gravity & planets. our ancestors envisioned the earth at the center of the universe and...
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Gravity & PlanetsGravity & Planets
Our ancestors envisioned the Our ancestors envisioned the earth at the center of the earth at the center of the universe and considered the universe and considered the stars to be fixed on a great stars to be fixed on a great revolving crystal sphere.revolving crystal sphere.
The observation of planetary The observation of planetary motions depends on the frame motions depends on the frame of reference of the viewer.of reference of the viewer.
Historical Background
Early Theories Early Theories
Geocentric Theory: Ptolemy~140 AD•Motion of planets from the point of view of the observer on EarthHeliocentric Theory: Copernicus, 1543.•Same motions but from the point of view of the observer on the SunWe use the heliocentric theory
due to its simpler mathematics.
Tycho BraheTycho Brahe (1546- (1546-1601)1601)
The Royal Astronomer for King Fredrick II of Denmark,
•measured accurately the positions of the planets• Charted planets motion for 20 years • Accurate to 1/60 of a degree • “Greatest Observational Genius”. • 35 years before the telescope.
Johannes KeplerJohannes Kepler
• Imperial Mathematician for Holy Roman Emperor Rudolf II•In 1600 became Brahe’s assistant. •Stole Brahe’s data after his death in 1601,
spent next 15 years analyzing it . • Deduced three laws of planetary motion• Apply to any system composed of a body
revolving about a more massive body•Examples: moons, satellites, comets.
Kepler’s 1Kepler’s 1stst Law Law
Kepler’s first law ( the Law of Orbits)Planets move in elliptical orbits with the Sun at one of the focal points.
Kepler’s 2Kepler’s 2ndnd Law Law
Kepler’s second law ( the Law of Areas)A line from the Sun to a planet sweeps out equal areas in equal lengths of time.
dA = k dt
Kepler’s 3Kepler’s 3rdrd Law Law
The Law of Periods or the Harmonic LawThe square of the orbital period of a planet is directly proportional to the cube of the average distance of the planet from the sun.
T2 r3
The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their mean distances from the sun.
(T1/ T2) 2 = ( r1/ r2)
3
Finding Kepler’s Third Finding Kepler’s Third Law with a Graphing Law with a Graphing
CalculatorCalculatorJames Metz, The Physics Teacher, April 2000James Metz, The Physics Teacher, April 2000 Kepler struggled for 15 years to learn the
relationship between the period (T) and the radius (r) of the orbit of the planets, but you can easily determine this relationship with a graphing calculator. Using a TI-83Press the STAT keySelect EDITEnter the radius of orbit of each planet in the table below in L1The corresponding period in L2 After entering the dataPress STATSelect CALCRequest the power regression equation
Kepler’s ValuesPlanet Radius (r) of orbit of
planet in A.U.Period (T) in days
Mercury 0.389 87.77
Venus 0.724 224.70
Earth 1.000 365.25
Mars 1.524 686.98
Jupiter 5.200 4332.62
Saturn 9.510 10 759.20
The calculator should return y = 364.2740081x1.5
Since the period is y and the radius is x, T = 364.2740081x3/2
Thus, the period of the a planet is directly proportional to the 3/2 power of its orbital radius, Kepler’s Third Law of Planetary Motion.
Isaac NewtonIsaac Newton
• Recognized that a force must be acting on the planets; otherwise, their paths would be straight lines.• Force Sun (Kepler’s 2nd Law) • Force decreases with the square of the distance (Kepler’s 3rd Law)
•Compared the fall of an apple with the fall of the moon. •The moon falls in the sense that it falls away from the straight line it would follow if there were no forces acting on it. •Therefore, the motion of the moon and the apple were the same motion.•Showed, everything in the universe follows a single set of physical laws.•Law of Universal Gravitation.
Newton Continued…Gravity
Universal Law of Universal Law of Gravitation ProofGravitation Proof
Assumptions:•Must conform to equations for circular motion•Used Kepler’s laws as evidence Fnet = ma
Newton’s 3rd Law symmetryForce of gravity ~ to both the massesFnet = Fgravity = G m1m2 = m1 a ac = v2
r2 r
Universal Law of Gravitation Universal Law of Gravitation ProofProof
Gravitational force centripetal force
=
Where So,
Rearranging, Law of Periods
r
vm 21
221
r
mGm
T
rv
2
2
21
221 4
T
rm
r
mGm
GmR
T 2
3
2 4
Law of Universal Law of Universal Gravitation Gravitation (1666) (1666)
Every particle in the universe attracts every other particle with a force. This force is: • proportional to the product of their masses• inversely proportional to the square of the distance between them. • Acts on a line joining the two particles.
G = 6.67 x 10-11 Nm2/kg2
Discovered by Henry Cavendish in 1798
221
r
mGmF
For a group of particles, the net effect is the sum of the individual effects.
Find the individual gravitational force Then find the net force vectorially.F = Σ F Gravitation Near Earth’s surfaceF = F
ma =2r
GMg
2r
GMm
Principle of SuperpositionPrinciple of Superposition
Note: a = g
Acceleration due to Gravity Acceleration due to Gravity on different Planets on different Planets
Consider an object thrown upwardConsider an object thrown upward
at the maximum height vat the maximum height vff=0=0
If the same object was thrown upward with the same If the same object was thrown upward with the same initial speed gd initial speed gd constant constant
So,So,
advv of 222
gdvo 22
2211 dgdg
1
2
2
1
d
d
g
g
Acceleration due to Gravity Acceleration due to Gravity on different Planetson different Planets
• ExampleExampleA person can jump 1.5m on the A person can jump 1.5m on the earth. How high could the person earth. How high could the person jump on a planet having the twice jump on a planet having the twice the mass of the earth and twice the mass of the earth and twice the radius of the earth? the radius of the earth?
Gravitational Potential Energy (GPE) of two point sources
For particles not on the Earth’s surface, the GPE decreases when the separation decreases
U = - GMm rE
Gravitational Field Strength Gravitational Field Strength (g.f.s)(g.f.s)
The force acting on a 1kg mass at a The force acting on a 1kg mass at a specific point in a gravitational specific point in a gravitational field.field.
Units: N/kgUnits: N/kg
From Newton’s 2From Newton’s 2ndnd law F/m = a law F/m = a
So, another name for g.f.s is the So, another name for g.f.s is the acceleration due to gravity (g)acceleration due to gravity (g)
Variation of g with distance Variation of g with distance from a point mass from a point mass
• g.f.s at a point p is g.f.s at a point p is
• Variation of g with distance from the Variation of g with distance from the center of a uniform spherical mass of center of a uniform spherical mass of radius, R radius, R
2r
GMg
point p is given by
Variation of g on a line Variation of g on a line joining the centers of two joining the centers of two
point masses point masses
• If mIf m11 > m > m22 then then
Gravitational Potential, V Gravitational Potential, V
The potential at a point in a gravitational field is The potential at a point in a gravitational field is equal to the work done bringing a equal to the work done bringing a 1kg 1kg mass mass fromfrom infinity infinity to that point.to that point.
• Units: J/kgUnits: J/kg
• W = Fd W = Fd (the force is not of constant magnitude).• The force varies with distance from the body (Newton’s The force varies with distance from the body (Newton’s
law of universal gravitation) and law of universal gravitation) and it can be shownit can be shown**
• where, M is the mass of the body (the earth, in this where, M is the mass of the body (the earth, in this
case).case).
*a very useful phrase if you a) don't know how to do something or b) can't be bothered to do it !*a very useful phrase if you a) don't know how to do something or b) can't be bothered to do it !
Gravitational Potential, V Gravitational Potential, V
• A body at infinity, has zero gravitational potential.
• A body normally falls to its lowest state of potential (energy) as r decreases, the potential decreases.
• We can do this by including a negative sign in the above equation.
• As r decreases, V decreases (becomes a greater negative quantity).
Source:http://www.saburchill.com/physics/chapters/0007.html
Escape VelocityEscape Velocity
• The velocity needed to overcome (escape) The velocity needed to overcome (escape) the gravitational pull of a planet.the gravitational pull of a planet.
• As the body is moving away from the planet, As the body is moving away from the planet, it is losingit is losing kinetic energy kinetic energy and gaining and gaining potential energy.potential energy.
• To completely escape from the gravitational attraction of the planet, the body must be given enough kinetic energy to take it to a position where its potential energy is zero.
Escape VelocityEscape Velocity
• The potential energy possessed by a body The potential energy possessed by a body of mass m, in a gravitational field is given of mass m, in a gravitational field is given byby
• If the field is due to a planet of mass M If the field is due to a planet of mass M and radius R, then the escape velocity can and radius R, then the escape velocity can be calculated as follows:be calculated as follows:
• So,So,
• as g = GM/R² as g = GM/R²
G.P.E. = Vm
ΔK.E. = ΔG.P.E.
E
escr
GMmmv 2
21R
GMvesc
2
gRvesc 2
SatellitesSatellites
Satellites: Orbits & EnergySatellites: Orbits & Energyusing Newton’s second law using Newton’s second law
(F=ma)(F=ma)
where vwhere v22/r is the /r is the
centripetal acceleration.centripetal acceleration.
Solving for vSolving for v22
So,So,
Total Mechanical Energy of a satellite is:Total Mechanical Energy of a satellite is:
r
GMmU
r
vm
r
GMmF
2
2
r
GMmmvKE
22
21 2
UKE
r
GMm
r
GMmUKEME
2 r
GMmME
2
r
GMv 2
Satellites: Orbits & EnergySatellites: Orbits & EnergyThis is for a circular This is for a circular
orbit.orbit.
• For a satellite in an elliptical orbit For a satellite in an elliptical orbit with a semi-major axis with a semi-major axis aa
where a was where a was substituted for rsubstituted for r
a
GMmE
2
ME = -KE
Energy Changes in a Gravitational Energy Changes in a Gravitational FieldField
• A mass placed in a gravitational field A mass placed in a gravitational field experiences a force. If no other force acts, the experiences a force. If no other force acts, the total energy will remain constant but energy total energy will remain constant but energy might be converted from g.p.e. to k.e.might be converted from g.p.e. to k.e.
• If the mass of the planet is M and the radius of If the mass of the planet is M and the radius of the orbit of the satellite is r, then it can easily the orbit of the satellite is r, then it can easily be shown that the speed of the satellite, v, is be shown that the speed of the satellite, v, is given bygiven by
• if r decreases, v must increase.if r decreases, v must increase. • If the satellite’s mass is m, then the kinetic If the satellite’s mass is m, then the kinetic
energy, K, possessed by the satellite is given byenergy, K, possessed by the satellite is given by
Energy Changes in a Gravitational Energy Changes in a Gravitational FieldField
• the potential energy, P, possessed by the the potential energy, P, possessed by the satellite is given bysatellite is given by
• These equations show:These equations show:• that if r decreases, that if r decreases, K increases but P decreasesK increases but P decreases
(becomes a bigger negative number) (becomes a bigger negative number) • the decrease in P is the decrease in P is greater thangreater than the increase in K. the increase in K.
• Therefore, to fall from one orbit to a lower orbit, Therefore, to fall from one orbit to a lower orbit, the total energy must decrease. In other words, the total energy must decrease. In other words, some some work must be donework must be done to decrease the to decrease the energy of the satellite if it is to fall to a lower energy of the satellite if it is to fall to a lower orbit. orbit.
Energy Changes in a Gravitational Energy Changes in a Gravitational FieldField
• The work done, w, is equal to the change in the The work done, w, is equal to the change in the total energy of the satellite, total energy of the satellite, w = w = ΔΔK + K + ΔΔPP..
• This work results in a conversion of energy from This work results in a conversion of energy from gravitational potential energy to internal gravitational potential energy to internal energy of the satellite (it makes it hot!)energy of the satellite (it makes it hot!)..
• Air resistance can thus reduce the speed of the Air resistance can thus reduce the speed of the satellite satellite along its orbitalong its orbit. This allows the . This allows the satellite to fall towards the planet. As it falls, it satellite to fall towards the planet. As it falls, it gains speedgains speed..
• So, if a viscous drag (air resistance) acts on a So, if a viscous drag (air resistance) acts on a satellite, it willsatellite, it will• decrease the radiusdecrease the radius of the orbit of the orbit • increase the speedincrease the speed of the satellite in it’s new orbit. of the satellite in it’s new orbit.
Energy Changes in a Gravitational Energy Changes in a Gravitational FieldField
• In principle, the satellite could settle in a In principle, the satellite could settle in a lower, faster orbit but in practice it will lower, faster orbit but in practice it will usually be falling to a region where the usually be falling to a region where the drag is drag is greatergreater. It will therefore continue . It will therefore continue to move towards the planet in a to move towards the planet in a spiralspiral path.path.