ISNS 3371 - Phenomena of Nature The first test will be next Thursday, Feb 8 at the regular class time. We will have two review sessions, one at 1:00 PM

ISNS 3371 - Phenomena of Nature

The first test will be next Thursday, Feb 8 at the regular class time.We will have two review sessions, one at 1:00 PM Monday and one at 1:00 PM Tuesday - both in FN 2.212 on the SW corner of Founder’s North. Bring questions.The test will cover any topic we have discussed in class. These include:

Exploration of Nature, ScienceFundamental quantitiesMeasurement unitsVectors, ScalarsMotion - distance, velocity, acceleration, projectilesForces - types, netMass, momentum, impulseNewton’s Laws of MotionLaw of Conservation of MomentumTorque, angular momentum, conservationMatterEnergy, WorkLaw of Conservation of EnergyForms of EnergyTransformation of energyPowerGravity - Law of, force, acceleration, weight, weightlessness, orbitsTides

ISNS 3371 - Phenomena of NatureThe Ballistic Pendulum

The ballistic pendulum is used to determine the speed of a projectile. Invented in the 18th century by Benjamin Robins to determine the speed of a bullet.

A bullet of mass m is fired at a block of wood (mass M) hanging from a string. The bullet embeds itself in the block, and causes the combined block plus bullet system to swing up a height h. Conservation of momentum and conservation of energy are used to determine the bullet’s speed.

ISNS 3371 - Phenomena of NatureConservation of momentum

(1)

b = before collision- mb and vb are for the ball/bulleta = after collision- ma and va are for the ball/bullet and pendulum

Conservation of energy

Kinetic Energy of ball and pendulum just after collision = Potential Energy of ball and pendulum at end of swing:

h = height of pendulum at end of swing

Substitute into (1):

€

mbvb = mava

vb =mavamb

€

1

2mava

2 = magh

€

va2 = 2gh

€

vb = 2ghmamb


Alternate Way Using Projectile Motion and g

h

x

Fire ball from top of table. Measure initial height of ball (h) and horizontal distance traveled (x).

€

h =1

2gt 2 ⇒ t =

2h

g

x = vt ⇒ v =x

t

v = xg

2h

Vertical motion

Horizontal motion

(1)

Substitute from (1)


• Law of Gravitation:

Objects in the universe attract each other with a force that varies directly as the product of their masses and inversely as the square of their distances from each other.

G is called the Universal Gravitation Constant

€

F =Gm1m2

d2

Gravity

Consider a light source (for instance the sun). Light travels in all directions from source. At 2 AU, is covers a patch 4 times as large as at 1 AU since the patch is twice as high and twice as wide. A 3 AU, it covers a patch 9 times as large as at 1 AU


Newton’s Law of Universal Gravitation.


The force on a body of mass m1 is:

(Newton’s Second Law)

If this force is due to gravity, then:

m1 cancels out, and:

€

F =m1a

€

m1a=Gm1m2

d2

€

a=Gm2

d2

Newton’s 2nd Law and the Acceleration Due to Gravity


The acceleration due to the force of gravity is:

Mass of the Earth (m2) = 5.97 X 1024 kgRadius of Earth (d) = 6.378 X 106 mG= 6.67 x 10-11 Nm2/ kg2

g= (6.67 x 10-11 Nm2/ kg2) X (5.97 X 1024 kg)/(6.378 X 106 m)2

g= 9.79 m/s2

g does not depend on the mass of the body m1 - so the feather falls at the same speed as the steel ball - Galileo learned this by experimentation (the Leaning Tower of Pisa experiment) - Newton showed why.

The mass of the Earth was calculated using this formula in the 18th century.

€

g=Gm2

d2


Weightlessness and Free-Fall

Remember: Weight is the result of the force of gravity on a body of mass m1:

Therefore all objects on earth having the same mass have the same weight.

Free-fall - condition of weightlessness - whenever nothing is preventing you from falling - e.g., an elevator floor drops away at same rate you fall

Apparent Weight Animation

€

W =m1g


Weightlessness - a state of being in free fall towards the Earth.

The Earth is round - its surface drops about 5 m for every 8 km of distance. If you were standing at sea level, you would only see the top of a 5-meter mast on a ship 8000 m away - remember the story of Columbus and the orange.

Given h=1/2gt2, if t=1 s then h = 5 m. So if a projectile is fired horizontally at 8 km/s, it will fall fast enough to keep “falling around” the Earth - becomes a satellite. So a spacecraft is in free fall around the Earth - free fall is not an absence of gravity. If a satellite is given a velocity greater than 8 km/s, it will overshoot a circular orbit and trace an elliptical path.

Cannonball Animation

Orbit and Velocity


Since gravity decreases with altitude (inversely proportional to square of distance from the center of the Earth), the orbital velocity varies with altitude.

Geosynchronous/Geostationary Orbits

A geostationary orbit is 42,000 km above the center of the Earth and the altitude is about 35,600 km

A geosynchronous orbit has a period the same as the rotational speed of the Earth - e.g., it orbits in the same amount of time that the Earth rotates - 1 sidereal day. A geostationary orbit is a geosynchronous orbit at the equator - it always stays above the same place on the Earth - communications satellites, satellite TV, etc…

http://www.geo-orbit.org/sizeimgs/orbitdefr.gif


Calculating Geosynchronous Orbit Altitude

€

Fcentripetal = Fcentrifugalmsatag = msatacag = ac

ac =ω2r

ag =MG

r2

ω2r =MG

r2

r3 =MG

ω2

r =MG

ω23

For an orbiting body, the inward and outward forces must equal each other (Newtons 3rd Law) - the centripetal force from orbital motion has to equal the centrifugal force from gravity:

is angular velocity - at geosynchronous orbit, of satellite is equal to the angular velocity of the Earth

= 2/86164 (length of sidereal day)M = 5.97 X 1024 kgG= 6.67 x 10-11 Nm2/ kg2

Plug in the numbers and you get

r = 42,164 km


Escape Velocity

If a projectile is fired straight up with a large enough velocity, it will escape the Earth’s gravity. It will travel slower and slower due to the Earth’s gravity, but never to zero. Escape velocity - velocity at which gravity can not stop outward motion. Note that the gravitational attraction of Earth never ceases, it just gets infinitesimally small.

Escape velocity is calculated by using conservation of energy - a body achieves escape velocity when the all of its initial gravitational energy is converted to kinetic energy.


€

PE(at Earth's surface) = mgh = mgRE

g =GME

RE

PE =GMEm

RE

KE =1

2mv 2

1

2mv 2 =

GMEm

RE

v =2GME

RE

Gravitational potential energy at the Earth’s surface

Acceleration due to gravity

Potential energy

Kinetic energy

KE=PE

Escape velocity

Starting from the surface of the Earth:

RE = 6.378 X 106 m, ME = 5.97 X 1024 kg, G= 6.67 x 10-11 Nm2/ kg2

v= 11,174 m/s


Center of Mass

Newton also showed that two objects attracted to each other by gravity actually orbit about their center of mass - the point at which the objects would balance if the were connected.

This idea is used to find planets orbiting other stars - massive planets cause star to move against background stars

Center of Mass - Binary Star

ISNS 3371 - Phenomena of Nature Tides

The gravitational attraction of the Moon varies as the square of the distance (Newton’s Law of Gravitation) - gravity stronger on side facing the Moon than on opposite side. The Moon pulls the ocean water towards it on facing side - creates tide - and pulls the Earth away from the ocean water on the other side - reason for tides twice a day. Time of tides varies by 50 min per day - Moon at its highest point every 24 hrs 50 min because Moon orbits Earth while Earth rotates.


The Sun also causes tides - why are they weaker than the Moons’ (by about 1/2)? The Sun’s gravitational attraction on the Earth is about 180 times that of the moon. But the Sun’s greater distance means the difference on opposite sides of the Earth is much smaller - only about 0.02% as opposed to 7% for the moon.

Neap tides - when Moon’s and Sun’s gravitational forces oppose each other

Spring tides - when Moon’s and Sun’s gravitational forces add up


Tides


Tidal Bulge

Because the Earth rotates, friction drags the tidal bulges off of the Earth-Moon line. This tidal friction causes the Earth’s rotation to slow and the Moon to move farther out.


The Moon pulls on tidal bulge - slows Earth’s rotationThe excess mass in Earth’s tidal bulge exerts a gravitational attraction on the Moon that pulls the Moon ahead in its orbit - Moon moves farther away -

Conservation of Angular Momentum!

ISNS 3371 - Phenomena of NatureTidal Heating in Jovian Moons

The four inner moons of Jupiter - Io, Europa, and Ganymede - all show evidence of geological activity - indicators of molten interiors. The heat source is tidal heating.Moons have elliptical orbit and synchonous rotation - one side always faces Jupiter

- as Ganymede completes one orbit, Europa completes exactly two orbits, and Io completes exactly four orbits - moons periodically line up - causes orbital ellipticity.

- tidal bulges are constantly being flexed in different directions - generates friction inside


Io

Jupiter’s tidal forces flex Io like a ball of silly putty.

- friction generates heat

- interior of Io is molten

Volcanoes erupt frequently.

- sulfur in the lava accounts for yellow color

- surface ice vaporizes and jets away

Evidence of tectonics and impact cratering is covered.

ISNS 3371 - Phenomena of NatureVolcanic Plumes


Lava fountain - active lava hot enough to cause "bleeding" in Galileo's camera - overloading of camera by the brightness of the target

Newly erupted hot lava flow. Dark, "L"-shaped lava flow marks the location of the November 1999 eruption.


A broad plume of gas and dust about 80 km high above a lava flow

Gas and Dust Plume

ISNS 3371 - Phenomena of NatureEuropa

Metallic core, rocky mantle, and a crust made of H2O ice

Its fractured surface tells a tale of tectonics.

- few impact craters seen

- double-ridged cracks

- jumbled icebergs

These provide photographic evidence of a subsurface ocean.

Europa has a magnetic field.

- implies liquid salt water beneath the icy crust

Where liquid water exists, there could be life!



Evidence of a Subsurface ocean

Jumbled crust with icebergs and surface cracks with double-ridged pattern - caused by tidal flexing of thick layer of ice on top of liquid ocean of water.

ISNS 3371 - Phenomena of NatureEuropa Ice Rafts

Thin, disrupted, ice crust in the Conamara region of Europa - white and blue colors outline areas blanketed by a fine dust of ice particles ejected at the time of formation of the large (26 kilometer in diameter) crater Pwyll 1000 kilometers to the south. - a few small craters - less than 500 meters in diameter were probably formed at the same time as the blanketing occurred by large, intact, blocks of ice thrown up in the impact explosion that formed Pwyll.

ISNS 3371 - Phenomena of NatureGanymede

Largest moon in the Solar System

Its surface has 2 types of terrain:

- heavily cratered, implies old

- long grooves, few craters, implies young like Europa

It also has a magnetic field.

Could it have subsurface ocean?

- case not as strong as Europa’s

- tidal heating would be weaker

- would need additional heating from radioactive decay

Documents

ISNS 3371 - Phenomena of Nature The first test will be next Thursday, Feb 8 at the regular class time. We will have two review sessions, one at 1:00 PM