Newton’s Laws

How far away is the Moon? The Greeks used a special configuration of Earth, Moon and Sun (link) in a lunar eclipselink Can measure EF in units of Moon’s diameter, then use geometry and same angular size of Earth and Moon to determine Earth-Moon distance See herehere for method

That means we can size it up! We can then take distance (384,000 km) and angular size (1/2 degree) to get the Moon’s size D = 0.5/360*2π*384,000km = 3,350 km

How far away is the Sun? This is much harder to measure! The Greeks came up with a lower limit, showing that the Sun is much further away than the Moon Consequence: it is much bigger than the Moon We know from eclipses: if the Sun is X times bigger, it must be X times farther away

Simple, ingenious idea – hard measurement

Timeline

Isaac Newton – The Theorist Key question: Why are things happening? Invented calculus and physics while on vacation from college His three Laws of Motion, together with the Law of Universal Gravitation, explain all of Kepler’s Laws (and more!) Isaac Newton (1642–1727)

Major Works: Principia (1687) [Full title: Philosophiae naturalis principia mathematica] Opticks [sic!] (1704) Later in life he was Master of the Mint, dabbled in alchemy, and spent a great deal of effort trying to make his enemies miserable

Newton’s first Law In the absence of a net external force, a body either is at rest or moves with constant velocity.force body –Contrary to Aristotle, motion at constant velocity (may be zero) is thus the natural state of objects, not being at rest. Change of velocity needs to be explained; why a body is moving steadily does not.

Mass & Weight Mass is the property of an object Weight is a force, e.g. the force an object of certain mass may exert on a scale

Newton’s second Law The net external force on a body is equal to the mass of that body times its acceleration F = ma. Or: the mass of that body times its acceleration is equal to the net force exerted on it ma = F Or: a=F/m Or: m=F/a

Newton II: calculate Force from motion The typical situation is the one where a pattern of Nature, say the motion of a planet is observed: –x(t), or v(t), or a(t) of object are known, likely only x(t) From this we deduce the force that has to act on the object to reproduce the motion observed

Calculate Force from motion: example We observe a ball of mass m=1/4kg falls to the ground, and the position changes proportional to time squared. Careful measurement yields: x ball (t)=[4.9m/s 2 ] t 2 We can calculate v=dx/dt=2[4.9m/s 2 ]t a=dv/dt=2[4.9m/s 2 ]=9.8m/s 2 Hence the force exerted on the ball must be F = 9.8/4 kg m/s 2 = 2.45 N –Note that the force does not change, since the acceleration does not change: a constant force acts on the ball and accelerates it steadily.

Newton II: calculate motion from force If we know which force is acting on an object of known mass we can calculate (predict) its motion Qualitatively: –objects subject to a constant force will speed up (slow down) in that direction –Objects subject to a force perpendicular to their motion (velocity!) will not speed up, but change the direction of their motion [circular motion] Quantitatively: do the algebra

Newton’s 3 rd law For every action, there is an equal and opposite reaction Does not sound like much, but that’s where all forces come from!

Newton’s Laws of Motion (Axioms) 1.Every body continues in a state of rest or in a state of uniform motion in a straight line unless it is compelled to change that state by forces acting on it (law of inertia) 2.The change of motion is proportional to the motive force impressed (i.e. if the mass is constant, F = ma) 3.For every action, there is an equal and opposite reaction (That’s where forces come from!)

Newton’s Laws a) No force: particle at rest b) Force: particle starts moving c) Two forces: particle changes movement Gravity pulls baseball back to earth by continuously changing its velocity (and thereby its position)  Always the same constant pull

Law of Universal Gravitation Force = G M earth M man / R 2 M Earth M man R

Orbital Motion

Cannon “Thought Experiment” ets/newt/newtmtn.htmlhttp:// ets/newt/newtmtn.html

From Newton to Einstein If we use Newton II and the law of universal gravity, we can calculate how a celestial object moves, i.e. figure out its acceleration, which leads to its velocity, which leads to its position as a function of time: ma= F = GMm/r 2 so its acceleration a= GM/r 2 is independent of its mass! This prompted Einstein to formulate his gravitational theory as pure geometry.

Applications From the distance r between two bodies and the gravitational acceleration a of one of the bodies, we can compute the mass M of the other F = ma = G Mm/r 2 (m cancels out) –From the weight of objects (i.e., the force of gravity) near the surface of the Earth, and known radius of Earth R E = 6.4  10 3 km, we find M E = 6  kg –Your weight on another planet is F = m  GM/r 2 E.g., on the Moon your weight would be 1/6 of what it is on Earth

Applications (cont’d) The mass of the Sun can be deduced from the orbital velocity of the planets: M S = r Orbit v Orbit 2 /G = 2  kg –actually, Sun and planets orbit their common center of mass Orbital mechanics. A body in an elliptical orbit cannot escape the mass it's orbiting unless something increases its velocity to a certain value called the escape velocity –Escape velocity from Earth's surface is about 25,000 mph (7 mi/sec)