nats 1311 - from the cosmos to earth billiard balls

NATS 1311 - From the Cosmos to Earth

Billiard Balls


Angular Momentum

Momentum associated with rotational or orbital motionangular momentum = mass x velocity x radius


Torque and Conservation of Angular Momentum

Conservation of angular momentum - like conservation of momentum -in the absence of a net torque (twisting force), the total angular momentum of a system remains constant

Torque - twisting force


A spinning skater speeds up as she brings her arms in and slows down as she spreads her arms because of conservation of angular momentum


The 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:

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F =m1a

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m1a=Gm1m2

d2

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a=Gm2

d2

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


The acceleration due to the force of gravity is called g, so:

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.

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.

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g=Gm2

d2

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W =m1g


The acceleration of gravity and therefore a person’s weight is dependent on a planet’s mass and radius.

Planetary Mass, Radius and Weight


Newton’s Formulation of Kepler’s Laws

As a planet moves around its orbit, it sweeps out equal areas in equal times - a planet moves slower when it is farther from the Sun and faster when it is closer

Kepler’s Laws were based on observation (experimentation). Newton’s lawsexplained Kepler’s Laws

Kepler’s Second Law


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F =m1a

For a circular orbit: (r = radius of orbit)

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a=v2

r

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F =m1

v2

r=Gm1m2

r2

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v2 =Gm2

r

Substitute (2) into (1):F is the force of gravity:

Cancel m1and r; then

(1)

(2)

(3)

(4)

The smaller the radius, the greater the speed.The orbital speed is independent of the mass of the orbiting body (m1). As the radius (the distance to the orbiting body) increases, the orbital speed decreases.

When you swing a ball around, the string exerts a force that pulls the ball inward (gravity for orbiting body). The acceleration is also inward.


The square of any planet's period of orbital revolution, P, is proportional to the cube of its mean distance, r, from the sun.

Kepler’s 3rd Law

Orbital Period vs Distance Animation


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v2 =Gm2

r

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v=2πrP

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v2 =4π 2r2

P2

From Kepler’s Second Law (previous slide):

Speed around orbit:Circumference (2r)/ timeP=period, time of 1 orbit

(1)

(2)

(3)

(4)

(5)

Combine (1) and (3):

Rearrange terms:

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Gm2

r=

4π 2r2

P2

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P2 =4π2

Gm2

r3

Square both sides:


A more complex derivation of this equation yields:

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P2 =4π 2

G(m1 +m2)r3

From this equation, if one knows the mass of the orbiting body (m1), the mass of the central body (m2) may be calculated.

What is the mass of the Sun?

MSun (m1) >> MEarth (m2) so: m1 + m2 m1

M1 = 42r3/GP2

G = 6.67 x 10-11 Nm2/ kg2

r = 1.5 x 1011mP = 3.15 x 1011 s

So:

Msun = 2 x 1030 kg


Geosynchronous/Geostationary OrbitsA 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…

What is the altitude of a geostationary orbit?

From Newton’s formulation of Kepler’s 3rd Law:

MEarth (m1) >> MSatellite (m2) so:

r = (GMEarth P2/42)1/3

G = 6.67 x 10-11 Nm2/ kg2

P = 3.15 x 1011 s

MEarth = 5.97 X 1024 kgSo:

R = 42,000 km above the center of the Earth and the altitude is about 35,600 km

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P2 =4π 2

G(m1 +m2)r3


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

NATS 1311 - From the Cosmos to Earth 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’?

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


Matter and Energy


DEFINITION:

• Anything that occupies space and has mass

PROPERTIES OF MATTER:

• Mass - a measure of a body’s resistance to a change in its state of motion - its inertia

• Density - mass per unit volume

• Dimensions - height, length, width

• Electric charge - positive/negative/neutral

• Heat content - everything above absolute 0 (-459.67º F) has heat - no such quantity as cold - only absence of heat

• Resistance to flow of electric current - flow of charged particles - electrons

• Pressure - exerted by moving molecules in all directions - resists compression

Matter


Energy

Definition of Energy:

• Anything that can change the condition of matter

• Ability to do work – the mover of substance (matter)

• Work is a force acting over a distance

• Force: The agent of change – push or pull on a body

Hence: Work is the change in the energy of a system resulting from the application of a force acting over a distance.


Four Types of Forces:

• Gravitational – holds the world together

• Electromagnetic – accounts for many observed forces - Push or pull on a body

• Strong Nuclear – hold nucleus together

• Weak Nuclear – involved in nuclear reactions

Power:Rate of change of energy


Energy Units

Energy:Joule = 1 kg m2/s2

1 Joule = 1/4184 Calorie, so2500 Cal = 1 x 107 J (average daily requirement for a human)

Power:1 watt = 1J/sThus for every second a 100 W light bulb is on, the electric company charges for 100 J of energy.The average daily power requirement for a human is about the same as for a 100-W light bulb.


Solar energy striking Earth’s surface per second = 2.5 x 1017 J.Energy released by burning 1 liter of oil = solar energy striking square 100 m on a side in 1 second

Energy Comparisons


Energy

Three basic categories:

Kinetic energy = energy of motion

Potential energy = stored energy

Radiative - energy carried by light


Types of Energy

Energy cannot be created or destroyed, only changed

– Mechanical –

• Potential- energy of position P.E.= mgh

• Kinetic- energy of motion K.E.=1/2mv2

– Electrical

– Chemical

– Elastic

– Gravitational

– Thermal

– Radiant

– Nuclear

nats 1311 - from the cosmos to earth billiard balls

Documents

body of mass m1

mass x velocity x radiusnats

planets mass

orbiting body m1

planetary mass

force of gravity

earththe force

nm2 kg2 x