Newton’s Laws of Motion8.01TSept 20, 2004Newton’s First LawEvery body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.Newton’s Second LawThe change of motion is proportional to the motive force impresses, and is made in the direction of the right line in which that force is impressed.Newton’s Third LawTo every action there is always opposed an equal reaction: or, the mutual action of two bodies upon each other are always equal, and directed to contrary parts.Inertial Mass• inertial mass is a ‘quantity of matter’ • standard body with mass msand SI units [kg]• the mass of all other bodies will be determined relative to the mass of our standard body. • apply the same action (force) to the standard body and an unknown body• Define the unknown mass in terms of ratio of accelerationsussumama=Standard Kilogram• Cylindrical alloy of 90 % platinum and 10 % iridium • Density and volume of the standard kilogram,• Constant volume for a cylinder • The surface area• minimize the area with respect to the radius • radius is one half the height, -3ρ = 21.56 g cm⋅2Vrhπ=22222 2VAr rhrrπππ=+=+2240dA Vrdr rπ=−=/2rh=-3 3/ 1000g/22g cm 46.38cmVmρ=≅ ⋅≅()13/ 2 1.95 cmrVπ=≅Definition: Momentum (Quantity of Motion)• Momentum is a vector quantity • Magnitude: product of the mass with the magnitude of the velocity • Direction: the direction of the velocity• In the SI system of units, momentum has units[kg-m-s-1]m=pvGGDefinition: Force• Force is a vector quantity. • Magnitude of the total force acting on the object is the product of the mass and the magnitude of the acceleration .• Direction of the total force on a body is the direction of the acceleration of the body. • SI units for force are [kg-m-s-2]• Unit has been named the Newton 1 N = 1 kg-m-s-2m≡FaGGSuperposition PrincipleApply two forces and on a single body Total force is the vector sum of the two forces 1FG2FG12total=+FFFGGGAverage ImpulseApply an average force for an interval of time Average impulse is the product of the average force and the time intervalUnits for impulse are the same as for momentum [kg-m-s-1] = [N-s]FG∆tt=∆IFGGImpulseWhen force is applied continuously over a time interval Impulse is the integral0,ftt⎡⎤⎣⎦fottdt=∫IFGGNewton’s Second LawThe change of momentum is equal to the applied average impulseFor an instantaneous action of the total force, force is equal to the product of mass with acceleration.When the mass remains constant in time,t=∆=∆IF pGGG0limtotaltdtdt∆→∆=≡∆ppFGGGtotalm=FaGGForce Law: Newtonian Induction• Definition of force has no predictive content • Need to measure the acceleration and the mass in order to define the force• Force Law: Discover experimental relation between force exerted on object and change in properties of object• Induction: Extend force law from finite measurements to all cases within some range creating a model• Second Law can now be used to predict motion!• If prediction disagrees with measurement adjust model.Hooke’s Law• Spring attached to a body • Stretch or compress spring by different amounts produces different accelerations • Magnitude:• Direction: restoring spring to equilibrium• Model holds within some reasonable range of extension or compression kl=∆FGContact Forces Between SurfacesComponent Perpendicular to surface is called the Normal Force Component tangent to the surface is called the friction force Vector sum is the total contact force≡+CNfGGGNGfGKinetic FrictionForce Law for kinetic friction: independent of surface area of contact, independent of velocity, proportional to the normal forceMagnitude:where different contact surfaces have different coefficients of friction µkDirection: opposes motionkkfNµ=Static FrictionVaries in direction and magnitude depending on applied forcesMagnitude varies betweenJust Slipping Condition: two objects in contact• accelerations are equal • Static friction is equal to it’s maximum value,max0ss sffNµ≤≤=12aa=,maxssfNµ=Models in Physics: Fundamental Laws of NatureForce laws are mathematical models of physical processes Search for fundamental forces• Electromagnetism• Weak Force• Strong Force•Gravity• Fifth Force?Universal Law of Gravitation• Gravitational force between two bodies with masses m1and m2• Direction: force points along the line connecting the bodies and is attractive • Magnitude: proportional to the product of the masses; and inversely proportional to square of the distance, r1,2, between the bodies • with 121,2 1,221,2ˆmmGr=−FrG11 2 -26.67 10 N-m kgG−=× ⋅Newton’s Third Law• To every action there is always opposed an equal reaction: or, the mutual action of two bodies upon each other are always equal, and directed to contrary parts.• Action-reaction pair of forces cannot act on same body; they act on different bodies. 1,2 2, 1=−FFGGTension in a RopeThe tension in a rope at a distance x from one end of the rope is the magnitude of the action -reaction pair of forces acting at that point ,,,() () ()left right right leftTx x x==FFGGFree Body Force Diagram• Isolate each object • Represent each force that is acting on the object by an arrow that indicates the direction of the force. • Use free body force diagram to make these vector decompositions of the individual forces.12...tota l=++FFFGGGˆˆˆtotal total total totalxyzFFF=++FijkGNewton’s Law in Components• Use Experimental Force Laws and Newton’s Second Law to predict the acceleration of the body totalxxFma=totalyyFma=totalzzFma=Methodology for Newton’s Second LawI. Understand – get a conceptual grasp of the problem• Think about the problem.• Sketch the system at some time when the system is in motion. • Choose a coordinate system and identify the position function of all objects.• Quantify the constraint conditions.• Draw free body diagrams for each body in the problem.II. Devise a Plan Draw free body diagrams for each body in the problem.• Include the set of unit vectors • Each force represented by an arrow indicating the direction of the force • Choose an appropriate symbol for the force Choosing directions for the forces• If you solve for a force and find that the force is negative then the force points in the opposite direction as your
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