Newton’s 2nd Law Explains the Feather and the Hammer

PHYS 3380 - Astronomy

Newton’s 2nd Law Explains the Feather and the Hammer

1 kg on the Earth weighs 9.8 N or 2.2 lbs

F = W = mg

W = 1kg X 9.8 m/s = 9.8 kg m/s = 9.8 N

Take a 1 kg rock and a 10 kg rock and drop them from the same height

a1 = F1/m1 = W1/m1 = 9.8 N/1 kg = 9.8 m/s2 = g

a2 = F2/m2 = W2/m2 = 98 N/10 kg = 9.8 m/s2 = g



LUNAR ATMOSPHERIC COMPOSITION EXPERIMENT (UTD)


• 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 thesquare of their distances from each other.

G is the Universal Gravitation ConstantG= 6.67 x 10-11 Nm2/ kg2

Gravity


The law of universal gravitation.


The force on a body of mass m1 is:

If this force is due to gravity, then:

m1 cancels out, and:

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)2g= 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 sameweight.


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

Planetary Mass, Radius and Weight


apparent weight - weight force that we actually sense not the downward force of gravity, but the normal (upward) force exerted by the surface we stand on

- opposes gravity and prevents us falling to the center of the Earth - what is measured by a weighing scale.

For a body supported in a stationary position, normal force exactly balances earth's gravitational force

- apparent weight has the same magnitude as actual weight.

If no contact with any surface to provide such an opposing force - no sensation of weight (no apparent weight).

- free-fall - experienced by sky-divers and astronauts in orbit who feel "weightless" even though their bodies are still subject to the force of gravity - also known as microgravity.

A degree of reduction of apparent weight occurs, for example, in elevators. In an elevator, a spring scale will register a decrease in a person's (apparent) weight as the elevator starts to accelerate downwards. This is because the opposing force of the elevator's floor decreases as it accelerates away underneath one's feet.

Apparent 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


Kepler’s 2nd Law Animation

http://kepler_2_area_and_time_int.htm


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

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.


What is the orbital velocity of a satellite near the Earth’s surface?


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 (false) 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

Another Way of Looking at Orbital Velocity


From Kepler’s 2nd Law

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

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


What is the mass of the Sun?

G = 6.67 x 10-11 Nm2/ kg2

r = 1.496 x 1011m - AUP = 3.147 x 107 s

So:Msun = 2 x 1030 kg


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…

What is the altitude of a geostationary orbit calculated from Newton’s formulation of Kepler’s 3rd Law:

G = 6.67 x 10-11 Nm2/ kg2P = 86,164 s (sidereal day)MEarth = 5.97 X 1024 kg

So:

r = 42,164 km above the center of the Earth and the altitude is 35,768 km.

Geosynchronous/Geostationary Orbits

PHYS 3380 - AstronomyAnother Way of Calculating Geosynchronous OrbitFor an orbiting body, the inward and outward forces must equal each other (Newtons 3rd Law) - the centrifugal force from orbital motion has to equal the centripetal force from gravity:

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

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

Plug in the numbers a you get

r = 42,164 km - same as when we used Kepler’s 3rd Law

PHYS 3380 - AstronomyEscape 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 potential energy is converted to kinetic energy.


Potential energy

Gravitational Force - vector r is the unit vector in outward

direction

Gravitational potential energy at distance r from reference point

Kinetic energy

Escape velocity

Starting from the surface of the Earth:r = 6.378 X 106 m, M = 5.97 X 1024 kg, G= 6.67 x 10-11 Nm2/ kg2

v= 11,174 m/s


EllipticalCircular

Parabolic Hyberbolic

Types of Orbits

PHYS 3380 - AstronomyCenter of Mass

Note: the previous calculations assumed that the mass of the orbiting body was much smaller that the central body - center of orbit at center of central body

Newton 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 they 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

PHYS 3380 - Astronomy Einstein1905 - “The Year of Physics”

Submitted doctoral thesis "A New Determination of Molecular Dimensions”

Published five pioneering papers in "Annalen der Physik" - revolutionized physics:

"On A Heuristic Point of View Concerning the Production and Transformation of Light"

- electromagnetic radiation must consist of quantums or photons - explained the photoelectric effect- became the foundation of quantum theory - what he received the Nobel Prize for in 1921

"On the Electrodynamics of Moving Bodies" - special relativity - new interpretation of the conception of space and time - observer can never detect their uniform motion except relative to other objects - coordinate systems- speed of light constant - independent of motion relative to light source

"Does the Inertia of a Body Depend upon its Energy Content?" - the equivalence of mass and energy - E = mc2

PHYS 3380 - AstronomyGeneral Relativity

1916 published "The Foundation of the General Theory of Relativity”

- generalized special theory of relativity- observer cannot distinguish between inertial forces due to acceleration and uniform gravitational forces- gravity is curvature of space-time

- curvature dependent on mass- acceleration of mass dependent on space-time curvature

Numerous implications on astronomy and astrophysics

- orbital motion- black holes- big bang- formation and structure of galaxies

PHYS 3380 - AstronomyProof of General Relativity

Einstein convinced that light deflection by the gravitational field of the sun could be observed during a total solar eclipse

- photograph section of sky where eclipse would occur- during eclipse, photograph same section and measure difference in positions- predicted deflection of 1.75 arcseconds for starlight grazing Sun’s surface

Theory predicted the deflection of light in a gravitational field

Several failed observations of total solar eclipses before proof in 1919- observed eclipse in island of Principe in the Gulf of Guinea in western Africa and Sobral, Brazil- found shift in stars outward from Sun

- 1.61±0.30 arcseconds at Principe- 1.98±0.12 arcseconds at Sobral



Einstein’s eclipse

.

PHYS 3380 - AstronomyProof of General Relativity

Precession of long axis of Mercury’s orbit

Newton’s formulation predicted precession of 5557.62 arcseconds per century

- actually 43.11 arcseconds more - about 29 km past position predicted by Newton per orbit - accumulative - 12,000 km per century - easily measured

Einstein predicted 43.03 arcseconds per century difference

Effect since observed on Venus, Earth, and asteroid Icarus

PHS 3380 - Astronomy

GPS Satellite clock drift - relativistic effect

• satellites move at 3874 m/s - relativistic time dilation means time runs slower on GPS satellite than on Earth

!~ 7.2 microseconds per day• satellite at 20000 km height exposed to a much weaker field of gravitation than the observer on Earth

!gravitational time dilation means clock on board of a satellite is running faster than one on Earth!this effect about six times stronger than time dilation

• frequency standard onboard each satellite given rate offset prior to launch

!makes clock run slightly slower than the desired frequency on Earth!at 10.22999999543 MHz instead of 10.23 MHz

•satellite clock errors are periodically corrected by Control Segment


Light

PHYS 3380 - AstronomyLight

What is light? - A vibration in an electromagnetic field through which energy is transported.

Light as a wave

Light as a particle E = hf photon

f l = c

The dual nature of light or wave-particle duality:

PHYS 3380 - Astronomy Properties of Waves

Period: time to complete one cycle of vibration - from crest to crest or trough to trough

Frequency (f): number of crests passing a fixed pointper second

Frequency= 1/period

Amplitude (a): maximum displacement fromequilibrium

Wave length (λ): distance between successive crests

Speed (of a wave) (s)= wave length x frequencys= λ x f


Light is a vibration in an electromagnetic field through which energy is transported

- a transverse wave - vibration or oscillation is perpendicular to direction of propagation of wave (vs. longitudinal wave -vibration or oscillation is in the same direction as propagation of wave)

So electrons can be manipulated by light. Electrons wiggle up and down as light passes by.

Light as a Wave


Visible light ranges through 7 major colors from long wavelengths (low frequency - red) to short wavelengths (high frequency - violet) - Red, orange, yellow, green, blue, indigo, violet (Roy G Biv)

•For a wave, its speed: s = l x f•But the speed of light is a constant, c.•For light: l x f = c

•The higher f is, the smaller l is, and vice versa. •Our eyes recognize f (or l) as color.


Light as a Particle (Photon)

• Light propagates as quanta of energy called photons• Photons

•move with speed of light•have no mass•are electrically neutral

• Energy of a photon or electromagnetic wave:E = hf = h c/ l

whereh = Planck’s constantf = frequency of a light wave - number of crests passing a fixed point in 1 secondc = velocity of lightl = wavelength of a light wave

• Higher frequency/shorter wavelength - higher energy


The Electromagnetic Spectrum

Most wavelengths of light can not be seen by the human eye.

The visible part of the electromagnetic spectrum lies between ultraviolet and infrared light (between about 400 and 700 nm). The higher the frequency (shorter the wavelength), the higher the photon energy. Radio waves are at the long wavelength end of the spectrum and gamma rays are at the short wavelength end of the spectrum.


Light as Information Bearer

Spectrum of a distant object - a spectrum is the amount of energy or intensity at different wavelengths.

By studying the spectrum of an object, we can learn its:1 Composition2 Temperature3 Velocity

We can separate light into its different wavelengths (spectrum).

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Newton’s 2nd Law Explains the Feather and the Hammer

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