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PHY 5200 Mechanical Phenomena Newton’s Laws of Motion Claude A Pruneau Physics and Astronomy Wayne State University Contents • • • • Classical Mechanics Space and Time - Notations Mass and Force Newton’s First and Second Laws: Inertial Frames • The third law and Conservation of Momentum • Applications of Newton’s Second Law in Cartesian Coordinates • Applications of Newton’s Second Law in Polar Coordinates 5/22/2017 C. Pruneau, W.S.U. 2 Classical Mechanics • Mechanics is the study of how/why things move. • The Greeks of the antiquity were the first to think about this - but their notions were seriously flawed. • Early Modern development of Mechanics are due to Galileo (1564-1642) and Newton (1642-1727) • Alternative formulations of mechanics due to Lagrange (1736-1813) and Hamilton (1805-1865) – The Newtonian, Lagrangian, and Hamiltonian formulations of Mechanics are equivalent. – These three formulations are regarded as “Classical Mechanics”. 5/22/2017 C. Pruneau, W.S.U. 3 20th Century Mechanics • But what about the 20th Century ? • Realization of inconsistencies between Maxwell’s Equations and Galilean Invariance lead to Special Relativity. – Discussed in PHY 6200 • Discovery of quantized phenomena lead to Quantum Mechanics. – Discussed in PHY 6300/PHY 6800. 5/22/2017 C. Pruneau, W.S.U. 4 Preamble to Newton’s Laws • Newton’s Laws required four fundamental concepts/notions – – – – 5/22/2017 Space Time Mass Force C. Pruneau, W.S.U. 5 Space (According to Newton) • We live in a 3 dimensional (3D) world. • Each point, P, of this 3D world can be labeled by a position vector, r, which specifies the distance, and the direction, relative to some arbitrary origin, O. and reference frame S. • Introduce the notion of coordinates. E.g. (x,y,z). z-axis P r z O x y-axis y x-axis 5/22/2017 C. Pruneau, W.S.U. 6 Coordinates and Vectors • Possibly many types of coordinate systems – Cartesian Coordinates – Polar, Cylindrical, Spherical Coordinates – Etc. • Various vector representations and notations will be used. • Cartesian Coordinates, Unit vectors, Representations x̂, ŷ, and ẑ r xx̂ yŷ zẑ iˆ, ĵ, and k̂ r x î yĵ zk̂ 3 ê1 , ê2 and ê3 r x, y, z 5/22/2017 r r1ê1 r2 ê2 r3ê 3 ri êi r r1 , r2 , r3 C. Pruneau, W.S.U. i 1 7 Vector Properties • • • • • • Addition, Subtraction Multiplication by a scalar Scalar Product (also called dot product) Magnitude (norm) of a vector Vector Product Differentiation of Vectors 5/22/2017 C. Pruneau, W.S.U. 8 Vector Addition & Subtraction • Given two vectors r r1 , r2 , r3 • s s1 , s2 , s3 Addition (Sum) r r s r1 s1 , r2 s2 , r3 s3 • r rs Subtraction (Difference) s r r s r1 s1 , r2 s2 , r3 s3 r • Null Vector r r 0 r r r1 r1 , r2 r2 , r3 r3 0, 0, 0 r s r rs 5/22/2017 C. Pruneau, W.S.U. 9 Multiplication by a Scalar Given a vector r r1 , r2 , r3 r Multiplication by a scalar, c r cr cr1 , cr2 , cr3 r cr Example: r r F ma 5/22/2017 C. Pruneau, W.S.U. c 1 10 Scalar Product Given two vectors r r1 , r2 , r3 s s1 , s2 , s3 Scalar Product r r gs r1s1 r2 s2 r3s3 3 ri si r i 1 rs cos s Magnitude of a vector 3 r r r r r r gr ri2 i 1 5/22/2017 C. Pruneau, W.S.U. r2 r r r r r gr 2 11 Vector Product (Cross Product) Given two vectors r r1 , r2 , r3 s s1 , s2 , s3 r rs Vector Product x̂ ŷ ẑ r r p r s det r1 r2 r3 s1 s2 s3 r p r2 s3 r3 s2 x̂ r3 s1 r1s3 ŷ r1s2 r2 s1 ẑ r s r r r r p r s r s sin 5/22/2017 C. Pruneau, W.S.U. 12 Important Properties • Commutability r r r r gs sgr r r r r s sr • Associability r r r r r r s q s r q • But Note r r r r s s r 5/22/2017 C. Pruneau, W.S.U. 13 Differentiation of Vectors • Many laws of Physics involve vectors • Most of these involve derivatives of vectors. – Different ways to differentiate a vector – Subject of Vector Calculus • Here, we start with time derivatives… • Example – Position function of time: – Velocity – Acceleration 5/22/2017 r (t) r dr v(t) dt r dv a(t) dt C. Pruneau, W.S.U. 14 Definition of Derivative of a Vector • Definition closely related to that of scalar functions. • For a scalar function: dx(t) x lim t 0 t dt where x x(t t) x(t) • Similarly, for a vector r r dr r lim dt t 0 t where 5/22/2017 r r r (t t) r r r r r (t t) r (t) C. Pruneau, W.S.U. r (t) 15 Some useful vector properties • Derivative of a sum equals the sum of the derivatives r r d r r dr ds r s dt dt dt • Derivative of a scalar function x vector function is obtained by the product rule r d r dr df r fr f r dt dt dt • For a vector represented in Cartesian coordinates, note that the unit vectors are considered constant (except when explicitly stated otherwise), one therefore has: r dr dx dy dz x̂ ŷ ẑ dt dt dt dt dx dt dy vy dt dz vz dt vx r dr v vx x̂ vy ŷ vz ẑ dt 5/22/2017 C. Pruneau, W.S.U. 16 Time • Classical (Newtonian) view of time is a single universal parameter, t, on which all observers agree. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. – Provided all are equipped with accurate clocks properly synchronized. • This view is actually incorrect and is modified under the theory of special relativity. • In the first part of this course, we will neglect effects associated with special relativity. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 5/22/2017 C. Pruneau, W.S.U. 17 Measurement of Time Time Measurement Standards and the definition of “the second” are based on measurements of the natural resonance frequency of the cesium atom (defined as 9,192,631,770 Hz), The US - National Institute of Standards and Technology holds the record in precision accuracy since 1999. The clock NIST-F1 operates with an uncertainty of 1.7 x 10-15, or accuracy to about one second in 20 million years. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. 5/22/2017 C. Pruneau, W.S.U. 18 Reference Frames • Description of motion in classical mechanics involves a choice of a reference frame (either implicitly or explicitly). • Involves choices of – Spatial origin, – Spatial Axes to label positions – Temporal Origin to measure time. 5/22/2017 C. Pruneau, W.S.U. 19 About Reference Frames • Two reference frames may differ in various ways – Different time origins – Different spatial origins or axes orientation – Relative velocity, or acceleration • Clever choice of reference frame may greatly facilitate the solution of a specific problem. • All reference frames are not physically equivalent. – The laws of mechanics can however be formulated in a simple form in inertial reference frames. – The laws of mechanics are independent of the inertial reference frame being used. 5/22/2017 C. Pruneau, W.S.U. 20 Mass & Force • Concepts central to the formulation classical mechanics • Subject of many treatises in physics and the philosophy of sciences… • We take a pragmatic approach. 5/22/2017 C. Pruneau, W.S.U. 21 Mass • The mass of an object is a measure of inertia, I.e. its resistance to being accelerated. • Measurement Unit (SI): kilogram. • Mass measurements – Inertial Balance – Weight (Gravitational) Balance 5/22/2017 C. Pruneau, W.S.U. 22 Mass Units - Kilogram • The kilogram or kilogramme, – symbol: kg – SI base unit of mass. • Defined as the mass of the international prototype of the kilogram – A chunk of platinum-iridium stored at the International Bureau of Weights and Measure (Paris, France). QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. • Notes: – Only SI base unit that employs a prefix, – Only SI unit that is still defined in relation to an artifact rather than to a fundamental physical property. 5/22/2017 C. Pruneau, W.S.U. 23 Force • Informally: A measure of push or pull acting on an object - to move it. • Formally: A measure of action required to move a reference mass with a given acceleration. • Measurement Unit (SI): Newton (N) – Newton defined as the force required to accelerate a standard kilogram mass at 1 m/s2. • Measurement Technique – Acceleration of a mass calibrated object: not too practical. – Spring balance. 5/22/2017 C. Pruneau, W.S.U. 24 Note • In this chapter, we discuss the notions of velocity, acceleration, force, mass, etc as applied to point-like objects, or particles. • We will discuss later how the laws of motions can be used for extended objects. • Description of point-like objects is sufficient to treat very many interesting, quite practical, physical cases. 5/22/2017 C. Pruneau, W.S.U. 25 Newton’s First Law A body remains at rest or in uniform motion unless acted on by a force. or In the absence of forces, a particle moves with constant velocity. 5/22/2017 C. Pruneau, W.S.U. 26 Newton’s Second Law A body acted upon by a force moves in such a manner that the time rate of change of its momentum equals the force. r dp F dt 5/22/2017 F Vector describing the magnitude and direction of the force acting on an object p Vector describing the magnitude and direction of the momentum or “quantity of motion” of an object. C. Pruneau, W.S.U. 27 2nd Law cont’d Momentum r p mv m Scalar quantity describing the amount of matter (or inertia ) of an object. v Vector quantity describing the speed and direction an object. r dr v dt r Vector quantity describing the distance and direction an object realtive to a chosen origin and axes of reference. The definition of force is complete and precise only when “mass” is defined. 5/22/2017 C. Pruneau, W.S.U. 28 Common Formulation of Newton’s 2nd Law Provided the mass, m, of the object considered is constant, r r one can write dp d dv r F dt dt mv m dt Usual Formulation of Newtons’ 2nd Law. r F ma where one defines the acceleration vector, a, as the rate of change of the velocity. r dv a dt 5/22/2017 C. Pruneau, W.S.U. 29 Comments/Notations We introduce the “dot” and “dot-dot” shorthand notations. r dr r dt r d r r 2 dt 2 Newton’s 2nd laws and various other quantities may then be written as F mvr& r r v r& r r a v& r & r& ar r r F m& r& r r F p& 5/22/2017 C. Pruneau, W.S.U. 30 Newton’s Equation as Differential Equation Newton’s 2nd law is 2nd order vector differential equation. r&& r mr F In one dimension, it reduces to x(t) F(t) / m Which is integrable 1 x(t) F(t)dt m x(t) x&(t)dt For constant forces, one gets the familiar results & v(t) x(t) 5/22/2017 F t vo m C. Pruneau, W.S.U. x(t) F 2 t vot xo 2m 31 Newton’s Third Law If two bodies exert forces on each other, these forces are equal and in opposite directions. r F12 F21 1 r F12 F21 2 5/22/2017 F12 C. Pruneau, W.S.U. 32 About the 3rd law • 3rd law is a law in its own right, an actual statement about the world. • 3rd law is NOT a general law. – Applies to central forces, and contact forces – Does not apply for some velocity & momentum dependent forces. • E.g. Lorentz force between moving charges - because of the finite velocity of light. 5/22/2017 C. Pruneau, W.S.U. 33 Reformulation of the 3rd Law • Assume two bodies constitute an ideal, isolated system. • Assume constant masses • Accelerations of two bodies in opposite directions. • Ratio of accelerations equals to the inverse ratio of the masses of the two bodies. 5/22/2017 C. Pruneau, W.S.U. r F1 F2 r r dp1 dp2 dt dt d d r r m1v1 m2 v2 dt dt r r dv1 dv2 m1 m2 dt dt m2 a1 m1 a2 34 Mass Measurement Methods • Measure relative acceleration of object with unknown mass, and reference object (with unit mass). a1 m2 a2 • Comparison of weights using a balance. – Make use of F= ma & W=mg – Where g is independent of a body’s mass. – Assumes the INERTIAL and GRAVITATIONAL masses are equal. 5/22/2017 C. Pruneau, W.S.U. 35 Principle of equivalence • Inertial Mass: – Mass determining the acceleration of a body under the action of a given force. • Gravitational Mass: – Mass determining the gravitational force between a body and other bodies. • The hypothesis that two masses are equal is called “principle of equivalence”. 5/22/2017 C. Pruneau, W.S.U. 36 Experimental Verification of the Principle of equivalence 1. 2. 3. 5/22/2017 First test of the principle of equivalence performed by Galileo (Pisa). Newton considered the problem (also) using pendulums of equal length but made of different materials. Recent experiments have tested the equivalence to a few parts in 1012. C. Pruneau, W.S.U. 37 Isolated system & 3rd Law • It is impossible to actually have an isolated system. • Yet… • The 3rd law implies: d r r p1 p2 0 dt r r p1 p2 constant Implies Conservation of Linear Momentum Believed to be strictly valid under all circumstances. Essentially used as Postulate/Foundation of Modern Physics. 5/22/2017 C. Pruneau, W.S.U. 38 Galilean Invariance • Laws of motion have a meaning only in inertial reference frame. • A reference frame can be considered inertial if a body subject to no external force, moves in a straight line with constant velocity in that frame. • If Newton’s laws are valid in a given reference frame, then they are also valid in any reference in uniform motion relative to that first frame. • A change of reference frame involving a constant velocity does not change the equation. r r r d(v(t) vo ) dv(t) Fm m dt dt Called Galilean Invariance, or Principle of Newtonian Relativity. 5/22/2017 C. Pruneau, W.S.U. 39 Implication/Comment on Galilean Invariance • There is no such a thing as “absolute rest”, or “absolute inertial reference frame.” • “Fixed” stars reference frames are a good approximation to inertial reference but NOT an absolute frame. • Both space and time are assumed/required to be homogeneous. 5/22/2017 C. Pruneau, W.S.U. 40 Equation of Motion for a single particle • 2nd law for fixed mass: r dp F dt r r d mv dv r& & F m mr dt dt 2nd order differential equation. If F is known, and initial or boundary conditions are supplied, it can be integrated to find the particle position vs time: r r r (t) 5/22/2017 C. Pruneau, W.S.U. 41 About Forces… • The force may in general be a function of any combination of position, velocity, time. r r • Generally denoted: F r, v,t • Seek to know r, v as a function of “t”. 5/22/2017 C. Pruneau, W.S.U. 42 Conservation Theorems • Conservation theorems can be “derived” from Newton’s Eqs. – If Newton’s laws are correct, then the conservations laws are also correct; and conversely… • The fact that conserved quantities are indeed observed in Nature is a confirmation of Newton’s laws. • The principles of conservation of momentum, angular momentum, and energy are sometimes presented as “more fundamental” - possibly because their formulation enable extension to other theories - but strictly speaking there are equivalent to Newton’s laws. 5/22/2017 C. Pruneau, W.S.U. 43 Conservation of Linear Momentum For a free particle (no net external force acting on the particle): F i,external r& p0 i p constant The total linear momentum p of a system of particles is conserved (constant) if the net external force acting on the system is zero. 5/22/2017 C. Pruneau, W.S.U. 44 Corollary • The linear momentum conservation theorem can be decomposed along the 3 coordinates (directions). • Consider a constant vector s. s constant r s& 0 Consider scalar product with a force satisfying r Fgs 0 independent of time. r r& r Fgs pg s0 r pgs constant Component(s) of linear momentum in a direction in which the (net) force vanishes is constant with time. 5/22/2017 C. Pruneau, W.S.U. 45 Newton’s Law in Cartesian Coordinates • Newton’s law is a 2nd order differential equation. • It is often possible (and sufficient) to express the position and force in terms of Cartesian, constant unit vectors r xx̂ yŷ zẑ F Fx x̂ Fy ŷ Fz ẑ r& & F mr Fx m& x& Fy m& y& Fz m& z& 5/22/2017 C. Pruneau, W.S.U. 46 Useful Problem-Solving Technique 1. 2. 3. 4. 5. Make a sketch of the problem at hand, indicating forces, velocities, etc Write down given quantities and information Write down useful equations Identify what is to be determined. Manipulate equations to find quantities you seek. o 6. 5/22/2017 Algebra, Differentiation, and Integration typically required… Plug in actual values to determine specific answer. C. Pruneau, W.S.U. 47 Example: Block Sliding on an Incline Given Initial position x(0)=0 Initial speed v(0)=0 y O f Fx m& x& mgsin f Fy m& y& N mg cos 0 N but r w mg N mg cos f N m& x& mgsin mg cos x x g sin cos x g sin cos t x 5/22/2017 1 g sin cos t 2 2 C. Pruneau, W.S.U. 48 Newton’s Law in Polar Coordinates • Usage of polar, cylindrical, or spherical coordinates may enable enable considerable simplification of solution of specific problems. • Definition By construction ̂ r̂ r y O 5/22/2017 x x r cos y r sin r x 2 y2 arctan(y / x) Definition: Unit radial and polar vectors r r r̂ r ̂ r̂ r C. Pruneau, W.S.U. 49 Newton’s Law in Polar Coordinates (cont’d) Force decomposition F Fr r̂ F̂ Newton’s Law r F& r& ̂ r̂ r Derivatives of r in polar coordinates r rr̂ ˆ r r&r̂ rr& r&r̂ r&̂ r O d & r&r̂ r&̂ & r& r̂ 2 r&&̂ r&&̂ r&ˆ dt r & r& r̂ 2r&&̂ r&&̂ r&2 r̂ & r& r&2 r̂ 2r&& r&&̂ Special Case: r=constant, Angular velocity: Angular acceleration: r a& r& r 2 r̂ r̂ Newton’s Law F 5/22/2017 m 2 r&& r&& Fr m & r& r&2 C. Pruneau, W.S.U. 50 O Example: An Oscillating Skateboard N A half-pipe for skateboarding Radius R= 5 m. Coordinates of the skateboard: (r,) with r=R m 2 r&& r&& Fr m & r& r&2 F Fr mR&2 F mR&& r w mg The forces acting on the skateboard are its weight, and the normal force due to the wall. r r F w N Which can be decomposed as follows: Fr mg cos N F mgsin mg cos N mR&2 mg sin mR&& N mg cos mR&2 g && sin R 5/22/2017 C. Pruneau, W.S.U. 51 Example: An Oscillating Skateboard (cont’d) O g R sin Consider “small” oscillation approximation N r w mg sin One gets g R Which may be written 2 0 2 with g R This differential equation has general solutions of the form (t) Asin( t) B cos( t) The frequency of the motion is f 5/22/2017 1 2 g R C. Pruneau, W.S.U. The period is T 2 R g 52 Example: An Oscillating Skateboard (cont’d) Determination of the constants A & B. We have: O (t) Asin( t) Bcos( t) &(t) Acos( t) Bsin( t) N r w mg Assume, for instance, the skateboard is initially at rest at an angle o. (0) o B B o A0 &(0) 0 A So the equations of motion are thus (t) o cos( t) &(t) o sin( t) Note again that the period of this motion is independent of the amplitude o and has a value T 2 5/22/2017 R 5m 2 4.5s g 9.8m / s 2 C. Pruneau, W.S.U. 53