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: xball(t)=[4.9m/s2] t2

• We can calculate v=dx/dt=2[4.9m/s2]t a=dv/dt=2[4.9m/s2]=9.8m/s2

• Hence the force exerted on the ball must be • F = 9.8/4 kg m/s2 = 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 3rd 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 restb) Force: particle starts movingc) Two forces: particle changes movement

Gravity pulls baseball back to earthby continuously changing its velocity(and thereby its position)

Always the same constant pull

Law of Universal Gravitation

Force = G Mearth Mman / R2

MEarth

Mman

R

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/r2

so its acceleration a= GM/r2 is independent of its mass!

• This prompted Einstein to formulate his gravitational theory as pure geometry.

Orbital Motion

Cannon “Thought Experiment”

• http://www.phys.virginia.edu/classes/109N/more_stuff/Applets/newt/newtmtn.html

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/r2 (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 RE = 6.4103 km, we find ME = 61024 kg

– Your weight on another planet is F = m GM/r2

• 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: MS = rOrbitvOrbit

2/G = 21030 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)

Activity: Newton’s Gravity Law

• Get out your worksheet books• Form a group of 3-4 people• Work on the questions on the sheet• Fill out the sheet and put your name on top• Hold on to the sheet until we’ve talked about

the correct answers• Hand in a sheet with the group member’s

names at the end of the lecture • I’ll come around to help out !

Intro to the Solar System

The Solar System

Contents of the Solar System

• Sun• Planets – 9 known (now: 8)

– Mercury, Venus, Earth, Mars (“Terrestrials”)– Jupiter, Saturn, Uranus, Neptune (“Jovians”)– Pluto (a Kuiper Belt object?)

• Natural satellites (moons) – over a hundred• Asteroids and Meteoroids

– 6 known that are larger than 300 km across– Largest, Ceres, is about 940 km in diameter

• Comets• Rings• Dust

Size matters: radii of the Planets

The Astronomical Unit

• A convenient unit of length for discussing the solar system is the Astronomical Unit (A.U.)

• One A.U. is the average distance between the Earth and Sun– About 1.5 108 km or 8 light-minutes

• Entire solar system is about 80 A.U. across

Homework: Distance to Venus

• Use: distance = velocity x travel time, where the velocity is the speed of light

• Remember that the radar signal travels to Venus and back!

The Terrestrial Planets• Small, dense and rocky

Mercury

Venus

Earth

Mars

The Jovian Planets

• Large, made out of gas, and low density

Jupiter

Uranus

Saturn

Neptune

Asteroids, Comets and Meteors

Debris in the Solar System

Asteroids

Asteroid Discovery

• First (and largest) Asteroid Ceres discovered New Year’s 1801 by G. Piazzi, fitting exactly into Bode’s law: a=2.8 A.U.

• Today more than 100,000 asteroids known

• Largest diameter 960 km, smallest: few km

• Most of them are named

• about 20 of them are visible with binoculars

How bright does a planet, moon, asteroid or comet appear?

• Apparent brightness of objects that reflect sunlight do depends on three things:– Size of the object (the bigger the brighter)– Distance to the object (the closer the brighter)– “Surface” properties of the object (the whiter the

brighter, the darker the dimmer)• Technical term: Albedo (Albedo =1.00 means 100% of

incoming radiation is reflected)