1Class 4Newton’s laws of motion Newton’s laws of motion Momentum… and a second way to look atNewton’s laws Frames of reference, symmetry and (Galilean)relativity… yet another way to look atNewton’s lawI : Newton’s laws of motion Newton’s first law : If a body is notacted upon by any forces, then itsvelocity remains constant Notes Remember that velocity is a vector quantity(it has direction as well as magnitude) This law sweeps away the idea that “beingat rest” is a natural state… this was a majorchange of thinking!2 Newton’s second law : If a body ofmass M is acted upon by a force F,then its acceleration a is given byF=Ma Notes Remember that both F and a are vectors This law defines the “inertial mass” as thedegree to which a body resists beingaccelerated by a force Newton’s third law - If a body Aexerts a force F on body B, thenbody B exerts a force -F on body A Notes This is the law of “equal and oppositereaction” We will see later that this law is closely tiedto conservation of momentum34Review of Goddard’spioneering work on rockets “Professor Goddard does not know therelation between action and reactionand the needs to have something betterthan a vacuum against, which to react.He seems to lack the basic knowledgeladled out daily in high schools.”…-1921 New York Times editorialII : Momentum Definition : If an object of mass m ismoving with velocity V, its momentum pis given by p=mV The total momentum ptot of a number ofobjects with masses m1, m2, … andvelocities V1, V2, … is just the (vector)sum of the objects’ separate momenta5 Conservation of momentum : The totalmomentum of a system of particles isconstant if no external forces act on thesystem Proof for a two particle system… Consider two particles with masses m1 and m2 They exert forces on each other, but there is no forcebeing applied to the pair as a whole At some instant in time, they have velocities V1 and V2 So momentum is p=m1V1+m2V2 Consider some instant in time Δt later… individualvelocities will have changed due to forces thatparticles exerted on each other… let new velocitiesbe V1’ and V2’ Difference between new and old momentum isNewton’s thirdlaw used here!6 Proof for a general (many particle) systemfollows very similar lines We now see that Newton’s laws can berephrased entirely in terms of momentum… Second law… the rate of change of momentum of abody is equal to the force applied to that body First law is special case of the Second law… themomentum of a body is unchanged if there are noforces acting on body Third law… the momentum of an isolated system ofobjects is conservedIII : Symmetries and frames ofreference The idea of symmetry is very important inmodern advanced physics! Let’s have aglimpse of symmetry in action… Consider… Two equal, connected masses M at rest. At some time, they are suddenly pushed apart by aspring They must fly apart with the same speed in oppositedirections (what else could possibly happen… whywould one mass “decide” to move faster?)7 Now think of same situation, but the two connectedmasses are initially moving at velocity V. Let’s turnthis into the above situation by “moving along with themasses at velocity V” Change perspective to bring masses to rest… Do same problem as before… Change back to the original perspective… You have “changed your frame of reference”. The “velocity addition” rule is called a Galileantransformation. We assume that, after changing our reference frame andusing a Galilean transformation, the laws of physics arethe same. This is called Galilean Relativity. Then find that momentum before = momentum after8 How do Newton’s laws fit into this picture? N1 comes directly from Galilean Relativity (there isno difference between a state of rest and a state ofmotion) N2 and N3 are exactly what’s needed to make surethat momentum is conserved and so is related tothe symmetry of space So… Newton’s laws are related to the symmetry ofspace and the way that different frames ofreference relate to each