Lecture 23: MOI

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Lecture 23 MOI Torque
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Kinetic energy of Rotation

• K sum of ½ m v2 for all parts of the body
• ? Moment of inertia I
• ? K ½ I ?2

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Example Two objects connected with a massless
rod ? I

• Mass of 1 kg at -2m, of 4kg at 1m
• ? I 1kg (-2m)2 4kg (1m)2
• 8 kg m2
• Different I around different axis!
• Example rotate around midpoint
• I 11.25 kg m2

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Post-lecture Exercise (10.1 10.6)  

• Two masses of 2 kg are connected by a massless 1m
  rod and rotated around their center of mass with
  a period of 2s. Calculate the rotational kinetic
  energy of this configuration.
• K   9.87 J
• Use Eqs. (10-33) (10-34) I 2kg(-0.5m)2
  2kg(0.5m)2 1 kg m2
• ? 2p/T p Hz, so K ½ I ? 2 ½ p2 J

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Determining MOI

• Integral
• Table
• Parallel-axis theorem

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Torque

• Torque is force times lever arm
• Lever arm is distance to rotation axis along a
  direction perpendicular to the force
• Later t r x F
• t r F sin f

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Pre-lecture Exercise (10.7 10.10)

• In the simulation Torque how large does the red
  force have to be (if the red position is negative
  1m and all other quantities at their initial
  values) such that the sum of the torques produced
  by the blue and the red forces is zero, i.e. that
  there is no net force, and hence no net angular
  acceleration, and hence no rotational motion of
  the bar? 
• Fred 10N
• Torque force times lever arm. If the lever arm
  is half as long, we need twice as much force f
  10 N

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Newton II for Rotation
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Work and Rotational Kinetic Energy

• W ? t d?
• P dW/dt t?

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Lecture 24 General Rotations
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Worksheet Torque and angular acceleration
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Rotation plus Translation

• Rolling is a combination of motion of the COM and
  rotation about the COM
• Point of contact remains stationary, while point
  on top of wheel moves with twice the velocity of
  the COM.

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Rolling as pure Rotation

• Rolling can also be viewed as a pure rotation
  around the point of contact with floor
• Need parallel axis theorem to calculate correct
  MOI
• Two parts represent contributions from rotation
  and translation to KE

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Forces of rolling

• Friction is static, since point of contact is
  stationary.
• Rolling down a ramp
• Can calculate is in linear coordinates static
  friction and mg sin ß determine linear acc.
• Torque due to static friction acting at radius R
  determines angular acc. torque I a

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Torque as a Vector

•  

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Pre-lecture Exercise (11.1 11.6)

• What is the direction of the torque produced by a
  force pointing in the SW direction and at a point
  2m directly below the origin?
• a. SW b. Down
• c. NE d. NW
• e. 45 degrees upward from West
• f. None of the above.
• Answer torque r x F, where r is in k, and F
  in i j direction, so ij or NW or d)

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A ladybug sits at the outer edge of a
merry-go-round, that is turning and slowing down
due to a force exerted on its edge. At the
instant shown, the torque on the disc is pointing
in

• -z direction
• -y direction
• y direction
• z direction

z
y
F
x
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Reminder Vector Product
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What is A x A ?

• Zero
• A
• -A
• A2

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Lecture 25 Angular Momentum
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Angular Momentum

•  

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A ladybug sits at the outer edge of a
merry-go-round, that is turning and slowing down
due to a force exerted on its edge. The angular
momentum of the bug is pointing in

• -z direction
• -y direction
• y direction
• z direction

z
y
F
x
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Pre-lecture Exercise (11.7-11.10)

• What is the direction of the angular momentum
  (around the origin) of a particle located 2m
  directly above the origin (i.e. 2m along the z
  axis) with a momentum vector in the SW (i.e.
  ij) direction?
• a. SW b. Down
• c. NE d. NW
• e. 45 degrees upward from West
• f. None of the above.
• Answer l r x p, where r is in k, and p in i
  j direction, so i-j or SE so f)

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Angular Momentum of Rigid Body about fixed axis

• L sum over angular momenta of parts
• Use l r p m r v m r (r ?)
• ? L I ?

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Dynamics Transliteration

• Mass ? Moment of inertia
• Force ? Torque
• Momentum ? Angular Momentum
• Newton II
• Momentum Conservation ? Angular Momentum
  Conservation

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Angular Momentum Conservation

• Angular momentum is conserved if no external net
  torque is present
• Demos turntable weights, turntable and
  bikewheel, bikewheel spinning on rope

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Post-lecture Exercise (11.7 11.10)

• In the sample problem on page 285, what are the
  magnitude and direction of the net angular
  momentum L about point O of the two-particle
  system if the velocity of particle two is
  reversed (180 degrees direction change)? The
  direction is going to be either out of the page
  (positive L) or into the page (negative L).
• Answer The only change is the sign of the vector
  l2, so L (10 8) kg m/s2 18 kg m/s2

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Lecture 26 Rotational Energy
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Pre-lecture Exercise (11.10 11.12)

• By what factor does the spinning volunteers
  period of rotation (p. 291) change if he is able
  to reduce is moment of inertia by a factor of
  1.5? (Hint your answer should be smaller than
  one if the period is reduced, and bigger than one
  if the period gets longer.)
• Answer I ? const, so if I goes down by 1.5, ?
  goes up by 1.5, so the period goes down by 1.5,
  or f 1/1.5 2/3 0.6667.

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Precession of a Gyroscope

• Demo Little Gyroscope
• Demo Bike Wheel on stick
• Rate of precession becomes larger as wheel slows
  down

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Gravitation
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History

• Ptolemy
• Copernicus
• Brahe
• Galilei
• Kepler
• Newton

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From Galileo to Newton - the Birth of Modern
Science
1609 1687
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Precursor Nicolas Copernicus (14731543)

• Rediscovers the heliocentric model of Aristarchus
• Planets on circles
• needs 48(!!) epicycles to explain
• different speeds of planets
• Not more accurate than Ptolemy
• Major Work De Revolutionibus Orbium
  Celestium
• (published posthumously)

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Geocentric vs Heliocentric How do we know?

• Is the Earth or the Sun the center of the solar
  system?
• How do we decide between these two theories?
• Invoke the scientific methods
• both theories make (different) predictions
• Compare to observations
• Decide which theory explains data

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Phases of Venus

• Heliocentric

Geocentric
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Sunspots

• MPEG video from Galileo Project (June 2 July 8,
  1613)

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Galileo and his Contemporaries

• Elizabeth I. (1533-1603) Queen of England
• Tycho Brahe (1546-1601) Danish Astronomer
• Francis Bacon (1561-1626) English Philosopher
• Shakespeare (1564- 1616) Poet Playwright
• Galileo Galilei (1564-1642) Italian PAM
• Johannes Kepler (1571-1630) German PAM
• Rene Descartes (1596 - 1650) French PPM
• Christiaan Huygens (1629-1695) Dutch PAM
• Isaac Newton (1643-1727) English PM
• Louis XIV (1638-1715) French Sun King

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Tycho Brahe Johannes Kepler Galileo Galilei

• Observations Phenomenology/Theory
  Experiment
• ? Data ? Predictions ? test
  predictions

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Johannes KeplerThe Phenomenologist

• Key question
• How are things happening?
• Major Works
• Harmonices Mundi (1619)
• Rudolphian Tables (1612)
• Astronomia Nova
• Dioptrice

Johannes Kepler (15711630)
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Keplers First Law

• The orbits of the planets are ellipses, with the
  Sun at one focus

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Ellipses

• a semimajor axis e eccentricity

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Lecture 27 Gravitation

• Piazza Convocation ? Short Lab!
• Starry Monday tonight 7pm, here (Sci 238)

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Keplers Second Law
An imaginary line connecting the Sun to any
planet sweeps out equal areas of the ellipse in
equal times
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Why is it warmer in the summer than in the winter
in the USA?

• Because the Earth is closer to the Sun
• Because the Sun is higher in the sky in the
  summer
• None of the above

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Axis Tilt earth as gyroscope

• The Earths rotation axis is tilted 23½ degrees
  with respect to the plane of its orbit around the
  sun (the ecliptic)
• It is fixed in space ? sometimes we look down
  onto the ecliptic, sometimes up to it

Rotation axis
Path around sun
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The Seasons

• Change of seasons is a result of the tilt of the
  Earths rotation axis with respect to the plane
  of the ecliptic
• Sun, moon, planets run along the ecliptic

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Animation

• TeacherTube video

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Position of Ecliptic on the Celestial Sphere

• Earth axis is tilted w.r.t. ecliptic by 23 ½
  degrees
• Equivalent ecliptic is tilted by 23 ½ degrees
  w.r.t. equator!
• ? Sun appears to be sometime above (e.g. summer
  solstice), sometimes below, and sometimes on the
  celestial equator

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• The vernal equinox happens when the sun enters
  the zodiacal sign of Aries, but is actually
  located in the constellation of Pisces.

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Precession of the Equinoxes

• 
  Precession period
• 
  about 26,000 years
• The dawning of the age of
• Aquarius

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Keplers Third Law

• The square of a planets orbital period is
  proportional to the cube of its orbital
  semi-major axis
• P 2 ? a3
• a
  P
• Planet Orbital Semi-Major Axis Orbital Period
  Eccentricity P2/a3
• Mercury 0.387 0.241 0.206 1.002
• Venus 0.723 0.615 0.007 1.001
• Earth 1.000 1.000 0.017 1.000
• Mars 1.524 1.881 0.093 1.000
• Jupiter 5.203 11.86 0.048 0.999
• Saturn 9.539 29.46 0.056 1.000
• Uranus 19.19 84.01 0.046 0.999
• Neptune 30.06 164.8 0.010 1.000
• Pluto 39.53 248.6 0.248 1.001
• (A.U.) (Earth years)

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Strange motion of the Planets

• Planets usually move from W to E relative to
  the stars, but sometimes strangely turn around in
  a loop, the so called retrograde motion.
  

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The heliocentric Explanation of retrograde
planetary motion
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The New Physics Astronomy in a Nutshell
Newtons Principia

• Newtons key question
• Why are things happening?
• Invented calculus and physics while on vacation
  from college
• His three Laws of Motion, together with the Law
  of Universal Gravitation, explain all of Keplers
  Laws (and more!)
• Principia (1687)
• Full title Philosophiae naturalis principia
  mathematica has his famous three laws on page 19
  of 443.

Isaac Newton (16421727)
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Newtons Synthesis Unify sub- and super-lunar
phenomena!

• Gravity on earth a g 9.8 m/s2
• Due to force of earth on object a earth radius R
  away
• Effect on Moon a v2 /r
• From length of month, distance to moon 384,000
  km 60 R (known to Greeks)
• Acceleration is a 0.00272 m/s2 g/3600
• Conclusion Force falls off like distance squared!

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Law of Universal Gravitation

• Force G Mearth Mman / r2
• Vector Fman,earth - G Mearth Mman rto man,
  from earth/ rm,e3
• 2

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Which of the following depends on the inertial
mass of an object (as opposed to its
gravitational mass)?

• The time it takes on object to fall from a
  certain height
• The weight of an object on a bathroom-type spring
  scale
• The acceleration given to the object by a
  compressed spring
• The weight of the object on an ordinary balance

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Orbital Motion
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Cannon Thought Experiment

• http//www.phys.virginia.edu/classes/109N/more_stu
  ff/Applets/newt/newtmtn.html

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Suppose Earth had no atmosphere, and a ball were
fired from the top of Mt. Everest in a direction
tangent to the ground. If the initial speed were
high enough to cause the ball to travel in a
circular trajectory around Earth, the balls
acceleration would be

• Much less than g (b/c the ball doesnt fall to
  the ground)
• Be approximately g
• Depend on the balls speed
• None of the above

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Lecture 28 Rest of Gravity
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Two satellites A and B of the same mass are going
around Earth in concentric orbits. The distance
of satellite B from Earths center is twice that
of satellite A. What is the ratio of centripetal
force acting on B to that acting on A?

• 1/8
• ¼
• ½
• 1

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Principle of Superposition

• Gravitational forces can be added together as
  vectors, of course
• Newtons shell theorem
• A uniform spherical shell of matter attracts a
  particle that is outside of the shell as if all
  the shells mass were concentrated at its center

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Gravitation near the surface

• FG Mm/r2
• F ma
• ? a GM/r2
• Approximations
• Earth is not uniform, not a sphere
• Is rotating

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Applications

• From the distance r between two bodies and the
  gravitational acceleration a of one of the
  bodies, we can compute the mass M of the other
• F ma G Mm/r2 (m cancels out)
• From the weight of objects (i.e., the force of
  gravity) near the surface of the Earth, and known
  radius of Earth RE 6.4?103 km, we find ME
  6?1024 kg
• Your weight on another planet is F m ? GM/r2
• E.g., on the Moon your weight would be 1/6 of
  what it is on Earth

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Applications (contd)

• The mass of the Sun can be deduced from the
  orbital velocity of the planets MS
  rOrbitvOrbit2/G 2?1030 kg
• actually, Sun and planets orbit their common
  center of mass
• Orbital mechanics. A body in an elliptical orbit
  cannot escape the mass it's orbiting unless
  something increases its velocity to a certain
  value called the escape velocity
• Escape velocity from Earth's surface is about
  25,000 mph (7 mi/sec)

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Gravity Inside the Earth

• A uniform shell of matter exerts no net
  gravitational force on a particle located inside
  of it

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Gravitational Potential Energy

• U -GM/r
• Proof by calculating work (integral)
• Force from potential energy
• Escape speed

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Einstein Gravity

• General relativity
• Equivalence principle
• Curvature of space

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General Relativity ?! Thats easy!

Rµ? -1/2 gµ? R 8pG/c4 Tµ?
What does that mean?

• (Actually, it took Prof. Einstein 10 years to
  come up with that!)

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The Idea behind General Relativity
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More General

• General Relativity is more general in the sense
  that we drop the restriction that an observer not
  be accelerated
• The claim is that you cannot decide whether you
  are in a gravitational field, or just an
  accelerated observer
• The Einstein field equations describe the
  geometric properties of spacetime

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Do bowling balls fall faster than apples?
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No!
Galileo In the absence of air, all objects
experience the same acceleration (change in
motion) near Earths surface
http//www.youtube.com/watch?v5C5_dOEyAfk
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Equivalence Principle

• A little reflection will show that the law of the
  equality of the inertial and gravitational mass
  is equivalent to the assertion that the
  acceleration imparted to a body by a
  gravitational field is independent of the nature
  of the body. For Newton's equation of motion in a
  gravitational field, written out in full, it is
• (Inertial mass) (Acceleration) (Intensity of
  the gravitational field) (Gravitational mass).
• It is only when there is numerical equality
  between the inertial and gravitational mass that
  the acceleration is independent of the nature of
  the body. Albert Einstein

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Meaning

• We cannot decide whether we live in an
  accelerated reference frame, or near a strong
  gravitational field.

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The Idea behind General Relativity

• We view space and time as a whole, we call it
  four-dimensional space-time.
• It has an unusual geometry
• Space-time is warped by the presence of masses
  like the sun, so Mass tells space how to bend
• Objects (like planets) travel in straight lines
  through this curved space (we see this as
  orbits), so
• Space tells matter how to move

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Planetary Orbits

• Sun
• Planets orbit

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Effects of General Relativity

• Bending of starlight by the Sun's gravitational
  field (and other gravitational lensing effects)