Lecture 23: MOI & Torque
Kinetic energy of Rotation
• K = sum of ½ m v2 for all parts of the body
•  Moment of inertia I
•  K = ½ I ω2
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 m^2
• Different I around different axis!
• Example: rotate around midpoint:
I’ = 11.25 kg m^2
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 m^2
ω = 2π/T = π Hz, so K = ½ I ω ^2 = ½ π^2 J
Determining MOI
• Integral
• Table
• Parallel-axis
theorem
Torque
• Torque is force times lever arm
• Lever arm is distance to rotation axis along
a direction perpendicular to the force
• Later: τ = r x F
• | τ | = |r| |F| sin φ
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
Newton II for Rotation
Work and Rotational Kinetic
Energy
• W = ∫ τ dθ
• P = dW/dt = τω
Lecture 24: General Rotations
Worksheet: Torque and angular
acceleration
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.
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
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 β determine linear acc.
– Torque due to static friction acting at radius R
determines angular acc.: torque = I α
Torque as a Vector
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 –i+j or NW or d)
A ladybug sits at the outer edge of a merry-goround, 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
Reminder: Vector Product
What is A x A ?
•
•
•
•
Zero
A
-A
A2
Lecture 25: Angular Momentum
Angular Momentum
A ladybug sits at the outer edge of a merry-goround, 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
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. –i–j)
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)
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 ω)
L=Iω
Dynamics Transliteration
•
•
•
•
•
Mass  Moment of inertia
Force  Torque
Momentum  Angular Momentum
Newton II
Momentum Conservation  Angular Momentum
Conservation
Angular Momentum
Conservation
• Angular momentum is conserved if no
external net torque is present
• Demos: turntable + weights, turntable and
bikewheel, bikewheel spinning on rope
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
Lecture 26: Rotational Energy
Pre-lecture Exercise (11.10 – 11.12)
• By what factor does the spinning
volunteer’s 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 ω = const, so if I goes down by
1.5, ω goes up by 1.5, so the period goes
down by 1.5, or f = 1/1.5 =2/3 = 0.6667.
Precession of a Gyroscope
• Demo: Little Gyroscope
• Demo: Bike Wheel on stick
– Rate of precession becomes larger as wheel
slows down
•
Gravitation
History
•
•
•
•
•
•
Ptolemy
Copernicus
Brahe
Galilei
Kepler
Newton
From Galileo to Newton - the Birth of
Modern Science
1609
1687
Precursor: Nicolas Copernicus
(1473–1543)
• 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)
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
Phases of
Venus
Heliocentric
Geocentric
Sunspots
• MPEG video
from Galileo
Project (June 2 –
July 8, 1613)
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”
Tycho Brahe Johannes Kepler
Galileo Galilei
Observations
 Data
Experiment
 test predictions
Phenomenology/Theory
 Predictions
Johannes Kepler–The Phenomenologist
• Key question:
How are things happening?
Major Works:
• Harmonices Mundi (1619)
• Rudolphian Tables (1612)
• Astronomia Nova
• Dioptrice
Johannes Kepler (1571–1630)
Kepler’s First Law
The orbits of the planets are ellipses, with
the Sun at one focus
Ellipses
a = “semimajor axis”; e = “eccentricity”
Lecture 27: Gravitation
• Piazza Convocation  Short Lab!
• Starry Monday tonight 7pm, here (Sci 238)
Kepler’s Second Law
An imaginary line connecting the Sun to any planet sweeps
out equal areas of the ellipse in equal times
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
Axis Tilt – earth as gyroscope
• The Earth’s 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
The Seasons
• Change of seasons
is a result of the tilt
of the Earth’s
rotation axis with
respect to the plane
of the ecliptic
• Sun, moon, planets
run along the
ecliptic
Animation
• TeacherTube video
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
The vernal equinox
happens when the
sun enters the
zodiacal sign of
Aries, but is actually
located in the
constellation of
Pisces.
Precession of the Equinoxes
Precession period
about 26,000 years
“The dawning of the age of
Aquarius”
Kepler’s Third Law
The square of a planet’s 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
Mercury
0.387
0.241
Venus
0.723
0.615
Earth
1.000
1.000
Mars
1.524
1.881
Jupiter
5.203
11.86
Saturn
9.539
29.46
Uranus
19.19
84.01
Neptune
30.06
164.8
Pluto
39.53
248.6
(A.U.)
(Earth years)
Eccentricity
0.206
0.007
0.017
0.093
0.048
0.056
0.046
0.010
0.248
P2/a3
1.002
1.001
1.000
1.000
0.999
1.000
0.999
1.000
1.001
“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.
The heliocentric Explanation of
retrograde planetary motion
The New Physics & Astronomy in a
Nutshell: Newton’s Principia
• Newton’s 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 Kepler’s Laws (and
more!)
• Principia (1687)
[Full title: Philosophiae naturalis
principia mathematica] has his famous
three laws on page 19 of 443.
Isaac Newton (1642–1727)
Newton’s Synthesis: Unify suband 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!
Vector:
Law of Universal Gravitation
Mman
MEarth
|r|
Force = G Mearth Mman / |r|2
Fman,earth = - G Mearth Mman rto man, from earth/ |rm,e|3
2
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
Orbital Motion
Cannon “Thought Experiment”
• http://www.phys.virginia.edu/classes/109N/more_stuff/Appl
ets/newt/newtmtn.html
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 ball’s acceleration
would be…
•Much less than g (b/c the
ball doesn’t fall to the
ground)
•Be approximately g
•Depend on the ball’s speed
•None of the above
Lecture 28: Rest of Gravity
Two satellites A and B of the same mass are
going around Earth in concentric orbits. The
distance of satellite B from Earth’s 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
Principle of Superposition
• Gravitational forces can be added together
as vectors, of course
• Newton’s shell theorem
– A uniform spherical shell of matter attracts a
particle that is outside of the shell as if all the
shell’s mass were concentrated at its center
Gravitation near the surface
•
•
•
•
F=G Mm/r2
F =ma
 a =GM/r2
Approximations:
– Earth is not uniform, not a sphere
– Is rotating
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
Applications (cont’d)
• 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)
Gravity Inside the Earth
• A uniform shell of matter exerts no net
gravitational force on a particle located
inside of it
Gravitational Potential Energy
•
•
•
•
U = -GM/r
Proof by calculating work (integral)
Force from potential energy
Escape speed
Einstein & Gravity
• General relativity
• Equivalence principle
• Curvature of space
General Relativity ?! That’s easy!
Rμν -1/2 gμν R = 8πG/c4 Tμν
What does that mean?
(Actually, it took Prof. Einstein 10 years to come up with that!)
The Idea behind General Relativity
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
Do bowling balls fall faster than apples?
No!
Galileo: In the absence of
air, all objects experience
the same acceleration
(change in motion) near
Earth’s surface
http://www.youtube.com/watch?v=5C5_dOEyAfk
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
Meaning
• We cannot decide whether we live in an
accelerated reference frame, or near a strong
gravitational field.
The Idea behind General Relativity
–
We view space and time as a whole, we call it fourdimensional 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”
Planetary Orbits
Sun
Planet’s orbit
Effects of General Relativity
Bending of starlight by the Sun's gravitational
field (and other gravitational lensing effects)