a vector space V and an operation AV A (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: - for every pair (p, q) of points in A there exists a unique element u in V such that p u q - for all p A, u, w V the following holds (p u) w p (u w) Examples lines, planes, space without the Euclidean structure (dot product and derived angle and length) 3 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 a b ), that satisfies the following a a 0 if and onlyif a 0 ab ba a (1b1 2 b 2 ) 1a b1 2a b 2 Definition: Length and Angle between | a | aa a b nonzero vectors are then defined by cos |a| |b| Euclidean Affine Space or Euclidean Space is an Affine Space whose Associated Vector Space has a Euclidean Vector Space structure 4 VECTOR ALGEBRA Vectors can be represented by their coordinates with respect to a choice of basis u u1b1 u 2 b 2 u ~ (u1 , u 2 ) b1 and then so can b2 vector operations u v ~ (u1 v1 , u 2 v 2 ) c u ~ (cu1, cu2 ) If the basis orthonormal, bi b j ij then u v u1v1 u 2 v2 5 KINEMATICS A trajectory in an affine space is a function f : [c, d ] A, Smooth trajectories in affine space define trajectories in the associated vector space, called velocities f : [c, d ] V , f ( t t ) f ( t ) f ( t ) lim t t 0 6 KINEMATICS Example 1 Choose p, q, r be points in A and construct f ( t ) r t (q p), f ( s s ) f ( s ) f ( s ) lim s s 0 ( r ( t t )( p q ) ( r t ( p q )) lim t t 0 t [c, d] t ( p q ) lim t 0 t lim t 0 (p q) p q 7 KINEMATICS Smooth trajectories in affine space define trajectories in the associated vector space, called accelerations ( t t ) f ( t ) f f( t ) lim t t 0 Remark Here, in contrast to the situation for velocity, the numerator is the difference between two vectors 8 KINEMATICS Example 2 Harmonic Oscillation of a small body, “particle”, along a line is described by f ( t ) r a cos(t b)( q p) f ( t ) a sin(t b)(q p) f( t ) a2 cos(t b)( q p) 9 KINEMATICS Example 3 Circular Motion of a small body is described with by orthogonal unit vectors u, v f (t ) r a cos(t )u a sin(t )v f (t ) a sin(t )u a cos(t )v ( f (t / 2 ) r ) f(t ) a 2 (cos(t )u sin(t )v) f (t ) 2 10 NEWTON’s FIRST LAW The velocity of an isolated body is constant 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 11 NEWTON’s FIRST LAW Inertial Frames are preferred coordinate systems for space-time for which Newtons laws hold 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 fnew ( t ) forig ( t ) u 0 ( t ) v0 fnew ( t ) forig ( t ) v0 f f ( t ) new orig ( t ) 12 NEWTON’s THIRD LAW 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 13 NEWTON’s SECOND and THIRD LAWS 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 {i , j} {i , j} is independent of the interaction ai /aj and for bodies i, j, k (interacting pairwise), these ratios satisfy the equation (this is not an algebraic identity) {i , j} {i , j} { j,k} { j,k} {k ,i} {k ,i} ai /aj aj / ak ak / ai 1 14 DEFINITION OF INERTIAL MASS 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 {i,cyl} {i ,cyl} mi (a cyl / a i ) kilograms {i , j} {i , j} then the three body equation implies a /aj i { j,cyl} { j,cyl} {i ,cyl} {i ,cyl} /a a /a a cyl j cyl i m j / mi 15 DEFINITION OF FORCE 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 F1on2 F2on1 Remark: this is a consequence of Newton’s laws, as discussed in Calkin’s book, together with the definitions of mass and force 16 NEWTON’s SECOND LAW Deals with the interaction of three or more bodies 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 a or, equivalently aj j ma Fnet Fj j 17 BOUND COMBINATIONS Suppose that particles 1, 2 and 3 interact. Then m1a1 F2on1 F3on1 m 2a 2 F3on2 F1on2 m3a 3 F1on3 F2on3 If particles 2 and 3 interact so that they are bound together as a single particle, then m 23a 23 F1on2 F1on3 F1on23 where m 23 m 2 m3 , a 23 a 2 a 3 18 MOMENTUM Suppose that particles 1 and 2 interact over time m1v1 ( t t ) m1v1 ( t ) m1a1 ( t ) t F2on1t F1on2 t m 2 a 2 ( t ) t ( m 2 v 2 ( t t ) m 2 v 2 ( t )) Therefore the momentum of the system, defined by p( t ) m1v1 ( t ) m2 v 2 ( t ) is constant or invariant. This is the case for any system of particles. 19 ENERGY Suppose that a particle accelerates under a force mv( t ) ma ( t ) F( x( t ), t ) 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 F grad U 1 Then energy E( t ) U( x ( t )) mv ( t ) v( t ) 2 is constant since ( t ) grad U( x( t )) v( t ) mv( t ) v ( t ) 0 E 20 HARMONIC OSCILLATIONS Consider an object attached to a spring that moves horizontally near equilibrium x 0 Then 2 1 F kx grad kx 2 2 2 1 1 E kx mx 2 2 x0 x x ( t ) a cos ( t ) a 2E k , k m, R 21 HARMONIC OSCILLATIONS Consider a pendulum - an object on a swinging lever. Then for small θ 2 2 Lm E gθ Lθ 2 θ L θ( t ) a cos (t ) a 2E , Lmg g , L R 22 IN LINE COLLISIONS Consider the collision of two objects m1 m2 v1 v2 v1 v2 Since momentum is conserved m1v1 m2 v 2 m1v1 m2 v2 Since kinetic energy is conserved 2 2 2 2 m1v1 m 2 v 2 m1v1 m 2 v2 23 INLINE COLLISIONS From the two equations we derive 1 1 v1 v 2 v1 m m v m v m v 2 2 2 2 1 1 1 m1 m2 2m2 v1 v1 v2 m1 m2 m1 m2 2m1 m2 m1 v2 v1 v2 m1 m2 m1 m2 24 STATICS Compute the force that each string exerts on the body mg Hint: The direction of each strings force is along the string and away from the body 25