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-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

x

yF

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-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

x

yF

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 = I ω

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 Phenomenology/Theory Experiment

Data Predictions test 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 LawAn 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

Path around sun

Rotation axis

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 LawThe 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 Eccentricity P2/a3

Mercury 0.387 0.241 0.206 1.002Venus 0.723 0.615 0.007 1.001Earth 1.000 1.000 0.017 1.000Mars 1.524 1.881 0.093 1.000Jupiter 5.203 11.86 0.048 0.999Saturn 9.539 29.46 0.056 1.000Uranus 19.19 84.01 0.046 0.999Neptune 30.06 164.8 0.010 1.000Pluto 39.53 248.6 0.248 1.001

(A.U.) (Earth years)

“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 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!

Law of Universal Gravitation

Force = G Mearth Mman / |r|2

Vector: Fman,earth = - G Mearth Mman rto man, from earth/ |rm,e|3

2

MEarth

Mman

|r|

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/Applets/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 = 6.4103 km, we find ME = 61024 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 = rOrbitvOrbit

2/G = 21030 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!

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

Rμν -1/2 gμν R = 8πG/c4 Tμν

What does that mean?

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 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”

Planetary Orbits

Sun Planet’s orbit

Effects of General Relativity

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