12.9: Removing the Homogeneity of Variance Assumption
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In our example, the homogeneity of variance assumption turned out to be a pretty safe one: the Levene test came back non-significant, so we probably don’t need to worry. However, in real life we aren’t always that lucky. How do we save our ANOVA when the homogeneity of variance assumption is violated? If you recall from our discussion of t-tests, we’ve seen this problem before. The Student t-test assumes equal variances, so the solution was to use the Welch t-test, which does not. In fact, Welch (1951) also showed how we can solve this problem for ANOVA too (the
Welch one-way test
). It’s implemented in R using the
oneway.test()
function. The arguments that we’ll need for our example are:
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formula
. This is the model formula, which (as usual) needs to specify the outcome variable on the left hand side and the grouping variable on the right hand side: i.e., something likeoutcome ~ group
. -
data
. Specifies the data frame containing the variables. -
var.equal
. If this isFALSE
(the default) a Welch one-way test is run. If it isTRUE
then it just runs a regular ANOVA.
The function also has a
subset
argument that lets you analyse only some of the observations and a
na.action
argument that tells it how to handle missing data, but these aren’t necessary for our purposes. So, to run the Welch one-way ANOVA for our example, we would do this:
oneway.test(mood.gain ~ drug, data = clin.trial)
##
## One-way analysis of means (not assuming equal variances)
##
## data: mood.gain and drug
## F = 26.322, num df = 2.0000, denom df = 9.4932, p-value = 0.000134
To understand what’s happening here, let’s compare these numbers to what we got earlier in Section 14.3 when we ran our original ANOVA. To save you the trouble of flicking back, here are those numbers again, this time calculated by setting
var.equal = TRUE
for the
oneway.test()
function:
oneway.test(mood.gain ~ drug, data = clin.trial, var.equal = TRUE)
##
## One-way analysis of means
##
## data: mood.gain and drug
## F = 18.611, num df = 2, denom df = 15, p-value = 8.646e-05
Okay, so originally our ANOVA gave us the result F(2,15)=18.6, whereas the Welch one-way test gave us F(2,9.49)=26.32. In other words, the Welch test has reduced the within-groups degrees of freedom from 15 to 9.49, and the F-value has increased from 18.6 to 26.32.


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"9+2"
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F-actin
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myosin ATPase
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\(\alpha \) tubulin
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F-actin polarity
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myosin
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A-band
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flagella
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myosin "heads"
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acetylcholine
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fluorescence microscopy
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neuromuscular junction
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acidic keratin
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force transduction
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nuclear lamina
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actin
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G-actin
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plus and minus ends
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actin-binding proteins
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hair, horn
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protofilaments
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actin-myosin interactions
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I-band
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psuedopodia
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actin-myosin paradox
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intermediate filaments
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sarcomere
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action potential
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intestinal microvilli
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sarcoplasmic reticulum
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aoeboid movement
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keratin
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sarcolemma
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ATPase
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keratin isoforms
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scales, feathers, fingernails
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axoneme
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lamins
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secretion vesicle transport
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\(\beta \) tubulin
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membrane depolarization
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skeletal muscle contraction
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basal body
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microfilaments
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skeletal muscle relaxation
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basic keratin
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microtubule assembly end
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sliding filament model
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Ca
2
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regulation of contraction
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microtubule disassembly end
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syncytium
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Ca
2
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release v. active transport
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microtubule doublets
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thick and thin filaments
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cell motility
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microtubule organizing center
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titin
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centriole
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microtubule polarity
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transverse (T) tubules
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cilia
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microtubule-associated proteins
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tread-milling
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contraction regulation
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microtubules
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tropomyosin
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cortical cellular microfilaments
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mitotic, meiotic spindle fibers
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troponin I
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creatine phosphate
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M-line
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troponin T
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cross-bridges
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motor proteins
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troponins
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cytoplasmic streaming
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MTOC
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troponin C
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cytoskeleton
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muscle cell
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tubulin heterodimer
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desmosomes
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muscle fiber
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tubulins
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dynein
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myocyte
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viscoelasticity
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evolution of actin genes
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myofiber
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Z-disks
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evolution of myosin genes
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myofibril
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Z-line
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1: Music Fundamentals
- Recognize a wide variety of sounds, comparing and contrasting them using musical elements of pitch, volume, articulation, and timbre.
- Aurally identify important performing forces (use of the voice and instruments) of Western music.
- Define basic elements of melody, harmony, rhythm, and texture and build a vocabulary for discussing them.
- Identify basic principles and types of musical form.
- Listen to music and describe its musical elements and forms.
- Compare and contrast categories of art music, folk music, and pop music.
- Identify ways in which humans have used music for social and expressive purposes
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- 1.2: Performing Forces for Music
- Music consists of the intentional organization of sounds by and for human beings. In the broadest classification, these sounds are produced by people in three ways: (1) through the human voice, the instrument with which most of us are born, (2) by using musical instruments, or (3) by using electronic and digital equipment to generate purely electronic sounds.
Thumbnail: String quartet performing for the Mozart Year 2006 in Vienna. (CC BY 2.5; Gryffindor via Wikipedia )


1.7.18: Compressions- Isothermal- Apparent Molar Compression
The volume of a given solution prepared at fixed temperature and fixed pressure using \(\mathrm{n}_{1}\) moles of water and \(\mathrm{n}_{j}\) moles of solute \(j\) is given by equation (a).
\[\mathrm{V}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
If the solution is prepared using \(1 \mathrm{~kg}\) of water(\(\ell\)),
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}(\mathrm{aq})+\mathrm{m}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) \nonumber \]
\(\mathrm{M}_{1}\) is the molar mass of the solvent, water(\(\ell\)); \(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) are the partial molar volumes of water and solute \(j\) respectively in the solution. As we change \(\mathrm{m}_{j}\) (for a fixed mass of solvent) so both \(\mathrm{V}_{1}(\mathrm{aq})\) and \(\mathrm{V}_{j}(\mathrm{aq})\) change. An important procedure rewrites equation (b) in the following form where \(\mathrm{V}_{1}^{*}(\ell)\) is the molar volume of pure solvent at the same \(\mathrm{T}\) and \(\mathrm{p}\). Thus,
\[\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{V}_{1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{V}_{\mathrm{j}}\right) \nonumber \]
\(\phi\left(\mathrm{V}_{\mathrm{j}}\right)\) is the apparent molar volume of the solute \(j\). The system, an aqueous solution, is displaced by a change in pressure (at fixed \(\mathrm{T}\)) along a path where the affinity for spontaneous change is zero. In other words the system is subjected to an equilibrium displacement. The isothermal differential dependence of volume \(\mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)\) is given by equation (d).
\[\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right.}{\partial \mathrm{p}}\right)_{\mathrm{T}}=\mathrm{M}_{1}^{-1} \,\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}}+\mathrm{m}_{\mathrm{j}} \,\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
By definition, the apparent molar (isothermal) compression of the solute, [1-3]
\[\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=-\left(\frac{\partial \phi\left(\mathrm{V}_{\mathrm{j}}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
Similarly for equilibrium molar compression of the pure solvent,
\[\mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=-\left(\frac{\partial \mathrm{V}_{1}^{*}(\ell)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
By definition,
\[\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=-\left(\frac{\partial \mathrm{V}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)}{\partial \mathrm{p}}\right)_{\mathrm{T}} \nonumber \]
Hence,
\[\mathrm{K}_{\mathrm{T}}\left(\mathrm{aq} ; \mathrm{w}_{1}=1 \mathrm{~kg}\right)=\mathrm{M}_{1}^{-1} \, \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)+\mathrm{m}_{\mathrm{j}} \, \phi\left(\mathrm{K}_{\mathrm{Tj}}\right) \nonumber \]
Moreover recalling that \(\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{V}_{\mathrm{j}}\right)=\phi\left(\mathrm{V}_{\mathrm{j}}\right)^{\infty}=\mathrm{V}_{\mathrm{j}}^{\infty}(\mathrm{aq})\),
\[\operatorname{limit}\left(\mathrm{m}_{\mathrm{j}} \rightarrow 0\right) \phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)^{\infty}=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq}) \nonumber \]
These equations combined with those yielding \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) from measured \(\kappa_{\mathrm{T} 1}^{*}(\ell)\) and \(\kappa_{\mathrm{T}}(\mathrm{aq})\) signal an attractive approach to the study of solvent-solute interactions via \(\mathrm{K}_{\mathrm{T} j}^{\infty}(\mathrm{aq})\). In this context Gurney [4] identified a cosphere of solvent around a solute molecule where the organization differs from that in the bulk solvent at some distance from a given solute molecule \(j\). For example, the limiting partial molar volume of solute \(j\) can be understood as the sum of two terms, \(\mathrm{V}\)(intrinsic) and \(\mathrm{V}\)(cosphere). Then \(\mathrm{V}\)(cosphere) is an indicator of the role of solvent-solute interaction, hydration in aqueous solution. Thus,
\[\left.V_{j}^{\infty}(a q)=V_{j} \text { (int rinsic }\right)+V_{j}(\cos p h e r e) \nonumber \]
Hence,
\[\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}} \text { (int rinsic) }+\mathrm{K}_{\mathrm{T}_{j}}(\cos \text {phere}) \nonumber \]
The argument is advanced that \(\mathrm{K}_{\mathrm{T}j}\)(intrinsic) for simple ions such as halide ions and alkali metal ions is zero. \(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})\) is an indicator of the hydration of a given solute in aqueous solution. \(\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})\) is obtained from the dependence of \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) on, for example, concentration \(\mathrm{c}_{j}\) using equation(l) for neutral solutes and equation (m) for salts, the latter being based on the DHLL.
\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})+\mathrm{a}_{\mathrm{KT}} \,\left(\mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right) \nonumber \]
\[\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)=\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}^{\infty}(\mathrm{aq})+\mathrm{b}_{\mathrm{KT}} \,\left(\mathrm{c}_{\mathrm{j}} / \mathrm{c}_{\mathrm{r}}\right)^{1 / 2} \nonumber \]
One might have expected an extensive scientific literature reporting \(\mathrm{K}_{\mathrm{T}_{j}}^{\infty}(\mathrm{aq})\) for a wide range of solutes. Unfortunately measurement of isothermal compressions of liquids is difficult at least to the precision required for the estimation of \(\mathrm{K}_{\mathrm{Tj}}^{\infty}(\mathrm{aq})\). Indeed direct measurement of the volume change of a liquid when compressed at constant temperature is difficult because the isothermal condition is difficult to satisfy. Two procedures have been adopted to over come this problem. In both cases isentropic compressibilities calculated from densities and speeds of sound have been used.
In one set of procedures, isentropic compressibilities, densities and isobaric heat capacities are used to calculate isothermal compressions for a given solution, molality \(\mathrm{m}_{j}\). For example, Bernal and Van Hook [5] use the Desnoyers-Philip Equation to evaluate \(\phi\left(K_{T_{j}}\right)^{\infty}\) for glucose, sucrose and fructose [5] in aqueous solutions at \(348 \mathrm{~K}\). An alternative procedure equates \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)^{\infty}\) with the experimentally accessible limiting apparent isentropic compression, ∞ φ(K ) Sj . In another approach, the starting point is equation (a) which is differentiated with respect to pressure at constant temperature to yield equation (n).
\[\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\mathrm{n}_{1} \, \mathrm{K}_{\mathrm{Tl}}(\mathrm{aq})+\mathrm{n}_{\mathrm{j}} \, \mathrm{K}_{\mathrm{T}_{\mathrm{j}}}(\mathrm{aq}) \nonumber \]
Equation (n) is divided by volume \(\mathrm{V}(\mathrm{aq})\). Hence
\[\begin{aligned}
\mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \,\left[1 / \mathrm{V}_{1}(\mathrm{aq})\right] \, \mathrm{K}_{1}(\mathrm{aq}) \\
&+\left[\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \,\left[1 / \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{j}}(\mathrm{aq})
\end{aligned} \nonumber \]
We use \(\phi_{1}\left[=\mathrm{n}_{1} \, \mathrm{V}_{1}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right] \text { and } \phi_{\mathrm{j}}\left[=\mathrm{n}_{\mathrm{j}} \, \mathrm{V}_{\mathrm{j}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})\right]\) to express volume fractions.
\[\kappa_{\mathrm{T}}(\mathrm{aq})=\phi_{1} \,\left[1 / \mathrm{V}_{1}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{T} 1}(\mathrm{aq})+\phi_{\mathrm{j}} \,\left[1 / \mathrm{V}_{\mathrm{j}}(\mathrm{aq})\right] \, \mathrm{K}_{\mathrm{Tj}}(\mathrm{aq}) \nonumber \]
The latter equation is not tremendously helpful. Although \(\kappa_{\mathrm{T}}(\mathrm{aq})\) can be measured, the right hand side involves six terms about which we have no information ‘a priori’ and which depend on the composition of the solution.
Footnotes
[1] F. T. Gucker, Chem. Rev.,1933, 14 ,127.
[2] F. T. Gucker, J. Am. Chem. Soc.,1933, 55 ,2709.
[3] Units; \(\phi\left(\mathrm{K}_{\mathrm{Tj}}\right)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right] ; \mathrm{K}_{\mathrm{T} 1}^{*}(\ell)=\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]\); \(\mathrm{K}_{\mathrm{T}}(\mathrm{aq})=\left[\mathrm{m}^{3} \mathrm{~Pa}^{-1}\right]\)
For the solution \(\kappa_{\mathrm{T}}(\mathrm{aq})=\mathrm{K}_{\mathrm{T}}(\mathrm{aq}) / \mathrm{V}(\mathrm{aq})=\left[\mathrm{Pa}^{-1}\right]\)
For the solvent \(\kappa_{\mathrm{T} 1}^{*}(\ell)=\mathrm{K}_{\mathrm{T} 1}^{*}(\ell) / \mathrm{V}_{1}^{*}(\ell)=\left[\mathrm{Pa}^{-1}\right]\)
Isothermal compressions have units of ‘volume per unit of pressure’ whereas compressibilities have units of ‘reciprocal pressure’. \(\phi\left(\mathrm{K}_{\mathrm{T}_{\mathrm{j}}}\right)\) is an apparent molar isothermal compression on the grounds that the units of this quantity are \(\left[\mathrm{m}^{3} \mathrm{~mol}^{-1} \mathrm{~Pa}^{-1}\right]\). Some reports use the term ‘apparent molar isothermal compressibility’ which should be avoided because in the present context this term corresponds to a different property; see J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998, 94 ,2385.
[4] R.W. Gurney, Ionic Processes in Solution, McGraw-Hill, New York, 1953.
[5] P. D. Bernal and W. A. Van Hook, J. Chem. Thermodyn., 1986, 18 ,955.


10: Earthlike Planets - Venus and Mars
The Moon and Mercury are geologically dead. In contrast, the larger terrestrial planets—Earth, Venus, and Mars—are more active and interesting worlds. We have already discussed Earth, and we now turn to Venus and Mars. These are the nearest planets and the most accessible to spacecraft. Not surprisingly, the greatest effort in planetary exploration has been devoted to these fascinating worlds. In the chapter, we discuss some of the results of more than four decades of scientific exploration of Mars and Venus. Mars is exceptionally interesting, with evidence that points to habitable conditions in the past. Even today, we are discovering things about Mars that make it the most likely place where humans might set up a habitat in the future. However, our robot explorers have clearly shown that neither Venus nor Mars has conditions similar to Earth. How did it happen that these three neighboring terrestrial planets have diverged so dramatically in their evolution?
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- 10.1: The Nearest Planets - An Overview
- Venus, the nearest planet, is a great disappointment through the telescope because of its impenetrable cloud cover. Mars is more tantalizing, with dark markings and polar caps. Early in the twentieth century, it was widely believed that the “canals” of Mars indicated intelligent life there. Mars has only 11% the mass of Earth, but Venus is nearly our twin in size and mass. Mars rotates in 24 hours and has seasons like Earth; Venus has a retrograde rotation period of 243 days.
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- 10.2: The Geology of Venus
- Venus has been mapped by radar, especially with the Magellan spacecraft. Its crust consists of 75% lowland lava plains, numerous volcanic features, and many large coronae, which are the expression of subsurface volcanism. The planet has been modified by widespread tectonics driven by mantle convection, forming complex patterns of ridges and cracks and building high continental regions such as Ishtar. The surface is extraordinarily inhospitable, with pressure of 90 bars and temperature of 730 K.
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- 10.3: The Massive Atmosphere of Venus
- The atmosphere of Venus is 96% CO2. Thick clouds at altitudes of 30 to 60 kilometers are made of sulfuric acid, and a CO2greenhouse effect maintains the high surface temperature. Venus presumably reached its current state from more earthlike initial conditions as a result of a runaway greenhouse effect, which included the loss of large quantities of water.
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- 10.4: The Geology of Mars
- Most of what we know about Mars is derived from spacecraft: highly successful orbiters, landers, and rovers. We have also been able to study a few martian rocks that reached Earth as meteorites. Mars has heavily cratered highlands in its southern hemisphere, but younger, lower volcanic plains over much of its northern half. The Tharsis bulge, as big as North America, includes several huge volcanoes; Olympus Mons is more than 20 kilometers high and 500 kilometers in diameter.
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- 10.5: Water and Life on Mars
- The martian atmosphere has a surface pressure of less than 0.01 bar and is 95% CO2. It has dust clouds, water clouds, and carbon dioxide (dry ice) clouds. Liquid water on the surface is not possible today, but there is subsurface permafrost at high latitudes. Seasonal polar caps are made of dry ice; the northern residual cap is water ice, whereas the southern permanent ice cap is made predominantly of water ice with a covering of carbon dioxide ice.
Thumbnail: This May 2004 image shows the tracks made by the Mars Exploration Spirit rover on the surface of the red planet. Spirit was active on Mars between 2004 and 2010, twenty times longer than its planners had expected. It “drove” over 7.73 kilometers in the process of examining the martian landscape. (credit: modification of work by NASA/JPL/Cornell).


3.6: Equilibrium Analysis for a Rigid Body
For an rigid body in static equilibrium—that is, a non-deformable body where forces are not concurrent—the sum of both the forces and the moments acting on the body must be equal to zero. The addition of moments (as opposed to particles, where we only looked at the forces) adds another set of possible equilibrium equations, allowing us to solve for more unknowns as compared to particle problems.
Moments, like forces, are vectors. This means that our vector equation needs to be broken down into scalar components before we can solve the equilibrium equations. In a two-dimensional problem, the body can only have clockwise or counterclockwise rotation (corresponding to rotations about the \(z\) axis). This means that a rigid body in a two-dimensional problem has three possible equilibrium equations; that is, the sum of force components in the \(x\) and \(y\) directions, and the moments about the \(z\) axis. The sum of each of these will be equal to zero.
For a two-dimensional problem, we break our one vector force equation into two scalar component equations. \[ \sum \vec{F} \, = \, 0 \]
\[ \sum F_x \, = \, 0\, ; \,\, \sum F_y \, = \, 0 \] The one moment vector equation becomes a single moment scalar equation. \[ \sum \vec{M} \, = \, 0 \]
\[ \sum M_z \, = \, 0 \]
If we look at a three-dimensional problem we will increase the number of possible equilibrium equations to six. There are three equilibrium equations for force, where the sum of the components in the \(x\), \(y\), and \(z\) directions must be equal to zero. The body may also have moments about each of the three axes. The second set of three equilibrium equations states that the sum of the moment components about the \(x\), \(y\), and \(z\) axes must also be equal to zero.
We break the forces into three component equations. \[ \sum \vec{F} \, = \, 0 \]
\[ \sum F_x \, = \, 0 \, ; \,\, \sum F_y \, = \, 0 \, ; \,\, \sum F_z \, = \, 0 \]
Then we also break the moments into three component equations. \[ \sum \vec{M} \, = \, 0 \]
\[ \sum M_x \, = \, 0 \, ; \,\, \sum M_y \, = \, 0 \, ; \,\, \sum M_z \, = \, 0 \]
Finding the Equilibrium Equations
As with particles, the first step in finding the equilibrium equations is to draw a free body diagram of the body being analyzed. This diagram should show all the force vectors acting on the body. In the free body diagram, provide values for any of the known magnitudes, directions, and points of application for the force vectors and provide variable names for any unknowns (either magnitudes, directions, or distances).
Next you will need to choose the \(x\), \(y\), and \(z\) axes. These axes do need to be perpendicular to one another, but they do not necessarily have to be horizontal or vertical. If you choose coordinate axes that line up with some of your force vectors you will simplify later analysis.
Once you have chosen axes, you need to break down all of the force vectors into components along the \(x\), \(y\) and \(z\) directions (see the vectors page in Appendix 1 page for more details on this process). Your first equation will be the sum of the magnitudes of the components in the \(x\) direction being equal to zero, the second equation will be the sum of the magnitudes of the components in the \(y\) direction being equal to zero, and the third (if you have a 3D problem) will be the sum of the magnitudes in the \(z\) direction being equal to zero.
Next you will need to come up with the the moment equations. To do this you will need to choose a point to take the moments about. Any point should work, but it is usually advantageous to choose a point that will decrease the number of unknowns in the equation. Remember that any force vector that travels through a given point will exert no moment about that point. To write out the moment equations, simply sum the moments exerted by each force (adding in pure moments shown in the diagram) about the given point and the given axis, and set that sum equal to zero. All moments will be about the \(z\) axis for two-dimensional problems, though moments can be about the \(x\), \(y\) and \(z\) axes for three-dimensional problems.
Once you have your equilibrium equations, you can solve these formulas for unknowns. The number of unknowns that you will be able to solve for will again be the number of equations that you have.
The car below has a weight of 1500 lbs with the center of mass 4 ft behind the front wheels of the car. What are the normal forces on the front and the back wheels of the car?
- Solution
A 5-meter-long beam has a fixed connection to a wall at point A and a force acting as shown at point B. What are the reaction forces acting on the beam at point A?
- Solution
A ladder with negligible mass is supporting a 120-lb person as shown below. If the contact point at A is frictionless, and the contact point at B is a rough connection, determine the forces acting at contact points A and B.
- Solution
Member ABC is 6 meters long, with point B being at its midpoint. Determine all forces acting on member ABC.
- Solution
While sitting in a chair, a person exerts the forces in the diagram below. Determine all forces acting on the chair at points A and B. (Assume A is frictionless and B is a rough surface).
- Solution
The trailer shown below consists of a deck with a weight of 250 lbs on an axle with wheels with a weight of 350 lbs. Assume the weight forces act in the center of each component. If we wish the tongue weight (\(F_T\)) of the unloaded trailer to be 50 lbs, what is the distance \(d\) from the front where we must place the axle?
- Solution
A 12-inch-by-24-inch flat steel sign is supported by two cables, each 6 inches from the edge of the sign. The sign has a weight of 10 lbs, and the wind is causing the sign to sit at an angle of 10 degrees from vertical (the \(y\) axis). If we treat the wind as a point force acting in the negative \(z\) direction on the center of the sign, how strong must the wind force be to cause this ten-degree angle?
- Solution
A sixty-kilogram acoustic panel is suspended by three cables as shown below. Assuming the panel has a uniformly distributed weight, what is the tension in each of the cables?
- Solution


17.5: Notes
-
- Last updated
- Save as PDF
- Marwan M. Kraidy
- University of California Press
Vicki Mayer—who alerted me to the work of Hans Joas—Michael Curtin, and Toby Miller have been key interlocutors on issues related to this chapter. I also thank Katerina Girginova for research assistance on creativity, Michael Curtin and Kevin Sanson for their useful feedback on the first draft of this chapter, and Marina Krikorian for editorial help.
1 Media stories and academic publications celebrating Arab revolutionary rap and graffiti have become so commonplace that we could talk of a Revolutionary Graffiti Index or an Arab Rap Index, following what Miller, referring to Richard Florida’s work, calls the Technological and Gay Indexes (Toby Miller, “A View from a Fossil: The New Economy, Creativity and Consumption—Two or Three Things I Don’t Believe In,” International Journal of Cultural Studies 7.1 [2004]: 60).
2 Clearly, this is only a small part of revolutionary labor at large, which includes demonstrating, confronting policy and security personnel, building barricades, feeding revolutionaries, tending to the wounded, and so on.
3 Marwan M. Kraidy, The Naked Blogger of Cairo: Creative Insurgency in the Arab World (Cambridge, MA: Harvard University Press, 2016).
4 James M. Jasper, The Art of Moral Protest: Culture, Biography and Creativity in Social Movements (Chicago: University of Chicago Press, 1997).
5 Vicki Mayer, Below the Line: Producers and Production Studies in the New Television Economy (Durham, NC: Duke University Press, 2011).
6 Hans Joas, The Creativity of Action (Chicago: University of Chicago Press, 1996).
7 Maurizio Lazzarato, “Immaterial Labour,” in Radical Thought in Italy: A Potential Politics , ed. Paolo Virno and Michael Hardt (Minneapolis: University of Minnesota Press, 1996), 133–147.
8 For example, Benjamin Shephard, Play, Creativity and Social Movement (New York: Routledge, 2011), pivots around the notion of play; while Glenda Ballantyne, Creativity and Critique: Subjectivity and Agency in Touraine and Ricoeur (Leiden: Brill, 2007), focuses on subjectivity and agency. The “culture-jamming” literature does not apply in this context because it concerns relatively low-risk subversion of consumer culture in relatively stable, relatively democratic, industrialized countries.
9 Jasper, Art of Moral Protest , 378–379.  The sociologist Doug McAdam has done extensive work on tactical innovation. For a summary introduction, see Doug McAdam, “Tactical Interaction and Innovation,” The Wiley-Blackwell Encyclopedia of Social and Political Movements , ed. David A. Snow et al. (Hoboken: Wiley-Blackwell, 2013).
12 Jasper, Art of Moral Protest , 65.
13 Ibid., 159.
14 Ibid., 219.
15 Ibid., 99. For an argument about upbringing and social support as key biographical enablers of creativity in famously “creative” people, see Howard Gardner, Creating Minds: An Anatomy of Creativity Seen through the Lives of Freud, Einstein, Picasso, Stravinsky, Eliot, Graham, and Gandhi (New York: Basic Books, 2011).
16 Political with a capital P connotes issues of state power and resistance to it, as opposed to cultural politics.
17 Media, cultural, and music production scholars have addressed labor issues, though mostly focusing on the exploitation of labor by those industries. See Toby Miller et al., Global Hollywood: Issue 2 (London: British Film Institute, 2004); Mark Andrejevic, Reality TV: The Work of Being Watched (Lanham, MD: Rowman and Littlefield, 2004); Matt Stahl, Unfree Masters: Recording Artists and the Politics of Work (Durham, NC: Duke University Press, 2013). The emerging literature on digital labor also focuses on the increasingly exploitative nature of capitalism; see Christian Fuchs, Digital Labour and Karl Marx (London: Routledge, 2014). This literature’s socio-economic focus is helpful, but only indirectly, for studying a revolutionary setting.
18 Mayer, Below the Line , 33.
19 Ibid., 32.
20 Miller, “A View from a Fossil.”
21 Mayer, Below the Line , 43.
22 Ibid.
23 See their YouTube channel at www.youtube.com/user/MasasitMati.
24 Mayer, Below the Line , 44.
25 Ibid.
26 Ibid., 47–51.
27 Jasper’s definition of creativity as an “extreme form of flexibility” ( The Art of Moral Protest , 94) has a matter-of-fact resonance in a revolutionary setting.
28 Mayer, Below the Line , 58.
29 Ibid., 56–57.
30 Ibid., 58.
31 Ibid., 59.
32 Michael Curtin, Playing to the World’s Biggest Audience: The Globalization of Chinese Film and TV (Berkeley: University of California Press, 2007).
33 One aspect of the compatibility of revolutionary work with digital labor is that they are both unpaid. Though Lazarrato’s elaboration of “immaterial labor” focused on the remunerated kind, there has been an active discussion of unwaged labor in the digital era at least since Terranova’s 2000 article: Tiziana Terranova, “Free Labor: Producing Culture for the Digital Economy,” Social Text 18.2 (2000): 33–58. Andrejevic’s critique of reality television highlighted the unpaid “work of being watched” (Reality TV, 2004). Most recently, see the special issue of TripleC , “Philosophers of the World Unite! Theorising Digital Labour and Virtual Work—Definitions, Dimensions and Form,” ed. Christian Fuchs et al.; especially Brian A. Brown, “Will Work for Free: The Biopolitics of Unwaged Digital Labour,” TripleC: Communication, Capitalism and Critique 12.2 (2014): 694–712, www.triple-c.at/index.php/tripleC/article/view/538.
34 Marwan M. Kraidy, “No Country for Funny Men,” Al-Jazeera America , February 26, 2014, http://america.aljazeera.com/opinion...-funnymen.html .
35 For more details, see Kraidy, The Naked Blogger of Cairo .
36 Joas, The Creativity of Action , 71 (emphasis in original).
37 I suspect that a systematic, theoretical, and comparative examination of “action” and “labor” would unearth fascinating overlaps and differences, but this falls outside the purview of this chapter.
38 Jasper, Art of Moral Protest , 154.
39 Lazzarato, “Immaterial Labour,” 139.
40 Marwan M. Kraidy, “The Revolutionary Body Politic: Preliminary Thoughts on a Neglected Medium in the Arab Uprisings,” Middle East Journal of Culture and Communication 5.1 (2012): 68–76; “The Body as Medium in the Digital Age: Challenges and Opportunities,” Communication and Critical-Cultural Studies 10.2–3 (2013): 285–290; “The Politics of Revolutionary Celebrity in the Contemporary Arab World,” Public Culture 27.1 (2014).
41 Mayer, Below the Line , 41–42.
42 I initially elaborated it in Marwan M. Kraidy, “Governance and Hypermedia in Saudi Arabia,” First Monday 11.9 (2006), http://firstmonday.org/issues/specia...idy/index.html . For an application of the concept in the context of political activism before the Arab uprisings, see Marwan M. Kraidy, Reality Television and Arab Politics: Contention in Public Life (Cambridge: Cambridge University Press, 2010).


12.1A: Passive Immunization
- Describe how artificial and natural passive immunity function to provide antibody protection against microorganisms
There are two types of passive immunity: artificial and natural. Artificial passive immunity is achieved by infusion of serum or plasma containing high concentrations of antibody. This form of passive immunity provides immediate antibody protection against microorganisms such as hepatitis A by administering preformed antibodies. These antibodies have been produced by another person or animal that has been actively immunized, but the ultimate recipient has not produced them. The recipient will only temporarily benefit from passive immunity for as long as the antibodies persist in their circulation.This type of immunity is short acting, and is typically seen in cases where a patient needs immediate protection from a foreign body and cannot form antibodies quickly enough independently.
Passive immunity can also be acquired naturally by the fetus due to the transfer of antibodies by the maternal circulation in utero through the placenta around the third month of gestation. Immunity in newborn babies is only temporary and starts to decrease after the first few weeks, or months. Breast milk also contains antibodies, which means that babies who are breastfed have passive immunity for longer periods of time. The thick, yellowish milk (colostrum) that is produced during the first few days after birth is particularly rich in antibodies. For the newborn to have lasting protection, active immunity must be received. The first immunisation, given when a baby is two months old, includes whooping cough and Hib (haemophilus influenza type b) because immunity to these diseases decreases the fastest. Passive immunity to measles, mumps and rubella (MMR) usually lasts for about a year, which is why the MMR is given just after the baby’s first birthday.
Key Points
- Passive immunization provides humoral immunity.
- Artificial passive immunization is the injection of preformed antibody solution when a patient is incapable of producing antibodies fast enough to combat a disease.
- Natural passive immunization is the transfer of antibodies through the placenta of a pregnant woman to the fetus. Immunity lasts for a couple of months after the baby is born, after which active immunization is required.
Key Terms
- in utero : Occurring or residing within the uterus or womb; unborn.


5.9: Kinematic analysis and coordinate systems
The kinematic analysis presented in the previous sections is valid for any coordinate system or constraint curve \(\overrightarrow{\boldsymbol{r}}_{s}\). We now show how to obtain the 3 dimensional motion in a specific coordinate system or a specific path curve \(\overrightarrow{\boldsymbol{r}}_{s}(s)\) that is expressed in Cartesian or cylindrical coordinates. An important case is motion along a straight line.
5.9.1 Rectilinear motion
Motion along a straight \(1 \mathrm{D}\) line is the most simple to analyse. If motion occurs along the \(x\)-coordinate axis in Cartesian coordinates, the path curve pointing in the positive \(x\)-coordinate direction is:
\[\overrightarrow{\boldsymbol{r}}_{s}(s)=s \hat{\boldsymbol{\imath}} \tag{5.53} \label{5.53}\]
A certain point mass \(i\) with path coordinate \(s_{i}\) is located at position vector \(\overrightarrow{\boldsymbol{r}}_{i}=\overrightarrow{\boldsymbol{r}}_{s}\left(s_{i}\right)=s_{i} \hat{\boldsymbol{\imath}}\). Projecting its position on the \(x\)-coordinate axis, we find that its \(x\)-coordinate is identical to its path coordinate \(x_{i}=\overrightarrow{\boldsymbol{r}}_{i} \cdot \hat{\boldsymbol{\imath}}=s_{i}\). So, essentially everything that we have derived for \(s_{i}(t)\) also holds for the \(x\)-coordinate \(x_{i}(t)\) of a point mass that moves on a rectilinear path along the \(x\)-coordinate axis, with \(s_{i}=x_{i}, v_{s, i}=\dot{x}_{i}\) and \(a_{s, i}=\ddot{x}_{i}\), from Eqs. (5.30)-(5.32).
5.9.2 Kinematics in Cartesian coordinates
Cartesian coordinates have the advantage that the direction of the unit vectors \(\hat{\boldsymbol{\imath}}, \hat{\boldsymbol{\jmath}}\) and \(\hat{\boldsymbol{k}}\) is independent of the coordinates in the system. If the coordinate system is not rotating (see also Sec. 6.12), the direction of the unit vectors is also independent of time \(\frac{\mathrm{d} \hat{\imath}}{\mathrm{d} t}=\frac{\mathrm{d} \hat{\jmath}}{\mathrm{d} t}=\frac{\mathrm{d} \hat{\boldsymbol{k}}}{\mathrm{d} t}=\overrightarrow{\mathbf{0}}\). This greatly simplifies the kinematic analysis and allows describing the motion of a point mass \(i\) using its 3 time-dependent coordinates \(x_{i}, y_{i}\) and \(z_{i}\) and Eqs. (5.28)-(5.29) as follows:
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{i}(t) & =x_{i}(t) \hat{\boldsymbol{\imath}}+y_{i}(t) \hat{\boldsymbol{\jmath}}+z_{i}(t) \hat{\boldsymbol{k}} \tag{5.54} \label{5.54}\\[4pt] \overrightarrow{\boldsymbol{v}}_{i}(t) & =\dot{x}_{i}(t) \hat{\boldsymbol{\imath}}+\dot{y}_{i}(t) \hat{\boldsymbol{\jmath}}+\dot{z}_{i}(t) \hat{\boldsymbol{k}} \tag{5.55} \label{5.55}\\[4pt] \overrightarrow{\boldsymbol{a}}_{i}(t) & =\ddot{x}_{i}(t) \hat{\boldsymbol{\imath}}+\ddot{y}_{i}(t) \hat{\boldsymbol{\jmath}}+\ddot{z}_{i}(t) \hat{\boldsymbol{k}} \tag{5.56} \label{5.56}\end{align}\]
It is also possible to integrate the individual functions \(\ddot{x}_{i}(t), \ddot{y}_{i}(t), \ddot{z}_{i}(t)\) to obtain the velocity and position of the point mass when the acceleration is known, similar to the integrals shown in Eqs. (5.35), (5.36), for the velocity and acceleration component along the \(x, y\) and \(z\)-axis:
\[\begin{align} & \overrightarrow{\boldsymbol{r}}_{i}\left(t_{2}\right)=\overrightarrow{\boldsymbol{r}}_{i}\left(t_{1}\right)+\int_{t_{1}}^{t_{2}} \dot{x}_{i}(t) \mathrm{d} t \hat{\boldsymbol{\imath}}+\int_{t_{1}}^{t_{2}} \dot{y}_{i}(t) \mathrm{d} t \hat{\boldsymbol{\jmath}}+\int_{t_{1}}^{t_{2}} \dot{z}_{i}(t) \mathrm{d} t \hat{\boldsymbol{k}} \tag{5.57} \label{5.57}\\[4pt] & \overrightarrow{\boldsymbol{v}}_{i}\left(t_{2}\right)=\overrightarrow{\boldsymbol{v}}_{i}\left(t_{1}\right)+\int_{t_{1}}^{t_{2}} \ddot{x}_{i}(t) \mathrm{d} t \hat{\boldsymbol{\imath}}+\int_{t_{1}}^{t_{2}} \ddot{y}_{i}(t) \mathrm{d} t \hat{\boldsymbol{\jmath}}+\int_{t_{1}}^{t_{2}} \ddot{z}_{i}(t) \mathrm{d} t \hat{\boldsymbol{k}} \tag{5.58} \label{5.58}\end{align}\]
Let us illustrate the kinematic analysis method in Cartesian coordinates by determining the acceleration and velocity vector of the balloon \(B\) in Figure 5.9, that moves along a parabolic path curve with \(y_{B}=c_{1} x_{B}^{2}\) in the \(x y\)-plane.
The motion of the balloon is described by:
\[\begin{align} x_{B}(t) & =c_{2} t \\[4pt] y_{B}(t) & =c_{1} c_{2}^{2} t^{2} \end{align}\]
The position vector of the balloon can be differentiated twice, like in Equation 5.54 and Equation 5.56, to obtain the velocity and acceleration vectors as follows:
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{B}(t) & =c_{2} t \hat{\boldsymbol{\imath}}+c_{1} t^{2} \hat{\boldsymbol{\jmath}} \\[4pt] \overrightarrow{\boldsymbol{v}}_{B}(t) & =c_{2} \hat{\boldsymbol{\imath}}+2 c_{1} t \hat{\boldsymbol{\jmath}} \\[4pt] \overrightarrow{\boldsymbol{a}}_{B}(t) & =2 c_{1} \hat{\boldsymbol{j}} \end{align}\]
The speed can be obtained by taking the absolute value of the velocity vector:
\[v_{B}=\left|\overrightarrow{\boldsymbol{v}}_{B}\right|=\sqrt{\overrightarrow{\boldsymbol{v}}_{B} \cdot \overrightarrow{\boldsymbol{v}}_{B}}=\sqrt{c_{2}^{2}+\left(2 c_{1} t\right)^{2}} \tag{5.59} \label{5.59}\]
5.9.3 Kinematics in cylindrical coordinates
Cylindrical coordinates, which were introduced in section 3.3.2 are often convenient when the kinematics involves rotation around an axis, like in Figure 5.10. The position, velocity and acceleration vector in cylindrical coordinates can be written as:
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{i} & =\rho_{i} \hat{\boldsymbol{\rho}}+z_{i} \hat{\boldsymbol{k}} \tag{5.60} \label{5.60}\\[4pt] \overrightarrow{\boldsymbol{v}}_{i} & =v_{i, \rho} \hat{\boldsymbol{\rho}}+v_{i, \phi} \hat{\boldsymbol{\phi}}+v_{i, z} \hat{\boldsymbol{k}} \tag{5.61} \label{5.61}\\[4pt] \overrightarrow{\boldsymbol{a}}_{i} & =a_{i, \rho} \hat{\boldsymbol{\rho}}+a_{i, \phi} \hat{\boldsymbol{\phi}}+a_{i, z} \hat{\boldsymbol{k}} \tag{5.62} \label{5.62}\end{align}\]
Concept. Kinematics in cylindrical coordinates
We will show that by taking the time derivatives of the position vector, the components of the velocity vector and acceleration vector in Eqs. (5.60-5.62) are found to be:
\[\begin{align} v_{i, \rho} & =\dot{\rho}_{i} \tag{5.63} \label{5.63}\\[4pt] v_{i, \phi} & =\rho_{i} \dot{\phi}_{i} \tag{5.64} \label{5.64}\\[4pt] v_{i, z} & =\dot{z}_{i} \tag{5.65} \label{5.65}\\[4pt] a_{i, \rho} & =\ddot{\rho}_{i}-\rho_{i} \dot{\phi}_{i}^{2} \tag{5.66} \label{5.66}\\[4pt] a_{i, \phi} & =2 \dot{\rho}_{i} \dot{\phi}_{i}+\rho_{i} \ddot{\phi}_{i} \tag{5.67} \label{5.67}\\[4pt] a_{i, z} & =\ddot{z}_{i} \tag{5.68} \label{5.68}\end{align}\]
Derivation.
The determination of these velocity and acceleration vectors in cylindrical coordinates is more difficult because the direction of the unit vectors \(\hat{\rho}\) and \(\hat{\boldsymbol{\phi}}\) is not constant. To determine their time derivatives, we use that the position vector can also be expressed in Cartesian unit vectors as:
\[\overrightarrow{\boldsymbol{r}}_{i}=\rho_{i} \cos \phi_{i} \hat{\boldsymbol{\imath}}+\rho_{i} \sin \phi_{i} \hat{\boldsymbol{\jmath}}+z_{i} \hat{\boldsymbol{k}} \tag{5.69} \label{5.69}\]
From the directions of the unit vectors, as shown in Figure 5.10, the cylindrical unit vectors can be expressed in Cartesian unit vectors as:
\[\begin{align} & \hat{\boldsymbol{\rho}}=\cos \phi_{i} \hat{\boldsymbol{\imath}}+\sin \phi_{i} \hat{\boldsymbol{\jmath}} \tag{5.70} \label{5.70}\\[4pt] & \hat{\boldsymbol{\phi}}=-\sin \phi_{i} \hat{\boldsymbol{\imath}}+\cos \phi_{i} \hat{\boldsymbol{\jmath}} \tag{5.71} \label{5.71}\end{align}\]
Now it can be seen that the time derivatives of the unit vectors obey (chain rule):
\[\begin{align} \frac{\mathrm{d} \hat{\boldsymbol{\rho}}}{\mathrm{d} t} & =\left(-\sin \phi_{i} \hat{\boldsymbol{i}}+\cos \phi_{i} \hat{\boldsymbol{j}}\right) \dot{\phi}_{i}=\dot{\phi}_{i} \hat{\boldsymbol{\phi}}=\overrightarrow{\boldsymbol{\omega}} \times \hat{\boldsymbol{\rho}} \tag{5.72} \label{5.72}\\[4pt] \frac{\mathrm{d} \hat{\boldsymbol{\phi}}}{\mathrm{d} t} & =\left(-\cos \phi_{i} \hat{\boldsymbol{i}}-\sin \phi_{i} \hat{\boldsymbol{j}}\right) \dot{\phi}_{i}=-\dot{\phi}_{i} \hat{\boldsymbol{\rho}}=\overrightarrow{\boldsymbol{\omega}} \times \hat{\boldsymbol{\phi}} \tag{5.73} \label{5.73}\end{align}\]
where the terms in brackets were replaced by \(\hat{\boldsymbol{\phi}}\) and \(\hat{\boldsymbol{\rho}}\) using Eqs. (5.70) and (5.71), and the equations were further simplified using the vector \(\overrightarrow{\boldsymbol{\omega}}=\dot{\phi}_{i} \hat{\boldsymbol{k}}\). We use these time derivatives of the unit vectors to find the velocity and acceleration in cylindrical coordinates, and the components given in Eqs. (5.63-5.68) by differentiating twice:
\[\begin{align} \overrightarrow{\boldsymbol{r}}_{i} & =\rho_{i} \hat{\boldsymbol{\rho}}+z_{i} \hat{\boldsymbol{k}} \tag{5.74} \label{5.74}\\[4pt] \overrightarrow{\boldsymbol{v}}_{i} & =\frac{\mathrm{d} \overrightarrow{\boldsymbol{r}}_{i}}{\mathrm{~d} t}=\dot{\rho}_{i} \hat{\boldsymbol{\rho}}+\rho_{i} \frac{\mathrm{d} \hat{\boldsymbol{\rho}}}{\mathrm{d} t}+\dot{z}_{i} \hat{\boldsymbol{k}} \tag{5.75} \label{5.75}\\[4pt] & =\dot{\rho}_{i} \hat{\boldsymbol{\rho}}+\rho_{i} \dot{\phi}_{i} \hat{\boldsymbol{\phi}}+\dot{z}_{i} \hat{\boldsymbol{k}} \tag{5.76} \label{5.76}\\[4pt] \overrightarrow{\boldsymbol{a}}_{i} & =\frac{\mathrm{d} \overrightarrow{\boldsymbol{v}}_{i}}{\mathrm{~d} t}=\ddot{\rho}_{i} \hat{\boldsymbol{\rho}}+\dot{\rho}_{i} \frac{\mathrm{d} \hat{\boldsymbol{\rho}}}{\mathrm{d} t}+\dot{\rho}_{i} \dot{\phi}_{i} \hat{\boldsymbol{\phi}}+\rho_{i} \ddot{\phi}_{i} \hat{\boldsymbol{\phi}}+\rho_{i} \dot{\phi}_{i} \frac{\mathrm{d} \hat{\boldsymbol{\phi}}}{\mathrm{d} t}+\ddot{z}_{i} \hat{\boldsymbol{k}} \tag{5.77} \label{5.77}\\[4pt] & =\left(\ddot{\rho}_{i}-\rho_{i} \dot{\phi}_{i}^{2}\right) \hat{\boldsymbol{\rho}}+\left(2 \dot{\rho}_{i} \dot{\phi}_{i}+\rho_{i} \ddot{\phi}_{i}\right) \hat{\boldsymbol{\phi}}+\ddot{z}_{i} \hat{\boldsymbol{k}} \tag{5.78} \label{5.78}\end{align}\]
The components of these vectors are identical to the equations (5.63-5.68) thus proving the correctness of those equations.
5.9.4 Circular motion
A special important case of constrained motion on a path curve is motion on a circle with constant radius \(\rho\). So, in Figure 5.10 this means \(\rho_{i}=\) constant. The components of the velocity and acceleration vector can be found by substituting \(\dot{\rho}_{i}=0\) and \(\ddot{\rho}_{i}=0\) into Eqs. (5.75-5.78) to obtain:
\[\begin{align} v_{i, \rho} & =0 \tag{5.79} \label{5.79}\\[4pt] v_{i, \phi} & =\rho \dot{\phi}_{i}=\dot{s}_{i} \tag{5.80} \label{5.80}\\[4pt] a_{i, \rho} & =-\rho \dot{\phi}_{i}^{2}=-\frac{v_{s, i}^{2}}{\rho} \tag{5.81} \label{5.81}\\[4pt] a_{i, \phi} & =\rho \ddot{\phi}_{i}=\dot{v}_{s, i}=\ddot{s}_{i} \tag{5.82} \label{5.82}\end{align}\]
It can be seen that the velocity vector only has a component in the \(\hat{\phi}\) direction, tangential to the circle. The acceleration has both a tangential component \(a_{i, \phi}\),
and a radial or centripetal component \(a_{i, \rho}\) that points towards the centre of the circle. The time derivatives of the angle \(\omega_{o, i}=\dot{\phi}_{i}\) and \(\alpha_{o, i}=\ddot{\phi}_{i}\) are called the orbital angular velocity and orbital angular acceleration of the point mass. The word orbital is used to distinguish this angular velocity from the angular velocity of a rigid body, which is sometimes called the spin angular velocity and will be discussed later in Ch. 9.
5.9.5 Natural \(t, n, b\) coordinates for curvilinear motion
Natural coordinates make use of the fact that even if a curved trajectory is not really a circle, it is possible to approximate it by a circle with radius \(\rho\) at every point of the trajectory, as shown in Figure 5.11. Like in the previous subsection, a cylindrical coordinate system can be chosen such that the centre of that circle (the centre of curvature \(O_{i}\) ) is at the origin and the point mass position is determined by a path coordinate \(s_{i}=\rho \phi_{i}\), where \(\rho\) is called the radius of curvature. The plane in which the instantaneous circle is positioned is called the osculating plane. For an instantaneous moment, the motion of the point mass can be described using natural coordinates or \(t, n, b\) coordinates, that are defined by a unit vector \(\hat{\boldsymbol{t}}\) that is tangential to the curvilinear path \({ }^{2}\), a normal unit vector \(\hat{\boldsymbol{n}}\) that points towards the centre of the instantaneous circle and a binormal unit vector \(\hat{\boldsymbol{b}}=\hat{\boldsymbol{t}} \times \hat{\boldsymbol{n}}\), that points perpendicular to the osculating plane. At every instant of time the path curve can have a different origin \(O\) and radius \(\rho\). Note that in Figure 5.11 the origin changes from point \(O_{1}\) to \(O_{2}\). As a consequence, at times \(t_{1}\) and times \(t_{3}\), the directions of both the \(\hat{\boldsymbol{n}}\) and \(\hat{\boldsymbol{b}}\) unit vectors have flipped because the motion changed from anti-clockwise to clockwise, even though the tangential unit vector \(\hat{\boldsymbol{t}}\) is the same. In \(t, n, b\) coordinates, the components of the instantaneous velocity vector are \(\overrightarrow{\boldsymbol{v}}_{i}=v_{i, t} \hat{\boldsymbol{t}}+v_{i, n} \hat{\boldsymbol{n}}\) and the acceleration vector is \(\overrightarrow{\boldsymbol{a}}_{i}=a_{i, t} \hat{\boldsymbol{t}}+a_{i, n} \hat{\boldsymbol{n}}\). Note the sign change \(a_{i, n}=-a_{i, \rho}\) compared to Equation 5.81, because \(\hat{\boldsymbol{n}}\) points inward, while \(\hat{\boldsymbol{\rho}}\) points away from the rotation axis. The velocity and acceleration components are:
\[\begin{align} v_{i, t} & =\dot{s}_{i} \tag{5.83} \label{5.83}\\[4pt] v_{i, n} & =0 \tag{5.84} \label{5.84}\\[4pt] a_{i, t} & =\dot{v}_{s, i}=\ddot{s}_{i} \tag{5.85} \label{5.85}\\[4pt] a_{i, n} & =v_{s, i}^{2} / \rho \tag{5.86} \label{5.86}\end{align}\]
Velocity and acceleration along the \(\hat{\boldsymbol{b}}\) direction are always zero. Note that these equations are very similar to those for circular motion, except for the minus sign in Equation 5.81. The radius \(\rho\) can change continuously along the curvilinear path and needs to be determined at every point of the curve in order to use these equations to determine the given velocity and acceleration components. Furthermore, because they do not have a fixed origin, natural coordinates are not used for describing positions, and are therefore not a replacement for Cartesian and cylindrical coordinates.
5.9.6 Summary parametrised motion and kinematics
After having discussed kinematics in different coordinate systems we conclude by summarising the different ways these coordinates allow us to parametrise motion in 3D and obtain the kinematics. Depending on the situation, the time dependent motion \(\overrightarrow{\boldsymbol{r}}(t)\) can be fully described either using 3 time dependent coordinate functions \(x(t), y(t), z(t)\), or using the path curve \(\overrightarrow{\boldsymbol{r}}_{s}(s)=x(s) \hat{\boldsymbol{\imath}}+\) \(y(s) \hat{\boldsymbol{j}}+z(s) \hat{\boldsymbol{k}}\) and a single time dependent coordinate function \(s(t):\)
Parametrisations of motion and determination of \(\vec{v}\) and \(\vec{a}\)
The time dependent coordinate functions:
-
Cartesian coordinate functions \(x(t)\) and \(y(t)\).
- Use methods from Sec. 5.9.2 to obtain \(\overrightarrow{\boldsymbol{v}}\) and \(\overrightarrow{\boldsymbol{a}}\).
-
Cylindrical coordinate functions \(\rho(t)\) and \(\phi(t)\).
- Use methods from Sec. 5.9.3 to obtain \(\overrightarrow{\boldsymbol{v}}\) and \(\overrightarrow{\boldsymbol{a}}\).
- First convert to Cartesian coordinates using \(x(t)=\rho(t) \cos \phi(t)\) and \(y(t)=\rho(t) \sin \phi(t)\), and proceed like in 1.
The path curve and one coordinate function of time:
-
Cartesian path curve \(x(s), y(s)\), and path coordinate \(s(t)\).
- Use methods from Sec. 5.8 to determine the path velocity \(v_{s}\), and tangential component of acceleration \(a_{s}\) from \(s(t)\).
- Use \(t, n, b\) coordinates (Sec. 5.9.5) to obtain the normal component of acceleration \(a_{n}\) by first determining the local path radius \(\rho\).
- Alternatively one can determine \(x(t)=x(s(t)), y(t)=y(s(t))\) and proceed like in 1.
-
Cartesian path curve \(y(x)\) and coordinate function \(x(t)\).
- Determine the function \(y(t)=y(x(t))\) and use method 1 .
-
Cylindrical path curve \(\rho(s), \phi(s)\) and path coordinate \(s(t)\).
- Proceed using \(s(t)\) and \(t, n, b\) coordinates like in 3 .
- Use \(x(t)=\rho(s(t)) \cos \phi(s(t))\) and \(y(t)=\rho(s(t)) \sin \phi(s(t))\), and proceed like in 1.
-
Cylindrical path curve \(\rho(\phi)\) and coordinate function \(\phi(t)\).
- Determine the function \(\rho(t)=\rho(\phi(t))\) and use method 2 .
As becomes clear from this list, there are often multiple methods to determine the velocity and acceleration from given kinematic information. However, some methods might be much more mathematically complex than others. It is therefore useful to become familiar with the different methods, and get experience in choosing the easiest one in a certain situation.
5.9.7 Determining the path coordinate*
We note that it is also possible to determine the path coordinate \(s(t)\) by going from parametrisation 4 to 3 . For that purpose you use that it holds from Pythagoras’ theorem that the distance \(\mathrm{d} s\) along a straight infinitesimal line element obeys \(\mathrm{d} s^{2}=\mathrm{d} x^{2}+\mathrm{d} y^{2}+\mathrm{d} z^{2}\), such that \(s(t)\) can be determined using the integral:
\[s(t)=\int_{0}^{s(t)} \mathrm{d} s=\int_{0}^{x(t)} \sqrt{1+\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^{2}+\left(\frac{\mathrm{d} z}{\mathrm{~d} x}\right)^{2}} \mathrm{~d} x \tag{5.87} \label{5.87}\]
For example if we deal with motion at constant speed along a straight tilted line, \(y=\alpha x\), and \(x(t)=v_{x} t\) then we have:
\[s(t)=\int_{0}^{v_{x} t} \sqrt{1+\alpha^{2}} \mathrm{~d} x=\sqrt{1+\alpha^{2}} \times v_{x} t \tag{5.88} \label{5.88}\]
Noting that the \(y\) velocity component is \(v_{y}=\dot{y}=\frac{\mathrm{d} y}{\mathrm{~d} x} \dot{x}=\alpha v_{x}\), we see that the path velocity \(v_{s}=\dot{s}=\sqrt{1+\alpha^{2}} \times v_{x}=\sqrt{v_{x}^{2}+v_{y}^{2}}\) as expected.
5.9.8 Determining \(\rho\), the radius of curvature*
When working with natural \(t, n, b\) coordinates, one needs to know the local radius of curvature \(\rho\) at each position to determine the velocity and acceleration components using (5.83-5.86). However, determining the radius of curvature \(\rho\) is not always easy. For a path curve in the \(x y\)-plane that can be written in the form \(y(x)\), the instantaneous radius \(\rho\) can be determined using this convenient function:
\[\rho=\frac{\left[1+\frac{\mathrm{d} y}{\mathrm{~d} x}\right]^{\frac{3}{2}}}{\left|\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right|} \tag{5.89} \label{5.89}\]
\[\begin{align} \rho^{2} & =\left(y-y_{0}\right)^{2}+\left(x-x_{0}\right)^{2} \tag{5.90} \label{5.90}\\[4pt] y & =y_{0}+\sqrt{\rho^{2}-\left(x-x_{0}\right)^{2}} \tag{5.91} \label{5.91}\\[4pt] \frac{\mathrm{d} y}{\mathrm{~d} x} & =-\frac{x-x_{0}}{y-y_{0}} \tag{5.92} \label{5.92}\\[4pt] \frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}} & =\frac{-1}{y-y_{0}}+\frac{\mathrm{d} y}{\mathrm{~d} x} \frac{x-x_{0}}{\left(y-y_{0}\right)^{2}}=-\frac{1+\frac{\mathrm{d} y^{2}}{\mathrm{~d} x}}{y-y_{0}} \tag{5.93} \label{5.93}\\[4pt] y-y_{0} & =-\frac{1+\frac{\mathrm{d} y}{\mathrm{~d} x}}{\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}} \tag{5.94} \label{5.94}\\[4pt] \rho^{2} & =\left(y-y_{0}\right)^{2}\left[1+\frac{\left(x-x_{0}\right)^{2}}{\left(y-y_{0}\right)^{2}}\right](\text { from Equation 5.90) } \tag{5.95} \label{5.95}\\[4pt] & =\left[\frac{1+\frac{\mathrm{d} y}{\mathrm{~d} x}}{\left|\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right|}\right]^{2}\left[1+\left(\frac{\mathrm{d} y}{\mathrm{~d} x}\right)^{2}\right](\text { used }(5.92) \&(5.94)) \tag{5.96} \label{5.96}\\[4pt] \rho & =\frac{\left(1+\frac{\mathrm{d} y^{2}}{\mathrm{~d} x}\right)^{3 / 2}}{\left|\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\right|} \tag{5.97} \label{5.97}\end{align}\]


1.1: Three Models of Spacetime
Learning Objectives
- Discuss the three models of spacetime
- Aristotelian spacetime
- Galilean spacetime
- Einstein’s spacetime
“ The test of a first-rate intelligence is the ability to hold two opposing ideas in mind at the same time and still retain the ability to function. ” —F. Scott Fitzgerald
Time and space together make spacetime, Figure \(\PageIndex{1}\), the stage on which physics is played out. Until 1905, physicists were trained to accept two mutually contradictory theories of spacetime. I’ll call these the Aristotelian and Galilean views, although my colleagues from that era would have been offended to be accused of even partial Aristotelianism.
Aristotelian Spacetime
Figure \(\PageIndex{2}\) shows an observer and two clocks, represented using the graphical conventions of Figure \(\PageIndex{1}\) (3). The existence of such a material object at a certain place and time constitutes an event, which we idealize as a point, Figure \(\PageIndex{2}\) (2). Spacetime consists of the set of all events. As time passes, a physical object traces out a continuous curve, a set of events known in relativistic parlance as its worldline. Since paper and computer screens are two-dimensional, the drawings only represent one dimension of space plus one dimension of time, which in relativity we call “\(1+1\) dimensions.” The real universe has three spatial dimensions, so real spacetime has \(3+1\) dimensions. Most, but not all, of the interesting phenomena in special relativity can be understood in \(1+1\) dimensions, so whenever possible in this book I will draw \(1+1\)-dimensional figures without apology or explanation.
The relativist’s attitude is that events and relationships between events are primary, while coordinates such as \(x\) and \(t\) are secondary and possibly irrelevant. Coordinates let us attach labels like (\(x,t\)) to points, but this is like God asking Adam to name all the birds and animals: the animals didn’t care about the names. Figure \(\PageIndex{2}\) (3) shows the use of vectors to indicate relationships between points. Vector \(o\) is an observer-vector, connecting two points on the world-line of the person. It points from the past into the future. The vector s connecting the two clocks is a vector of simultaneity. The clocks have previously been synchronized side by side, and if we assume that transporting them to separate locations doesn’t disrupt them, then the fact that both clocks read two minutes after three o’clock tells us that the two events occur at the same time.
The Aristotelian model of spacetime is characterized by a set of rules about what vectors are valid observer- and simultaneity vectors. We require that every \(o\) vector be parallel to every other, and likewise for \(s\) vectors. But, as is usual with vectors, we allow the arrow to be drawn anywhere without considering the different locations to have any significance; that is, our model of spacetime doesn’t allow different regions to have different properties.
When Einstein was a university student, these rules (phrased differently) were the ones he was taught to use in describing electricity and magnetism. He later recalled imagining himself on a motorcycle, riding along next to a light wave and trying to imagine how his observations could be reconciled with Maxwell’s equations. I don’t know whether he was ever brave enough to describe this daydream to his professors, but if he had, their answer would have been essentially that his hypothetical \(o\) vector was illegal. The good \(o\) vectors were thought to be the ones that represented an observer at rest relative to the ether, a hypothetical all-pervasive medium whose vibrations were electromagnetic waves. However silly this might seem to us a hundred years later, it was in fact strongly supported by the evidence. A vast number of experiments had verified the validity of Maxwell’s equations, and it was known that if Maxwell’s equations were valid in coordinates (\(x,t\)) defined by an observer \(o\), they would become invalid under the transformation \((x', t') = (x + vt, t)\) to coordinates defined by an observer \(o'\) in motion at velocity \(v\) relative to \(o\).
Galilean Spacetime
But the Aristotelian model was already known to be wrong when applied to material objects. The classic empirical demonstration of this fact came around 1610 with Galileo’s discovery of four moons orbiting Jupiter, Figure \(\PageIndex{4}\). Aristotelianism in its ancient form was originally devised as an explanation of why objects always seemed to settle down to a natural state of rest according to an observer standing on the earth’s surface. But as Jupiter ﬂew across the heavens, its moons circled around it, without showing any natural tendency to fall behind it like a paper cup thrown out the window of a car. Just as an observer \(o_1\) standing on the earth would consider the earth to be at rest, \(o_2\) hovering in a balloon at Jupiter’s cloud tops would say that the Jovian clouds represented an equally “natural” state of rest.
Figure \(\PageIndex{4}\):
A simulation of how Jupiter and its moons might appear at intervals of three hours through a telescope. Because we see the moons’ circular orbits edge on, their world-lines appear sinusoidal. Over this time period, the innermost moon, \(Io\), completes half a cycle.
We are thus led to a different, Galilean, set of rules for \(o\) and \(s\) vectors. All \(s\) vectors are parallel to one another, but any vector \(t\) is not parallel to an \(s\) vector is a valid \(o\) vector. (We may wish to require that it point into the future rather than the past, but Newton’s laws are symmetric under time-reversal, so this is not strictly necessary.)
Galilean spacetime, unlike Aristotelian spacetime, has no universal notion of “same place.” I can drive to Gettysburg, Pennsylvania, and stand in front of the brass plaque that marks the site of the momentous Civil War battle. But am I really in the same place? An observer on another planet would say that our planet had moved through space since 1863.
Parallelism
Note that our geometrical description includes a notion of parallelism, but not of angular measure. We don’t know or care whether the “angle” between an \(s\) and an \(o\) is \(90\) degrees. One represents a distance, while the other represents an interval of time, and we can’t define the angle between a distance and a time. The same was true in the Aristotelian model; the vectors in Fgure \(\PageIndex{3}\) were drawn perpendicular to one another simply as a matter of convention, but any other angle could have been used.
The Galilean Twin Paradox
Alice and Betty are identical twins. Betty goes on a space voyage, traveling away from the earth along vector \(o_1\) and then turning around and coming back on \(o_2\). Meanwhile, Alice stays on earth. Because this is an experiment involving material objects, and the conditions are similar to those under which Galilean relativity has been repeatedly verified by experiment, we expect the results to be consistent with Galilean relativity’s claim that motion is relative. Therefore it seems that it should be equally valid to consider Betty and the spaceship as having been at rest the whole time, while Alice and the planet earth traveled away from the spaceship along \(o_3\) and then returned via \(o_4\). But this is not consistent with the experimental results, which show that Betty undergoes a violent acceleration at her turnaround point, while Alice and the other inhabitants of the earth feel no such effect.
The paradox is resolved by realizing that Galilean relativity defines unambiguously whether or not two vectors are parallel. It’s true that we could fix a frame of reference in which \(o_1\) represented the spaceship staying at rest, but \(o_2\) is not parallel to \(o_1\), so in this frame we still have a good explanation for why Betty feels an acceleration: she has gone from being at rest to being in motion. Regardless of which frame of reference we pick, and regardless of whether we even fix a frame of reference, \(o_3\) and \(o_4\) are parallel to one another, and this explains why Alice feels no effect.
Einstein’s Spacetime
We have two models of spacetime, neither of which is capable of describing all the phenomena we observe. Because of the relatively crude state of technology ca. 1900, it required considerable insight for Einstein to piece together a fragmentary body of indirect evidence and arrive at a consistent and correct model of spacetime. Today, the evidence is part of everyday life. For example, every time you use a GPS receiver, you’re using Einstein’s theory of relativity. Somewhere between 1905 and today, technology became good enough to allow conceptually simple experiments that students in the early \(20^{th}\) century could only discuss in terms like “ Imagine that we could... ”
A good jumping-on point is 1971. In that year, J.C. Hafele and R.E. Keating brought atomic clocks aboard commercial airliners, Fgure \(\PageIndex{7}\), and went around the world, once from east to west and once from west to east. Hafele and Keating observed that there was a discrepancy between the times measured by the traveling clocks and the times measured by similar clocks that stayed home at the U.S. Naval Observatory in Washington. The east-going clock lost time, ending up off by \(-59 \pm 10\) nanoseconds, while the west-going one gained \(273 \pm 7\) ns.
We are used to thinking of time as absolute and universal, so it is disturbing to find that it can ﬂow at a different rates for different observers. Nevertheless, the effects that Hafele and Keating observed were small. This makes sense: Galilean relativity had already been thoroughly verified for material objects such as clocks, planets, and airplanes, so a new theory like Einstein’s had to agree with Galileo’s to a good approximation, within the Galilean theory’s realm of applicability. This requirement of backward-compatibility is known as the correspondence principle.
It’s also reassuring that the effects on time were small compared to the three-day lengths of the plane trips. There was therefore no opportunity for paradoxical scenarios such as one in which the east going experimenter arrived back in Washington before he left and then convinced himself not to take the trip. A theory that maintains this kind of orderly relationship between cause and effect is said to satisfy causality.
Hafele and Keating were testing specific quantitative predictions of relativity, and they verified them to within their experiment’s error bars. Let’s work backward instead, and inspect the empirical results for clues as to how time works. The disagreements among the clocks suggest that simultaneity is not absolute: different observers have different notions of simultaneity, as suggested in Figure \(\PageIndex{9}\). Just as Galilean relativity freed the \(o\) vectors from the constraint of being parallel to one another, Einstein frees the \(s\) vectors. Galileo made “same place” into an ambiguous concept, while Einstein did the same with “simultaneous.” But because a particular observer does have methods of synchronizing clocks (e.g., Einstein synchronization, example \(\PageIndex{4}\)), the definition of simultaneity isn’t completely arbitrary. For each \(o\) vector we have a corresponding \(s\) vector, which represents that observer’s opinion as to what constitutes simultaneity. Because the convention on a Cartesian \(x - t\) graph is to draw the axes at right angles to one another, we refer to such a pair of vectors as orthogonal, but the word is not to be interpreted literally, since we can’t define an actual angle between a time interval and a spatial displacement.
What, then, are the rules for orthogonality? Figure \(\PageIndex{10}\) shows three possibilities. In each case, we have an initial pair of vectors \(o_1\) and \(s_1\) that we assume are orthogonal, and we then draw a new pair \(o_2\) and \(s_2\) for a second observer who is in motion relative to the first. The Galilean case, where \(s_2\) remains parallel to \(s_1\), has already been ruled out. The second case is the one in which \(s\) rotates in the same direction as \(o\). This one is forbidden by causality, because if we kept on rotating, we could eventually end up rotating \(o\) by \(180\) degrees, so by a continued process of acceleration, we could send an observer into a state in which her sense of time was reversed. We are left with only one possibility for Einstein’s spacetime, which is the one in which a clockwise rotation of \(o\) causes a counterclockwise rotation of \(s\), like closing a pair of scissors.
Now there is a limit to how far this process can go, or else the \(s\) and \(o\) would eventually lie on the same line. But this is impossible, for a valid \(s\) vector can never be a valid \(o\), nor an \(o\) a valid \(s\). Such a possibility would mean that an observer would describe two different points on his own world-line as simultaneous, but an observer for whom no time passes is not an observer at all, since observation implies collecting data and then being able to remember it at some later time. We conclude that there is a diagonal line that forms the boundary between the set of possible \(s\) vectors and the set of valid \(o\) vectors. This line has some slope, and the inverse of this slope corresponds to some velocity, which is apparently a universal and fixed property of Einstein’s spacetime. This velocity we call \(c\), and the correspondence principle tells us that \(c\) must be very large, because otherwise Einsteinian, or “relativistic,” effects such as time distortion would have been large even for motion at everyday speeds; in the Hafele-Keating experiment they were quite small, even at the high speed of a passenger jet.
Although \(c\) is a large number when expressed in meters per second, for convenience in relativity we will always choose units such that \(c = 1\). The boundary between \(s\) and \(o\) vectors then appears on spacetime diagrams as a diagonal line at \(\pm 45\) degrees. In more than one spatial dimension, this boundary forms a cone,Ffigure \(\PageIndex{11}\), and for reasons that will become more clear in a moment, this cone is called the light cone. Vectors lying inside the light cone are referred to as timelike, those outside as spacelike, and those on the cone itself as lightlike or null.
An important advantage of Einstein’s relativity over Galileo’s is that it is compatible with the empirical observation that some phenomena travel at a certain fixed speed. Light travels at a fixed speed, and so do other phenomena such as gravitational waves (first directly detected in 2016). So do all massless particles (subsection 4.3.4). This fixed speed is \(c\), and all observers agree on it. In 1905, the only phenomenon known to travel at \(c\) was light, so \(c\) is usually described as the “ speed of light, ” but from the modern point of view it functions more as a kind of conversion factor between our units of measurement for time and space. It is a property of spacetime, not a property of light.
More fundamentally, \(c\) is the maximum speed of cause and effect. If we could propagate cause and effect, e.g., by transmitting a signal, at a speed greater than \(c\), then the following argument shows that we would be violating either causality or the principle that motion is relative. If a signal could be propagated at a speed greater than \(c\), then the vector \(r\) connecting the cause and the effect would be spacelike. By opening and closing the “ scissors ” of Figure \(\PageIndex{10}\), we can always find an observer \(o\) who considers \(r\) to be a vector of simultaneity. Thus if faster-than-light propagation is possible, then instantaneous propagation is possible, at least for some observer. Since motion is relative, this must be possible for all observers, regardless of their state of motion. Therefore any spacelike vector is one along which we can send a signal. But by adding two spacelike vectors we can make a vector lying in the past timelike light cone, so by relaying the signal we could send a message into the past, violating causality.
In interpreting this argument, note that neither the relativity of motion nor causality is a logical necessity; they are both just generalizations based on a body of evidence. For more on causality, and its uncertain empirical status, see section 2.1.
The Ring Laser Gyroscope
If you’ve ﬂown in a jet plane, you can thank relativity for helping you to avoid crashing into a mountain or an ocean. Figure \(\PageIndex{12}\) shows a standard piece of navigational equipment called a ring laser gyroscope . A beam of light is split into two parts, sent around the perimeter of the device, and reunited. Since the speed of light is constant, we expect the two parts to come back together at the same time. If they don’t, it’s evidence that the device has been rotating. The plane’s computer senses this and notes how much rotation has accumulated.
No Frequency - Dependence
Relativity has only one universal speed, so it requires that all light waves travel at the same speed, regardless of their frequency and wavelength. Presently the best experimental tests of the invariance of the speed of light with respect to wavelength come from astronomical observations of gamma-ray bursts, which are sudden outpourings of high-frequency light, believed to originate from a supernova explosion in another galaxy. One such observation, in 2009, found that the times of arrival of all the different frequencies in the burst differed by no more than \(2\) seconds out of a total time in ﬂight on the order of ten billion years!
Einstein's Train
The Fgure \(\PageIndex{13}\) shows a famous thought experiment devised by Einstein. A train is moving at constant velocity to the right when bolts of lightning strike the ground near its front and back. Alice, standing on the dirt at the midpoint of the ﬂashes, observes that the light from the two ﬂashes arrives simultaneously, so she says the two strikes must have occurred simultaneously. Bob, meanwhile, is sitting aboard the train, at its middle. He passes by Alice at the moment when Alice later figures out that the ﬂashes happened. Later, he receives ﬂash \(2\), and then ﬂash \(1\). He infers that since both ﬂashes traveled half the length of the train,ﬂash \(2\) must have occurred first. How can this be reconciled with Alice’s belief that the ﬂashes were simultaneous?
Figure \(\PageIndex{14}\) shows the corresponding spacetime diagram. It seems paradoxical that Alice and Bob disagree on simultaneity, but this is only because we have an ingrained prejudice in favor of Galilean relativity. Alice’s method of determining that \(1\) and \(2\) were simultaneous is valid, and is known as Einstein synchronization. The dashed line connecting \(1\) and \(2\) is orthogonal to Alice’s worldline. But Bob has a different opinion about what constitutes simultaneity. The slanted dashed line is orthogonal to his worldline. According to Bob, \(2\) happened before the time represented by this line, \(1\) after.
Example \(\PageIndex{4}\) is of course impractical as described, since real trains don’t travel at speeds anywhere near \(c\) relative to the dirt. We say that their speeds are “ nonrelativistic .” Because Einstein coined the term “ relativity ,” and his version of relativity superseded Galileo’s, the unmodified word is normally understood to refer to Einsteinian relativity. A physicist who studies Einstein-relativity is a relativist. A material object moving at a speed very close to \(c\) is described as ultrarelativistic. One often hears lay people describing relativity in terms of certain effects that would happen “ if you went at the speed of light. ” In fact, as we’ll see in chapter 3 and chapter 4, it is not possible to accelerate material objects to \(c\), and in any case that isn’t necessary. Relativistic effects exist at all speeds, but they’re weak at speeds small compared to \(c\).
Numerical value of c
In this book we’ll use units in which \(c = 1\). However, many beginners are vexed by the question of why \(c\) has the particular value it does in a given system of units such as the SI. Related to this is the question of whether \(c\) could ever change, so that measuring it today and measuring it tomorrow would give slightly different results. In a system of units where \(c\) has units, its value is what it is only because of our choice of units, and there is no meaningful way to test whether it changes.
Let’s take the SI as an example of a system of units. The SI was originally set up so that the meter and the second were defined in terms of properties of our planet. The meter was one forty-millionth of the earth’s circumference, and the second was \(1/86,400\) of a mean solar day. Thus when we express \(c\) as \(3.0 \times 10^8 m/s\), we are basically specifying the factor by which \(c\) exceeds the speed at which a point on the equator goes around the center of the earth (with additional conversion factors of \(40,000,000\) and \(86,400\) thrown in). Since the properties of our planet are accidents of the formation of the solar system, there is no physical theory that can tell us why \(c\) has this value in the original French-Revolutionary version of the SI.
The base units of the SI were redefined over the centuries. Today, the second is defined in terms of an atomic standard, and the meter is defined as \(1/299,792,458\) of a light-second. Therefore \(c\) has a defined value of exactly \(299,792,458 m/s\). Again, we find that the numerical value of \(c\) has no fundamental significance; it is merely a matter of definition. It is possible to form the unitless ratio \(\alpha = e^2/\tilde{h} c \approx 1/137\), called the fine structure constant. Its value does not depend on our choice of units, so it is possible to do experiments to look for changes in its value over time, e.g., by comparing the spectrum of hydrogen on earth with the spectrum of distant stars, whose light has taken billions of years to get to us. Claims have even been made to the effect that these observations do show a change, although this appears to have been a mistake. If such a change did occur, we would not be able to attribute it unambiguously to a change in \(c\) rather than a change in \(\tilde{h}\) or the fundamental charge. The standards used to define our units could change over time. The platinum-iridium standard for the kilogram in Paris is suspected to have lost about \(50\) µg of mass over the last century. Even the atomic standard used to define the second could be changing due to physics beyond our present knowledge. A change in \(c\) might produce such a change, but any such change could also be produced by changes in other physical constants, such as the others occurring in the fine structure constant . Such issues are discussed at greater length in Section 9.6.
Discussion question \(\PageIndex{1}\)
The machine-gunner in the Fgure \(\PageIndex{15}\) sends out a spray of bullets. Suppose that the bullets are being shot into outer space, and that the distances traveled are trillions of miles (so that the human figure in the diagram is not to scale). After a long time, the bullets reach the points shown with dots which are all equally far from the gun. Their arrivals at those points are events \(A\) through \(E\), which happen at different times. The chain of impacts extends across space at a speed greater than \(c\). Does this violate special relativity?


5.7: §37. The Latin suffixes -ANUS (> E -an) and -INUS (> E -ine)
So far, we have met two Latin suffixes (- ā lis and – īlis ) that create 3rd declension adjectives. In this section we see two (- ā nus and – īnus ) that form adjectives of the 1st and 2nd declension, as we can recognize by the reliable ending – us .
The – ā nus group is very small and will be dismissed with two examples:
urbs , urb-is > urb anus (E urban and urbane )
The Latin word hūmānus (E human ) is an odd case. Regardless of appearances, it is not related to hŭmus , “earth,” but is an irregular derivative of the 3rd declension noun homo , hominis , “mankind,” “humankind,” “person” (to be distinguished from vir , the male human being).
The – ī nus group is somewhat larger and more productive, from the English point of view. Here are some representative examples:
sal , sal-is (“salt”) > salinus (E saline )
mare , mar-is (“sea”) > marinus (E marine )
divus = deus (“god”) > divinus (E divine )
femina (“woman”) > femininus (E feminine )
The corresponding word masculine is derived from masculinus , whose etymology we’ll learn a little later in the course. Notice that the -ine of these -inus derivatives can be pronounced at least three different ways in English.
You will be familiar with at least some Latin-derived adjectives that refer to animals. This was a consistent use of the suffix -inus , forming words that regularly came into English as derivatives in – ine (here pronounced to rhyme with “mine”):
canis (“dog”) > caninus (E canine , “pertaining to a dog”)
feles (“cat”) > felinus (E feline , “pertaining to a cat,” “catlike”)
equus (“horse”) > equinus (E equine , “pertaining to a horse”)
porcus (“pig”) > porcinus (E porcine , “pertaining to a pig,” “like a pig”)
These animal words are mainly clinical and technical. We say “canine distemper,” but “doggy bag”; and no one would ever reverse the two adjectives. On the other hand, we can say that a person moves with feline grace, or that someone’s appearance is porcine . When we’re being uncomplimentary or rude, we generally use Germanic adjectives. To call a man “piggy” or “piggish” is a more direct insult than to call him “porcine”—unless he’s a Latin scholar. You might describe a woman as “horsey,” referring either to her interests or to her appearance; in the second case, at least, that would be a “catty” remark.
How many English animal adjectives in -ine can you think of? How do they differ in connotation from their Germanic counterparts, such as “catty” or “piggish” or “doglike”? Search your own vocabulary honestly, and then consult the Bestiary in § 41.


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People value biodiversity in many ways. The reasons vary from person to person, and from region to region. But generally, nature’s contributions to people, also called ecosystem services, are divided into three overlapping and interdependent categories, namely material contributions, regulating services, and nonmaterial contributions.
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Material contributions include benefits people get from consuming natural resources (e.g. drinking water or burning wood for cooking) or using natural resources in production and trade (e.g. timber to build homes or other structures).
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Biodiversity also provides a large variety of regulating services that enable people to benefit from nature’s material contributions. Some of these contributions include ecosystem productivity, water and soil protection, climate regulation, pollination, seed dispersal, and disaster prevention and detection.
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People attach nonmaterial values to biodiversity which are difficult to quantify, and thus to account for, in modern economic systems. These values include support for inspiration and learning, support for psychological and physical experiences, and support for personal and group identities.
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Environmental economics studies the implications of economic transactions, environmental policies, and other decisions that impact the environment. This field has highlighted how damage to the environment, such as pollution caused by industry, are not always fully considered when making political and development decisions, leading to unsustainable economic practices and market failure. Accounting for negative externalities and perverse subsidies can help policymakers design incentives that promote sustainable practices.


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3.1: Evaluating OERs
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When evaluating Open Educational Resources (OERs) for use in a course, educators should consider several factors to ensure that the selected materials meet the educational needs of their students. The biggest draw to OER is cost savings to students, but beyond that a faculty needs to evaluate other factors to ensure the resource is of a quality that can be maintained.
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3.2: Combining Licenses When Adapting
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When adapting a Creative Commons (CCP) work to create a new work of your own, you may use resources that have different types of CC licenses. Combining works with different CC licenses can sometimes be tricky.
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3.3: Adapting and Adopting an OER
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Why Adapt OER? There are several reasons you may want to adapt or curate a resource. Perhaps existing OER do not have all the content you would like to include in your course. Or there may not be any cohesive OER for your course but there may be available materials you can pick and choose from to build the perfect resource for your class.


9.1: The First Isomorphism Theorem
A very powerful theorem, called the First Isomorphism Theorem , lets us in many cases identify factor groups (up to isomorphism) in a very slick way. Kernels will play an extremely important role in this. We therefore first provide some theorems relating to kernels.
Theorem \(\PageIndex{1}\)
Let \(G\) and \(G'\) be groups, let \(\phi\) be a homomorphism from \(G\) to \(G'\text{,}\) and let \(K=\text{Ker}\; \phi\text{.}\) Then for \(a,b\in G\text{,}\) \(aK=bK\) if and only if \(\phi(a)=\phi(b)\text{.}\)
- Proof
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Let \(a,b∈G\). Then
\( \begin{array} &ϕ(a)=ϕ(b)&⇔ϕ(b)^{−1}ϕ(a)=e_{G′}\\&⇔ϕ(b^{−1}a)=e_{G′}\\&⇔b^{−1}a∈K\\&⇔a∈bK\\&⇔aK=bK, \end{array}\)
as desired.
Corollary \(\PageIndex{1}\)
Let \(\phi\) be a homomorphism from group \(G\) to group \(G'\text{.}\) Then \(\phi\) is one-to-one (hence a monomorphism) if and only if \(\text{Ker} \phi=\{e_G\}\text{.}\)
- Proof
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Clearly, if \(ϕ\) is one-to-one then \(\text{Ker}ϕ=\{e_G\}\). Conversely, assume \(\text{Ker}ϕ=\{e_G\}\).
If \(a,b∈G\) with \(ϕ(a)=ϕ(b)\), then by the above theorem, \(a\text{Ker}ϕ=b\text{Ker}ϕ\). But \(a\text{Ker}ϕ=a\{e_G\}=\{a\}\) and \(b\text{Ker}ϕ=b\{e_G\}=\{b\}\). Thus, \(a=b\), and we see that \(ϕ\) is one-to-one.
We now prove a theorem that provides the meat and potatoes of the First Isomorphism Theorem.
Theorem \(\PageIndex{2}\): Factorization Theorem
Let \(G\) and \(G'\) be groups, let \(\phi\) be a homomorphism from \(G\) to \(G'\text{,}\) let \(K=\text{Ker} \phi\text{,}\) let \(N\) be a normal subgroup of \(G\) with \(N\subseteq K\text{,}\) and let \(\Psi\) be the canonical epimorphism from \(G\) to \(G/N\text{.}\) Then the map \(\overline{\phi}: G/N \to G'\) defined by \(\overline{\phi}(aN)=\phi(a)\) is a well-defined homomorphism, with \(\overline{\phi} \circ \Psi=\phi\text{.}\)
We summarize this in the following picture:
- Proof
-
We define \(\overline{\phi}:G/N→G′\), as indicated above, by \(\overline{\phi}(aN)=ϕ(a)\) for every \(aN∈G/N\). Since \(\overline{\phi}\) is defined using coset representatives, we must first show that \(\overline{\phi}\) is well-defined. So let \(aN=bN∈G/N\). Then \(\overline{\phi}(aN)=ϕ(a)\) and \(\overline{\phi}(bN)=ϕ(b)\), so we must show that \(ϕ(a)=ϕ(b)\). Since \(aN=bN\) and \(N⊆K\), we have that \(aK=bK\) by Statement 6 of Theorem \(7.2.3\); thus \(ϕ(a)=ϕ(b)\) (using Theorem \(9.1.1\)). So \(\overline{\phi}\) is well-defined.
Next, we show that \(\overline{\phi}\) is a homomorphism. Let a\(aN,bN∈G/N\). Then
\(\overline{\phi}((aN)(bN))= \overline{\phi}(abN)=ϕ(ab)=ϕ(a)ϕ(b)=\overline{\phi}(aN)\overline{\phi}(bN)\).
Finally, for every \(a∈G\),
\((\overline{\phi} \circ Ψ)(a)=\overline{\phi}(Ψ(a))=\overline{\phi}(aN)=ϕ(a)\);
so \(\overline{\phi} \circ Ψ=ϕ\) , as desired.
Note that the above theorem does not state that \(\overline{\phi}\) is a monomorphism or an epimorphism. This is because in general it may be neither! We do have the following theorem:
Theorem \(\PageIndex{3}\)
Let \(G\text{,}\) \(G'\text{,}\) \(\phi\text{,}\) \(N\text{,}\) and \(\overline{\phi}\) be as defined in the Factorization Theorem. Then
- \(\overline{\phi}\) is onto (an epimorphism) if and only if \(\phi\) is onto (an epimorphism); and
- \(\overline{\phi}\) is one-to-one (a monomorphism) if and only if \(N=\text{Ker} \phi\text{.}\)
- Proof
-
This is clear, since \(\overline{\phi}(G/N)=ϕ(G)\).
By Corollary \(9.1.1\), \(\overline{\phi}\) is one-to-one if and only if \(\text{Ker}\overline{\phi}=\{N\}\). But
\(\begin{array}& \text{Ker}\overline{\phi}&=\{aN∈G/N:\overline{\phi}(aN)=e_{G′}\} \\&=\{aN∈G/N:ϕ(a)=e_{G′}\} \\&=\{aN∈G/N:a∈\text{Ker}ϕ\}. \end{array}\)
So \(\text{Ker}\overline{\phi}=\{N\}\) if and only if
\(\{aN∈G/N:a∈\text{Ker}ϕ\}=\{N\}\),
in other words if and only if \(aN=N\) for all \(a∈\text{Ker}ϕ\). But
\(\begin{array} &&3&&aN=N &\text{ for all } a∈\text{Ker}ϕ \\&& ⇔&a∈N &\text{ for all } a∈\text{Ker}ϕ \\&&⇔&\text{Ker}ϕ⊆N \\&&⇔&\text{Ker}ϕ=N, \end{array}\)
since we are given that \(N⊆\text{Ker}ϕ\). Thus, \(\overline{\phi}\) is one-to-one if and only if \(N=\text{Ker}ϕ\), as desired.
We are now ready to state the all-important First Isomorphism Theorem, which follows directly from the Factorization Theorem and Theorem \(9.1.3\).
Theorem \(\PageIndex{4}\): First Isomorphism Theorem
Let \(G\) and \(G'\) be groups, with homomorphism \(\phi:G \rightarrow G'\text{.}\) Let \(K=\text{Ker} \phi\text{.}\) Then \(G/K \simeq \phi(G)\text{.}\) In particular, if \(\phi\) is onto, then \(G/K\simeq G'\text{.}\)
So to prove that a factor group \(G/N\) is isomorphic to a group \(G'\text{,}\) it suffices to show there exists an epimorphism from \(G\) to \(G'\) that has \(N\) as its kernel.
Example \(\PageIndex{1}\)
Letting \(n\in \mathbb{Z}^+\text{,}\) let's identify a familiar group to which \(GL(n,\mathbb{R})/SL(n,\mathbb{R})\) is isomorphic. As in Example \(8.2.3\), the map \(\phi:GL(n,\mathbb{R})\to \mathbb{R}^*\) defined by \(\phi(A)\) is a homomorphism with kernel \(SL(n,\mathbb{R})\text{.}\) Moreover, \(Phi\) clearly maps onto \(\mathbb{R}^*\text{:}\) indeed, given \(\lambda \in \mathbb{R}^*\text{,}\) the diagonal matrix having \(\lambda\) in the uppermost left position and \(1\)'s elsewhere down the diagonal gets sent to \(\lambda\) by \(\phi\text{.}\) So by the First Isomorphism Theorem, we have \(GL(n,\mathbb{R})/SL(n,\mathbb{R}) \simeq \mathbb{R}^*\text{.}\)
Example \(\PageIndex{2}\)
Let \(G=S_3\times \mathbb{Z}_{52}\) and let \(N=S_3 \times \{0\}\subseteq G\text{.}\) It is straightforward to show that \(N\) is normal in \(G\text{.}\) What is the structure of \(G/N\text{?}\) Well, define \(\phi:G\to \mathbb{Z}_{52}\) by \(\phi((\sigma, a))=a\text{.}\) Then \(\phi\) is clearly an epimorphism and \(\text{Ker} \phi=\{(\sigma,a)\in G\,:a=0\}=N\text{.}\) So \(G/N\) is isomorphic to \(\mathbb{Z}_{52}\text{.}\)
Generalizing the above example, we have the following theorem, whose proof we leave to the reader.
Theorem \(\PageIndex{5}\)
Let \(G=G_1\times G_2 \times \cdots \times G_k\) (where \(k\in \mathbb{Z}^+\)) and let \(N_i\) be a normal subgroup of \(G_i\) for each \(i=1,2,\ldots, k\text{.}\) Then \(N=N_1 \times N_2 \times \cdots \times N_k\) is a normal subgroup of \(G\text{,}\) with \(G/N \simeq G_1/N_1 \times G_2/N_2 \cdots \times G_k/N_k.\)
We provide one more cool example of using the First Isomorphism Theorem. Clearly, since \(\mathbb{R}\) is abelian, \(\mathbb{Z}\) is a normal subgroup of \(\mathbb{R}\text{.}\) What is the structure of \(\mathbb{R}/\mathbb{Z}\text{?}\) Well, in modding \(\mathbb{R}\) out by \(\mathbb{Z}\) we have essentially identified together all real numbers that are an integer distance apart. So we can think of the canonical epimorphism from \(\mathbb{R}\) to \(\mathbb{R}/\mathbb{Z}\) as wrapping up \(\mathbb{R}\) like a garden hose! Thus, one might guess that \(\mathbb{R}/\mathbb{Z}\) has some circle-like structure—but if we want to think of it as a group, we have to figure out what the group structure on such a “circle” would be!
We leave, for a moment, our group \(\mathbb{R}/\mathbb{Z}\text{,}\) and look at how we can consider a circle to be a group.
Recall that for every \(\theta \in \mathbb{R}\text{,}\) \(e^{i\theta}\) is defined to be \(\cos \theta + i\sin \theta\text{.}\) It is clear then that the set \(S^1=\{e^{i\theta} \,:\, \theta\in \mathbb{R}\}\) is the unit circle in the complex plane.
Remark
Note that if \(\theta_1, \theta_2\in \mathbb{R}\text{,}\) then \(e^{i\theta_1}=e^{i\theta_2}\) if and only if \(\theta_1-\theta_2 \in 2\pi \mathbb{Z}\text{.}\)
Theorem \(\PageIndex{6}\)
\(S^1\) is a group under the multiplication \(e^{i\theta_1}e^{i\theta_2}=e^{i(\theta_1+\theta_2)}\text{.}\)
- Proof
-
The tricky part is showing that the operation is well-defined. Suppose \(θ_1,θ_2,t_1\), and \(t_2\) are in \(R\), with \(e^{iθ_1}=e^{it_1}\) and \(e^{iθ_2}=e^{it_2}\). We want to show that
\(e^{iθ_1}e^{iθ_2}=e^{it_1}e^{it_2}\),
i.e., that
\(e^{i(θ_1+θ_2)}=e^{i(t_1+t_2)}\).
Now, \(e^{iθ_1}=e^{it_1}\) and \(e^{iθ_2}=e^{it_2}\) imply that \(θ_1−t_1=2πm\) and \(θ_2−t_2=2πn\) for some \(m\) a nd \(n\) in \(\mathbb{Z}\); hence,
\(θ_1+θ_2−(t_1+t_2)=2π(m+n)∈2π\mathbb{Z}\).
Thus, \(e^{i(θ_1+θ_2)}=e^{i(t_1+t_2)}\), so our operation is well-defined.
Since products of elements in \(S^1\) are in \(S^1\), we have that \(S^1\) is a binary structure; moreover, associativity of the operation in \(S^1\) follows from the associativity of addition in \(\mathbb{R}\). Moreover, \(e^{i0}=1\) clearly acts as an identity element in \(S^1\), and if \(e^{iθ}∈S^1\), then \(e^{iθ}\) has inverse \(e^{i(−θ)}∈S^1\). So \(S^1\) is a group under the described operation.
Beautifully, it turns out that our group \(\mathbb{R}/\mathbb{Z}\) is isomorphic to \(S^1\text{.}\)
Theorem \(\PageIndex{7}\)
\(\mathbb{R}/\mathbb{Z} \simeq S^1\text{.}\)
- Proof
-
We will define an epimorphism \(ϕ\) from \(\mathbb{R}\) to \(S^1\) with \(\text{Ker}ϕ=\mathbb{Z}\); then we'll have \(\mathbb{R}/\mathbb{Z}≃S^1\), by the First Isomorphism Theorem.
Define \(ϕ:\mathbb{R}→S^1\) by \(ϕ(r)=e^{i2πr}\). We have that \(ϕ\) is a homomorphism, since for every \(r,s∈\mathbb{R}\), we have
\(ϕ(r+s)=e^{i2π(r+s)}=e^{i2πr+i2πs}=e^{i2πr}e^{i2πs}=ϕ(r)ϕ(s)\).
Moreover, \(ϕ\) is clearly onto, since if \(e^{iθ}∈S^1\), then
\(e^{iθ}=e^{i2π(θ2π)}=ϕ(θ2π)\).
Finally, \(\text{Ker}ϕ=\mathbb{Z}\): indeed,
\(\begin{array}& r∈\text{Ker}ϕ&⇔ϕ(r)=1\\&⇔e^{i2πr}=1\\&⇔\cos2πr+i\sin2πr=1\\&⇔\cos2πr=1 \text{ and } \sin2πr=0\\&⇔r∈\mathbb{Z}. \end{array}\)
Thus, \(\text{Ker}ϕ=\mathbb{Z}\), and hence \(\mathbb{R}/\mathbb{Z}≃S^1\), as desired.