Simple Nature. Workbook

57 Pages·2002·1.526 MB·English

by  Crowell B.

Simple Nature
Workbook
Benjamin Crowell
www.lightandmatter.com
Light and Matter
Fullerton, California
www.lightandmatter.com
Copyright (cid:13)c 2001 Benjamin Crowell
All rights reserved.
rev. September 13, 2002
Permission is granted to copy, distribute and/or modify this document under the terms of the GNU
Free Documentation License, Version 1.1, with no invariant sections, no front-cover texts and no
back-cover texts. A copy of the license is provided in the appendix titled GNU Free Documentation
License. The license applies to the entire text of this book, plus all the illustrations that are by
Benjamin Crowell. All the illustrations are by Benjamin Crowell except as noted in the photo
credits or in parentheses in the caption of the figure. This book can be downloaded free of charge
from www.lightandmatter.com in a variety of formats, including editable formats.
Contents
0 Preliminaries
0.1 Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0.2 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0.3 Kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1 Conservation of Mass
1.1 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.1.1. Problem-solving techniques, 13.—1.1.2. Delta notation, 13.
1.2 Equivalence of Gravitational and Inertial Mass . . . . . . . . . . . . . . . . . . . . 13
1.3 Galilean Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1. Applications of calculus, 14.
1.4 A Preview of Some Modern Physics . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Conservation of Energy
2.1 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1. The energy concept, 15.—2.1.2. Logical issues, 15.—2.1.3. Kinetic energy, 15.—2.1.4. Power,
15.—2.1.5. Gravitational energy, 15.—2.1.6. Equilibrium and stability, 15.—2.1.7. Predicting the
direction of motion, 15.
2.2 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Gravitational Phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1. Kepler’s laws, 15.—2.3.2. Circular orbits, 15.—2.3.3. The sun’s gravitational field, 15.—2.3.4.
Gravitational energy in general, 15.—2.3.5. The shell theorem, 15.
2.4 Atomic Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1. Heat is kinetic energy., 15.—2.4.2. All energy comes from particles moving or interacting., 15.
3 Conservation of Momentum
3.1 Momentum in One Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.1. Mechanical momentum, 17.—3.1.2. Nonmechanical momentum, 17.—3.1.3. Momentum com-
pared to kinetic energy, 17.—3.1.4. Collisions in one dimension, 17.—3.1.5. The center of mass,
17.—3.1.6. Momentum and Galilean relativity, 17.—3.1.7. The center of mass frame of reference,
17.—3.1.8. Momentum transfer, 17.—3.1.9. Newton’s laws, 17.—3.1.10. Forces between solids, 18.—
3.1.11. Work, 18.—3.1.12. Simple machines, 18.—3.1.13. Force related to interaction energy, 18.
3.2 Resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1. Damped, free motion, 18.—3.2.2. The quality factor, 18.—3.2.3. Driven motion, 18.
3.3 Motion in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.1. The cartesian perspective, 18.—3.3.2. Rotational invariance, 19.—3.3.3. Vectors, 19.—3.3.4.
Calculus with vectors, 20.—3.3.5. The dot product, 21.—3.3.6. Gradients and line integrals, 21.
4 Conservation of Angular Momentum
4.1 Angular Momentum in Two Dimensions. . . . . . . . . . . . . . . . . . . . . . . 27
4.1.1. Angular momentum, 27.—4.1.2. Application to planetary motion, 27.—4.1.3. Two Theorems
AboutAngularMomentum,27.—4.1.4. Torque,27.—4.1.5. Applicationstostatics,28.—4.1.6. Proof
of Kepler’s elliptical orbit law, 28.
4.2 Rigid-Body Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.1. Kinematics,28.—4.2.2. Relationsbetweenangularquantitiesandmotionofapoint,28.—4.2.3.
Finding moments of inertia by integration, 28.
4.3 Angular Momentum in Three Dimensions . . . . . . . . . . . . . . . . . . . . . . 28
4.3.1. Angularmomentuminthreedimensions,28.—4.3.2. Rigid-bodydynamicsinthreedimensions,
28.
5 Thermodynamics
5.1 Pressure and Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.1.1. Temperature, 31.
5.2 Microscopic Description of an Ideal Gas . . . . . . . . . . . . . . . . . . . . . . 31
5.2.1. Evidence for the kinetic theory, 31.—5.2.2. Pressure, volume, and temperature, 31.
5.3 Entropy as a Macroscopic
Quantity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3.1. Efficiency and grades of energy, 31.—5.3.2. Heat engines, 31.—5.3.3. Entropy, 31.
5.4 Entropy as a Microscopic
Quantity (Optional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.4.1. A microscopic view of entropy, 31.—5.4.2. Phase space, 31.—5.4.3. Microscopic definitions of
entropy and temperature, 31.—5.4.4. The arrow of time, or “This way to the Big Bang”, 31.—5.4.5.
Quantum mechanics and zero entropy, 31.
6 Waves
6.1 Free Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.1.1. Wave motion, 33.—6.1.2. Waves on a string, 33.—6.1.3. Sound and light waves, 33.—6.1.4.
Periodic waves, 33.—6.1.5. The Doppler effect, 33.
6.2 Bounded Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.2.1. Reflection, transmission, and absorption, 33.—6.2.2. Quantitative treatment of reflection,
4 Contents
34.—6.2.3. Interference effects, 34.—6.2.4. Waves bounded on both sides, 34.
7 Relativity
7.1 Basic Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.1.1. The principle of relativity, 35.—7.1.2. Distortion of time and space, 35.—7.1.3. Applications,
35.
7.2 The Lorentz Transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
7.2.1. Coordinate transformations in general, 35.—7.2.2. Derivation of the Lorentz transformation,
35.—7.2.3. Spacetime, 35.
7.3 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7.3.1. Invariants, 37.—7.3.2. Combinationofvelocities, 37.—7.3.3. Momentumandforce, 37.—7.3.4.
Kinetic energy, 37.—7.3.5. Equivalence of mass and energy, 37.
8 Atoms and Electromagnetism
8.1 The Electric Glue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
8.1.1. The quest for the atomic force, 43.—8.1.2. Charge, electricity and magnetism, 43.—8.1.3.
Atoms, 43.—8.1.4. Quantization of charge, 43.—8.1.5. The electron, 43.—8.1.6. The raisin cookie
model of the atom, 43.
8.2 The Nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
8.2.1. Radioactivity, 43.—8.2.2. The planetary model of the atom, 44.—8.2.3. Atomic number,
44.—8.2.4. The structure of nuclei, 44.—8.2.5. The strong nuclear force, alpha decay and fission,
44.—8.2.6. The weak nuclear force; beta decay, 44.—8.2.7. Fusion, 44.—8.2.8. Nuclear energy and
binding energies, 44.—8.2.9. Biological effects of ionizing radiation, 44.—8.2.10. The creation of the
elements, 44.
9 DC Circuits
9.1 Current and Voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9.1.1. Current, 45.—9.1.2. Circuits, 45.—9.1.3. Voltage, 45.—9.1.4. Resistance, 45.—9.1.5. Current-
conducting properties of materials, 45.
9.2 Parallel and Series Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9.2.1. Schematics,45.—9.2.2. Parallelresistancesandthejunctionrule,45.—9.2.3. Seriesresistances,
45.
10 Fields
10.1 Fields of Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
10.1.1. Why fields?, 47.—10.1.2. The gravitational field, 47.—10.1.3. The electric field, 47.
10.2 Voltage Related to Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
10.2.1. One dimension, 47.—10.2.2. Two or three dimensions, 47.
10.3 Fields by Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
10.3.1. Electricfieldofacontinuouschargedistribution,47.—10.3.2. Thefieldnearachargedsurface,
Contents 5
47.
10.4 Energy in Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
10.4.1. Electric field energy, 47.—10.4.2. Gravitational field energy, 48.—10.4.3. Magnetic field
energy, 48.
10.5 LRC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
10.5.1. Capacitance and inductance, 48.—10.5.2. Oscillations, 49.—10.5.3. Voltage and current,
49.—10.5.4. Decay, 49.—10.5.5. Impedance, 49.—10.5.6. Power, 49.—10.5.7. Impedance Matching,
50.—10.5.8. Complex Impedance, 50.
10.6 Fields by Gauss’ Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
10.6.1. Gauss’ law, 50.—10.6.2. Additivity of flux, 50.—10.6.3. Zero flux from outside charges, 50.—
10.6.4. ProofofGauss’theorem,50.—10.6.5. Gauss’lawasafundamentallawofphysics,50.—10.6.6.
Applications, 50.
10.7 Gauss’ Law in Differential Form. . . . . . . . . . . . . . . . . . . . . . . . . . 50
11 Electromagnetism
11.1 More About the Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11.1.1. Magneticforces,51.—11.1.2. Themagneticfield,51.—11.1.3. Someapplications,51.—11.1.4.
No magnetic monopoles, 51.—11.1.5. Symmetry and handedness, 51.
11.2 Magnetic Fields by Superposition. . . . . . . . . . . . . . . . . . . . . . . . . 51
11.2.1. Superposition of straight wires, 51.—11.2.2. Energy in the magnetic field, 51.—11.2.3. Su-
perposition of dipoles, 51.—11.2.4. The Biot-Savart law (optional), 51.
11.3 Magnetic Fields by Ampe`re’s Law. . . . . . . . . . . . . . . . . . . . . . . . . 51
11.3.1. Amp`ere’s law, 51.—11.3.2. A quick and dirty proof, 51.—11.3.3. Maxwell’s equations for
static fields, 51.
11.4 Ampe`re’s Law in Differential Form (optional) . . . . . . . . . . . . . . . . . . . . 51
11.4.1. The curl operator, 51.—11.4.2. Properties of the curl operator, 51.
11.5 Induced Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11.5.1. Faraday’s experiment, 51.—11.5.2. Why induction?, 51.—11.5.3. Faraday’s law, 51.
11.6 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
11.6.1. Induced magnetic fields, 51.—11.6.2. Light waves, 52
6 Contents
Chapter 0
Preliminaries
0.1 Conversions 1 2
x x
A Each of the following conversions contains
an error. In each case, explain what the error
t t
is. (a) 1000 kg × 1 kg = 1 g 3 4
1000 g x x
(b) 50 m × 1 cm = 0.5 cm
100 m
(c) “Nano” is 10−9 , so there are 10−9 nm in a
t t
meter.
5 6
(d) “Micro” is 10−6 , so 1 kg is 106µg. x x
B How many square centimeters are there in t t
a square inch? (1 inch=2.54 cm) First find an
approximate answer by making a drawing, then
DiscussionquestionA
derive the conversion factor more accurately us-
ing the symbolic method.
0.2 Scaling 30
25
A A toy fire engine is 1/30 the size of the
realone, butisconstructedfromthesamemetal 20
x
with the same proportions. How many times
15
(m)
smaller is its weight? How many times less red
10
paint would be needed to paint it?
5
B Single-celledanimalsmustpassivelyabsorb
0
nutrients and oxygen from their surroundings,
0 2 4 6 8 10
unlike humans who have lungs to pump air in
t (s)
andoutandahearttodistributetheoxygenated
blood throughout their bodies. Even the cells
composing the bodies of multicellular animals DiscussionquestionB
mustabsorboxygenfromanearbycapillarythr-
ough their surfaces. Based on these facts, ex-
C Two physicists duck out of a boring scien-
plain why cells are always microscopic in size.
tificconferencetogogetbeer. Onthewaytothe
bar, they witness an accident in which a pedes-
0.3 Kinematics trian is injured by a hit-and-run driver. A crim-
inal trial results, and they must testify. In her
A An ant walks forward, pauses, then runs
testimony, Dr. Transverz Waive says, “The car
quickly ahead. It then suddenly reverses direc-
wasmovingalongprettyfast,I’dsaythevelocity
tion and walks slowly back the way it came.
was +40 mi/hr. They saw the old lady too late,
Which graph could represent its motion?
and even though they slammed on the brakes
B If an object has a wavy motion graph like theystillhitherbeforetheystopped. Thenthey
the one in the figure, which are the points at made a U turn and headed off at a velocity of
which the object reverses its direction? What is about −20 mi/hr, I’d say.” Dr. Longitud N.L.
true about the object’s velocity at these points? Vibrasheun says, “He was really going too fast,
maybe his velocity was −35 or −40 mi/hr. Af-
ter he hit Mrs. Hapless, he turned around and
left at a velocity of, oh, I’d guess maybe +20 or
+25 mi/hr.” Is their testimony contradictory?
Explain.
D Discuss anything unusual about the three
graphs in the figure.
1 2
x x
t t
3
x
t
DiscussionquestionD
E In the graph shown in the figure, describe
how the object’s velocity changes.
x
t
DiscussionquestionE
8 Chapter 0 Preliminaries
Exercise: Models and Idealization
Equipment:
coffee filters
ramps (one per group)
balls of various sizes
sticky tape
vacuum pump and “guinea and feather” apparatus (one)
The motion of falling objects has been recognized since ancient times as an important piece of
physics, but the motion is inconveniently fast, so in our everyday experience it can be hard to tell
exactly what objects are doing when they fall. In this exercise you will use several techniques to get
around this problem and study the motion. Your goal is to construct a scientific model of falling. A
model means an explanation that makes testable predictions. Often models contain simplifications
or idealizations that make them easier to work with, even though they are not strictly realistic.
1. One method of making falling easier to observe is to use objects like feathers that we know from
everyday experience will not fall as fast. You will use coffee filters, in stacks of various sizes, to test
the following two hypotheses and see which one is true, or whether neither is true:
Hypothesis 1A: When an object is dropped, it rapidly speeds up to a certain natural falling speed,
and then continues to fall at that speed. The falling speed is proportional to the object’s weight.
(A proportionality is not just a statement that if one thing gets bigger, the other does too. It says
that if one becomes three times bigger, the other also gets three times bigger, etc.)
Hypothesis 1B: Different objects fall the same way, regardless of weight.
Test these hypotheses and discuss your results with your instructor.
2. Asecondwaytoslowdowntheactionistoletaballrolldownaramp. Thesteepertheramp, the
closer to free fall. Based on your experience in part 1, write a hypothesis about what will happen
when you race a heavier ball against a lighter ball down the same ramp, starting them both from
rest.
Hypothesis:
Show your hypothesis to your instructor, and then test it.
You have probably found that falling was more complicated than you thought! Is there more than
one factor that affects the motion of a falling object? Can you imagine certain idealized situations
that are simpler?Try to agree verbally with your group on an informal model of falling that can
make predictions about the experiments described in parts 3 and 4.
3. You have three balls: a standard “comparison ball” of medium weight, a light ball, and a heavy
ball. Suppose you stand on a chair and (a) drop the light ball side by side with the comparison
ball, then (b) drop the heavy ball side by side with the comparison ball, then (c) join the light and
heavy balls together with sticky tape and drop them side by side with the comparison ball.
Use your model to make a prediction:
Test your prediction.
4. Your instructor will pump nearly all the air out of a chamber containing a feather and a heavier
object, then let them fall side by side in the chamber.
Use your model to make a prediction:
Section 0.3 Kinematics 9
Exercise: Scaling Applied to Leaves
Equipment:
leaves of three sizes
balance
1. Each group will have one leaf, and should measure its surface area and volume, and determine
its surface-to-volume ratio (surface area divided by volume). For consistency, use units of cm2 and
cm3, and only count the area of one side of the leaf. The area can be found by tracing the area of
the leaf on graph paper and counting squares. The volume can be found by weighing the leaf and
assuming that its density is 1 g/cm3, which is nearly true since leaves are mostly water.
Write your results on the board for comparison with the other groups’ numbers.
2. Both the surface area and the volume are bigger for bigger leaves, but what about the surface to
volumeratios? Whatimplicationswouldthishavefortheplants’survivalindifferentenvironments?
10 Chapter 0 Preliminaries