page 1 lecture 3: motion and gravity claire max april 7, 2009 astro 18: planets and planetary...

Lecture 3: Lecture 3: Motion and GravityMotion and Gravity

Claire Max

April 7, 2009

Astro 18: Planets and Planetary Systems

UC Santa Cruz

PracticalitiesPracticalities

• Homework 2 is due Thursday April 9:Homework 2 is due Thursday April 9:– If you have not yet downloaded Homework 2, If you have not yet downloaded Homework 2,

be sure to get a handout in class today. be sure to get a handout in class today. – I will put all lectures, assignments, I will put all lectures, assignments,

homeworks, etc on class web page homeworks, etc on class web page http://www.ucolick.org/~max/Astro18.2009/Astro18.html

• Signing up for this course:Signing up for this course:– If you are not yet officially enrolled but want If you are not yet officially enrolled but want

to be, see me during the break. to be, see me during the break.

Sections start this weekSections start this week

• Section this week: Wed 4 - 5:10 pmSection this week: Wed 4 - 5:10 pm– Interdisciplinary Science Bldg 165Interdisciplinary Science Bldg 165– Topic: Review of math and exponential Topic: Review of math and exponential

notation, order of magnitude estimatesnotation, order of magnitude estimates

• If you can’t make it at that time, see If you can’t make it at that time, see Stefano during the breakStefano during the break

• Bring your own scientific calculatorsBring your own scientific calculators– Must have exponents and logs and powersMust have exponents and logs and powers– If you are not sure, email Stefano: If you are not sure, email Stefano:

[email protected]@ucolick.org

Outline of this lectureOutline of this lecture

• Gravity: historical development of Gravity: historical development of conceptsconcepts

• Velocity and accelerationVelocity and acceleration

• Gravity: Newton’s laws, orbitsGravity: Newton’s laws, orbits

• Thursday: Energy and The Scientific Thursday: Energy and The Scientific MethodMethod

Please remind Please remind me to take a me to take a break at 2:45 break at 2:45

pm! pm!

The Main PointThe Main Point

• Motions of planets, their moons, Motions of planets, their moons, asteroids, and comets can be very asteroids, and comets can be very accurately predicted because of accurately predicted because of underlying physical lawsunderlying physical laws

– Laws of planetary motion discovered by Laws of planetary motion discovered by KeplerKepler

– Laws of gravity and conservation of Laws of gravity and conservation of momentum discovered by Newtonmomentum discovered by Newton

» Kepler’s Laws can be derived from Kepler’s Laws can be derived from Newton’s Laws, which are more generalNewton’s Laws, which are more general

History: How did astronomical History: How did astronomical observations benefit ancient observations benefit ancient societies?societies?

• Keeping track of time and seasons

– for practical purposes ( e.g. agriculture)

– for religious and ceremonial purposes

• Aid to navigation• Ancient Polynesians a spectacular example

Ancient Polynesian Ancient Polynesian NavigatorsNavigators

• Ancient polynesians Ancient polynesians used celestial used celestial navigation (positions navigation (positions of the stars) to sail of the stars) to sail thousands of miles thousands of miles over open ocean over open ocean

• Society Islands, the Society Islands, the Marquesas, Easter Marquesas, Easter Island in the east, the Island in the east, the Hawaiian Islands in Hawaiian Islands in the north, and New the north, and New Zealand in the Zealand in the southwest.southwest.

Ancient people of central Africa (6500 BC) could predict seasons from the orientation of the crescent moon!

Another stunning exampleAnother stunning example

What did ancient What did ancient civilizations achieve in civilizations achieve in astronomy?astronomy?

• Daily timekeeping

• Tracking the seasons and calendar

• Monitoring lunar cycles

• Monitoring planets and stars

• Predicting eclipses

• And more…

– Naming the stars and

constellations

– ....

China: Earliest known records of China: Earliest known records of supernova explosions (1400 B.C.) supernova explosions (1400 B.C.)

Bone or tortoise shell inscription from the 14th century BC (!)

"On the Xinwei day the new star dwindled."

"On the Jisi day, the 7th day of the month, a big new star appeared in the company of the Ho star."

• Greeks were the first people known to make models of nature.

• They tried to explain patterns in nature without resorting to myth or the super-natural. Greek geocentric

model of the Solar System (c. 400 B.C.)

Why does modern science Why does modern science trace its roots to the Greeks?trace its roots to the Greeks?

Underpinnings of the Greek geocentric* model:

Plato

Aristotle

How did the Greeks explain How did the Greeks explain planetary motion?planetary motion?

Earth at the center of the universe

Heavens must be “perfect”: Objects moving on perfect spheres or in perfect circles.

* Sun, stars, planets rotate around the Earth

The Ptolemaic model:

• Hugely successful

• Sufficiently accurate to remain in use for 1,500 years!

• Arabic translation of Ptolemy’s work named Almagest (“the greatest compilation”)

Ptolemy (A.D. 100-170)

The most sophisticated The most sophisticated geocentric model was that of geocentric model was that of PtolemyPtolemy

Ptolemy: lived 100 - 170 AD in Ptolemy: lived 100 - 170 AD in Alexandria Egypt (Greek city at Alexandria Egypt (Greek city at the time)the time)

• Earth at center of universeEarth at center of universe

• Stars moved on spheresStars moved on spheres

• Planets moved on spheres Planets moved on spheres within sphereswithin spheres

• Seems implausible today, Seems implausible today, but gave but gave veryvery good good predictions of positions of predictions of positions of planets, moon, etc given planets, moon, etc given the quality of data the quality of data availableavailable

• Golden Age of Arabic-Islamic science (8th to 13th centuries C.E.)

• Al-Mamun’s House of Wisdom in Baghdad a great center of learning around A.D. 800

• Arabs invented algebra, made major strides in medicine, anatomy, pharmacology, astronomy

• With the fall of Constantinople (Istanbul) in 1453, Eastern scholars headed west to Europe, carrying knowledge that helped ignite the European Renaissance.

The Islamic world built on and The Islamic world built on and enhanced Greek knowledge for 800 enhanced Greek knowledge for 800 yearsyears

Copernicus Copernicus

• Nicolaus Copernicus (1473 – Nicolaus Copernicus (1473 – 1543), Poland1543), Poland

• First European astronomer to First European astronomer to formulate a modern heliocentric formulate a modern heliocentric theory of the solar system. theory of the solar system.

• A mathematician, astronomer, A mathematician, astronomer, jurist, physician, classical jurist, physician, classical scholar, Catholic cleric, scholar, Catholic cleric, governor, administrator, governor, administrator, diplomat, economist and diplomat, economist and military leader (!)military leader (!)

Copernicus, continuedCopernicus, continued

• Amid Copernicus' extensive other Amid Copernicus' extensive other responsibilities, astronomy was little more responsibilities, astronomy was little more than an avocation.than an avocation.

• Heliocentric (sun at the center) theory had Heliocentric (sun at the center) theory had been formulated by Greeks and Muslims been formulated by Greeks and Muslims centuries before Copernicus. centuries before Copernicus.

• But his reiteration that the sun (rather than But his reiteration that the sun (rather than the Earth) is at the center of the solar system the Earth) is at the center of the solar system is considered among the most important is considered among the most important landmarks in the history of western science.landmarks in the history of western science.

1500’s AD: Two models existed 1500’s AD: Two models existed that made predictions about that made predictions about orbits of planetsorbits of planets

• Ptolemy: Earth at center of “universe”Ptolemy: Earth at center of “universe”

• Copernicus: Sun at center of “universe”Copernicus: Sun at center of “universe”

Copernicus’ Sun-centered model Copernicus’ Sun-centered model was was notnot much more accurate than much more accurate than Ptolemy’sPtolemy’s

• By that time, Ptolemaic model was noticeably By that time, Ptolemaic model was noticeably inaccurateinaccurate

• Copernicus concluded that a Sun-centered Copernicus concluded that a Sun-centered Solar System could predict planetary motions Solar System could predict planetary motions more easilymore easily

• But he believed that planetary orbits must be But he believed that planetary orbits must be circles (they aren’t)circles (they aren’t)

• His model’s predictions were His model’s predictions were notnot much more much more accurate than Ptolemy!accurate than Ptolemy!

What was needed What was needed was high quality was high quality datadata

• Tycho Brahe - Danish, 1546-1601Tycho Brahe - Danish, 1546-1601– VeryVery accurate naked-eye observations of accurate naked-eye observations of

positions of planets and starspositions of planets and stars– Persisted for three decades, kept careful Persisted for three decades, kept careful

recordsrecords– Couldn’t explain why his data looked the way Couldn’t explain why his data looked the way

they did, but he hired a young apprentice they did, but he hired a young apprentice who who did did explain it:explain it:

• Johannes Kepler - German, 1571-1630Johannes Kepler - German, 1571-1630– Realized that if he tried to predict position of Realized that if he tried to predict position of

Mars using circular orbits, it didn’t fit dataMars using circular orbits, it didn’t fit data– Abandoned circular orbits, developed a Abandoned circular orbits, developed a

theory using ellipsestheory using ellipses

Kepler’s first lawKepler’s first law

• The orbit of The orbit of each planet each planet around the Sun around the Sun is an ellipse is an ellipse with the Sun at with the Sun at one focusone focus

© Nick Strobel

Difference between ellipse Difference between ellipse and circleand circle

• Eccentricity: Eccentricity: – (distance from (distance from

center to focus) ÷ center to focus) ÷ (semi-major axis)(semi-major axis)

• Eccentricity = 0:Eccentricity = 0:– a circlea circle

• Eccentricity = 1:Eccentricity = 1:– a very flat ovala very flat oval

• Ellipse is specified by Ellipse is specified by its semi-major axis its semi-major axis and eccentricity and eccentricity

© Nick Strobel

More about ellipsesMore about ellipses

• Ellipse applet

• The equation of an ellipse, centered on the point (0,0) which passes through the points (a,0) and (0,b) and whose major and minor axes are parallel to the x and y axes is

• The eccentricity is

x2

a2+

y2

b2 =1

e= 1−b2

a2

Kepler’s second lawKepler’s second law

• As a planet moves around its orbit, it sweeps out equal As a planet moves around its orbit, it sweeps out equal areas in equal timesareas in equal times

• Consequence: planet or comet moves fastest when it is Consequence: planet or comet moves fastest when it is closest to Sunclosest to Sun

– Kepler did not have a way to explain WHYKepler did not have a way to explain WHY

Speed of a planet in an Speed of a planet in an elliptical orbitelliptical orbit

• Moves fastest when close to Sun, Moves fastest when close to Sun, slowest when farther awayslowest when farther away

• Intuitively: gravity is weaker when Intuitively: gravity is weaker when farther from the Sunfarther from the Sun

• Velocity applet

Kepler’s third lawKepler’s third law

• Describes how a planet’s orbital period (in years) is Describes how a planet’s orbital period (in years) is related to its average distance from the Sun (in related to its average distance from the Sun (in Astronomical Units, or AU)Astronomical Units, or AU)

– Earth-Sun average distance is 1 AUEarth-Sun average distance is 1 AU

(orbital period in years)(orbital period in years)2 2 = (average dist. in AU) = (average dist. in AU)3 3

The more distant a planet is from the Sun, the longer its The more distant a planet is from the Sun, the longer its orbital period. orbital period.

– Distant planets move at slower speedsDistant planets move at slower speeds

p2 =a3

Many good web sites for Many good web sites for interactive animations of interactive animations of Kepler’s lawsKepler’s laws

• http://www.astro.utoronto.ca/~zhu/ast210/kepler.html

• http://www.abdn.ac.uk/physics/ntnujava/Kepler/Kepler.html

• http://csep10.phys.utk.edu/guidry/java/kepler/kepler.html

• Try out at least one of these!

ConcepTest 1ConcepTest 1

• Kepler’s 3rd Law (that the period Kepler’s 3rd Law (that the period squared is proportional to the semi-squared is proportional to the semi-major axis cubed) does NOT apply to major axis cubed) does NOT apply to the motion ofthe motion of

a) a satellite around the Eartha) a satellite around the Earthb) one star around another in a binary b) one star around another in a binary

systemsystemc) one galaxy around anotherc) one galaxy around anotherd) all of the above applyd) all of the above apply

How did Galileo solidify the How did Galileo solidify the Copernican revolution?Copernican revolution?

Galileo (1564-1642) overcame 3 major objections to Copernican view: 1. Earth could not be moving

because objects in air would be left behind.

2. Non-circular orbits are not “perfect” as heavens should be.

3. If Earth were really orbiting Sun,we’d detect stellar parallax.

• We would see nearby stars appear to move relative to background stars.

Galileo’s experiments showed that objects in air would stay with a moving Earth.

Overcoming the first Overcoming the first objection (nature of objection (nature of motion) motion)

• Aristotle thought all objects naturally come to rest.

– Experience based on horse pulling a heavy cart over a rutted ancient road!

• Galileo showed that objects will stay in motion unless a force acts to slow them down

– Became Newton’s first law of motion

Overcoming the second Overcoming the second objection (heavenly objection (heavenly perfection)perfection)

• Using his telescope, Galileo saw:

• Sunspots on Sun (“imperfections”)

• Mountains and valleys on the Moon (proving it is not a perfect sphere)

• Tycho thought he had measured stellar distances, so lack of parallax seemed to rule out an orbiting Earth.

• Galileo showed that stars must be much farther than Tycho thought — in part by using his telescope to see that the Milky Way is countless individual stars.

If stars were much farther away, then lack of detectable parallax was no longer so troubling.

Overcoming the third Overcoming the third objection (parallax)objection (parallax)

Galileo saw four moons orbiting Jupiter, proving that not all objects orbit the Earth.

Felt that this justified the hypothesis that the Earth orbits the Sun.

He kept good notes and published his results!

Galileo and the moons of Galileo and the moons of JupiterJupiter

• Realized that the same Realized that the same physical laws that operate physical laws that operate on Earth also operate in the on Earth also operate in the heavensheavens one one universeuniverse

• Discovered laws of motion Discovered laws of motion and gravityand gravity

– Explain Explain whywhy Kepler’s laws Kepler’s laws workwork

• Much more: Experiments Much more: Experiments with light; first reflecting with light; first reflecting telescope, calculus… telescope, calculus…

Sir Isaac Newton (1642-1727)

How did Newton change our How did Newton change our view of the universe?view of the universe?

Three ways to describe motionThree ways to describe motion

• Speed: Rate at which object moves

example: 10 meters/sec

• Velocity: Speed and direction

example: 10 meters/sec, due east

• Acceleration: Change in velocity example: speed/time

(meters/sec2) with direction

Speed = distance

time

First, some definitions and First, some definitions and conceptsconcepts

• SpeedSpeed: rate of change of distance with time: rate of change of distance with time – miles/hourmiles/hour– km/seckm/sec– furlongs per fortnightfurlongs per fortnight

• VelocityVelocity: speed in a given : speed in a given directiondirection– where and are vectorswhere and are vectors– a vector quantity has a magnitude and a directiona vector quantity has a magnitude and a direction

• AccelerationAcceleration: rate of change of velocity with time: rate of change of velocity with time– – units: km/secunits: km/sec22, m/sec, m/sec22, miles/hour, miles/hour22 in a specific direction in a specific direction– in other words, change in (miles/hour) per hourin other words, change in (miles/hour) per hour

rv =Δ

rx / Δt

rv

rx

ra =Δ

rv / Δt

v =Δ x / Δt

Vector

The Acceleration of GravityThe Acceleration of Gravity

• All falling objects accelerate at the same rate (not counting friction of air resistance).

• On Earth, acceleration of gravity is g ≈ 10 meters/sec2

– Speed increases by 10 meters/sec with each second of falling.

The Acceleration of Gravity (The Acceleration of Gravity (gg))

• Galileo showed that Galileo showed that gg is the is the samesame for all for all falling objects, falling objects, regardless of their regardless of their mass.mass.– Dropped bodies from Dropped bodies from

the leaning tower of the leaning tower of PisaPisa

• Heavy bodies fall the Heavy bodies fall the same way as light same way as light bodiesbodies (if friction (if friction doesn’t matter)doesn’t matter)

The Acceleration of Gravity (The Acceleration of Gravity (gg))

• Galileo showed that Galileo showed that gg is the is the samesame for all for all falling objects, falling objects, regardless of their regardless of their mass.mass.– Dropped bodies from Dropped bodies from

the leaning tower of the leaning tower of PisaPisa

• Heavy bodies fall the Heavy bodies fall the same way as light same way as light bodiesbodies (if friction (if friction doesn’t matter)doesn’t matter)

Aristotle Galileo

Galileo said all three of these Galileo said all three of these balls will hit the ground at balls will hit the ground at the same timethe same time

Trajectory when you throw a Trajectory when you throw a ball up into the airball up into the air

• Path is in shape of Path is in shape of a parabolaa parabola

• First goes up, but First goes up, but more and more more and more slowlyslowly

• Then turns around Then turns around and falls down to and falls down to the groundthe ground

• Horizontal speed Horizontal speed stays stays constantconstant

Acceleration always points Acceleration always points downwards towards grounddownwards towards ground

Velocity changes throughout Velocity changes throughout the fallthe fall

• Horizontal Horizontal velocity stays velocity stays constantconstant

• Vertical Vertical velocity velocity continually continually decreases due decreases due to gravitational to gravitational acceleration acceleration downwardsdownwards

How is mass different from How is mass different from weight?weight?

• Mass – the amount of matter in an object

• Weight – the force that acts upon an object

You are weightless in free-fall!

You fall at same rate as the elevator

Concept Test 2Concept Test 2

A.A. My weight is the same, my mass is less.My weight is the same, my mass is less.

B.B. My weight is less, my mass is the same.My weight is less, my mass is the same.

C.C. My weight is more, my mass is the My weight is more, my mass is the same.same.

D.D. My weight is more, my mass is less.My weight is more, my mass is less.

On the Moon: On the Moon:

• There is gravity in space

• Weightlessness is due to a constant state of free-fall

Why are astronauts weightless Why are astronauts weightless in space?in space?

What are Newton’s three laws of What are Newton’s three laws of motion?motion?

Newton’s first law of motion:Newton’s first law of motion: An object moves at An object moves at constant velocity unless constant velocity unless a net force acts to a net force acts to change its speed or change its speed or direction.direction.

Newton’s second law of motionNewton’s second law of motion

Force = mass Force = mass accelerationacceleration

rF =m

ra

Newton’s third law of motion:Newton’s third law of motion:

For every For every force, there is force, there is an an equal and equal and oppositeopposite reaction force.reaction force.

Newton’s Universal Law of Newton’s Universal Law of GravitationGravitation

1.1. Every mass attracts every other mass.Every mass attracts every other mass.

2.2. Attraction is Attraction is directlydirectly proportional to the proportional to the product of their masses.product of their masses.

3.3. Attraction is Attraction is inverselyinversely proportional to the proportional to the squaresquare of the distance between their centers. of the distance between their centers.

Center of MassCenter of Mass

• Because of momentum Because of momentum conservation, two conservation, two objects orbit around objects orbit around their “center of mass”their “center of mass”

Newton’s universal law of Newton’s universal law of gravitationgravitation

• Force of gravity between two bodies, 1 and 2Force of gravity between two bodies, 1 and 2

• G is a constant of nature, the “gravitational G is a constant of nature, the “gravitational constant”constant”

• mm11 and m and m22 are the masses of two bodies whose are the masses of two bodies whose gravity we are consideringgravity we are considering

• d is the distance between the two bodiesd is the distance between the two bodies

Fgravity =Gm1m2

d2

⎛⎝⎜

⎞⎠⎟

Implications of “inverse Implications of “inverse square law”square law”

• Gravitational force decreases as you get Gravitational force decreases as you get farther away from an objectfarther away from an object

• Falls off as the Falls off as the squaresquare of the distance away of the distance away

• If you go twice as far away, force is 4 x If you go twice as far away, force is 4 x smallersmaller

Fgravity =Gm1m2

d2

⎛⎝⎜

⎞⎠⎟∝ d−2

Why do all objects fall at the same Why do all objects fall at the same rate? rate?

arock =Fg

Mrock

Fg =GMEarthMrock

REarth2

arock =Fg

Mrock

= GMEarthMrock

REarth2

⎛

⎝⎜⎞

⎠⎟ Mrock=G

MEarth

REarth2

• The gravitational acceleration of an object The gravitational acceleration of an object like a rock does not depend on its mass like a rock does not depend on its mass because because MMrockrock in the equation for acceleration in the equation for acceleration cancels cancels MMrockrock in the equation for gravitational in the equation for gravitational forceforce

Newton’s version of Newton’s version of Kepler’s third lawKepler’s third law

• In this form, applies to any pair of orbiting In this form, applies to any pair of orbiting objects with period objects with period pp and average separation and average separation aa

• So Newton provided a So Newton provided a physicalphysical reasonreason why the why the period and semi-major axis of a planet are period and semi-major axis of a planet are relatedrelated

p2 =4π 2

G m1 + m2( )

⎡

⎣⎢

⎤

⎦⎥×a3

Remarkable consequenceRemarkable consequence

• We can calculate mass of the Sun just by We can calculate mass of the Sun just by knowing the length of a year and the size of knowing the length of a year and the size of the Earth’s orbit (150 million km)the Earth’s orbit (150 million km)

(mEarth + msun) ≈msunbecausemEarth<<msun

pEarth( )2=

4π 2

Gmsun

aEarth( )3

msun =4π 2

GaEarth( )

3

pEarth( )2 =2 ×1030 kg=2 ×1033g

Acceleration due to gravity at Acceleration due to gravity at the surface of the Earththe surface of the Earth

• mm11 = mass of Earth, d = radius of Earth R = mass of Earth, d = radius of Earth REE

• At surface of Earth, acceleration due to At surface of Earth, acceleration due to gravity is gravity is

Fgravity =Gm1m2

d2

⎛⎝⎜

⎞⎠⎟=

Gmearth

RE2

⎛

⎝⎜⎞

⎠⎟m2 =m2g

g =Gmearth

RE2

⎛

⎝⎜⎞

⎠⎟=9.8meters/sec2

ConcepTest 3ConcepTest 3

• You hold a ball in your hand at a fixed height and release it. Its initial acceleration after you let go is

– 1. up– 2. zero– 3. down

What have we learned?What have we learned?

• What determines the strength of gravity?What determines the strength of gravity?

– Directly proportional to the Directly proportional to the productproduct of the of the masses (M masses (M xx m) m)

– InverselyInversely proportional to the proportional to the squaresquare of the of the separationseparation

• How does Newton’s law of gravity allow us to How does Newton’s law of gravity allow us to extend Kepler’s laws?extend Kepler’s laws?

– Applies to other objects, not just planets.Applies to other objects, not just planets.– Includes unbound orbit shapes: parabola, Includes unbound orbit shapes: parabola,

hyperbolahyperbola– Can be used to measure mass of orbiting Can be used to measure mass of orbiting

systems.systems.

Newton’s second lawNewton’s second law

• Definition: momentum = mass x Definition: momentum = mass x velocityvelocity– – momentum has a momentum has a directiondirection because does because does

• Newton’s second law:Newton’s second law:

Force = mass x acceleration = rate of change of Force = mass x acceleration = rate of change of momentummomentum

rp =m

rv

rv

rF =m

ra=rateofchangeof m

rv( )withtime=

Δ(mrv)

Δt=m

Δrv

Δt

Consequences of Newton’s Consequences of Newton’s second lawsecond law

• F = m aF = m a

• If there is no force (F = 0), there is no If there is no force (F = 0), there is no acceleration (mv = momentum = constant)acceleration (mv = momentum = constant)

• Momentum is conserved if there are no forces Momentum is conserved if there are no forces actingacting– Implies that objects move with constant velocity (!)Implies that objects move with constant velocity (!)

• If force is due to gravity, for example near If force is due to gravity, for example near surface of the Earth, surface of the Earth, – acceleration = acceleration due to gravityacceleration = acceleration due to gravity– a = ga = g– Force on an object of mass m = m gForce on an object of mass m = m g

Conservation of MomentumConservation of Momentum

• The total The total momentum of momentum of interacting interacting objects cannot objects cannot change unless an change unless an external force is external force is acting on them acting on them

What keeps a planet rotating What keeps a planet rotating and orbiting the Sun?and orbiting the Sun?

Conservation of Angular Conservation of Angular MomentumMomentum

• The angular momentum of an object The angular momentum of an object cannot change unless an external cannot change unless an external twisting force (torque) is acting on ittwisting force (torque) is acting on it

• Earth experiences no twisting force as Earth experiences no twisting force as it orbits the Sun, so its rotation and it orbits the Sun, so its rotation and orbit will continue indefinitelyorbit will continue indefinitely

• Intuitive explanation of Kepler’s law: Intuitive explanation of Kepler’s law: planets that are close to the Sun (small planets that are close to the Sun (small radius) must move faster in their orbitsradius) must move faster in their orbits

angular momentum = mass x velocity x radius

Angular momentum conservation also explains why skater spins faster as she pulls in her arms

angular momentum = mass x velocity x radius

See you Thursday!See you Thursday!

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