2: Drawing Materials
THE MATERIALS
Papers
https://en.wikibooks.org/wiki/Papermaking/History_of_paper
Paper is the most popular surface for drawing. Paper can be made out of different substances with each combination having certain characteristics. Some papers are best for graphite pencils, others are well-suited to pen-and-ink, while some are made specifically for charcoal, Conté and pastels. So it is wise to select the appropriate paper for the specific medium.
Paper Characteristics
Paper is measured by the weight of a ream of around 500 uncut sheets. For instance, if a 500 sheet uncut ream weighs 90 pounds, the paper is identified as 90 pound paper. The weight also corresponds to the paper thickness. Typical printer paper is 20 pounds, whereas a quality drawing paper is usually 80 pounds, with sketching paper weighing in around 60 pounds.
https://www.strathmoreartist.com/blog-reader/determining-paper-weight.html
Another important quality of paper is the amount of acid in the contents of the manufactured paper. If you have ever observed a newspaper left outside in the elements, or exposed to direct sunlight, you'll see that the paper turns yellow and begins to break down. This is due to the amount of acid in this low-quality, thin, and inexpensive paper. Artists use this paper for quick sketching exercises. The paper is called Newsprint . Newsprint paper is often used in the drawing studios where models pose and students create multiple, quick drawings, called "gesture drawings." This paper often has a nice "tooth" to the surface making drawing in soft pencils, charcoal and Conté easier and more aesthetic. The archival qualities of Newsprint paper is limited, so this paper is not a wise choice for a finished drawing of quality that may be kept for any future needs.
For more finished drawing, an acid-free drawing paper is the logical choice. For pencil drawing, the Strathmore Series 400 paper pad, or equivalent, is recommended because of the combination of weight (80 pounds), eggshell paper color, and archival qualities. It has a slight texture (tooth) to the surface allowing for detail and enough thickness to take multiple changes and drawing revisions using erasers.
The highest quality drawing papers will be handmade and acid-free. This paper is the preferred choice of professional fine artists desiring a combination of specific surface and weight characteristics and documented archival specifications. Handmade papers come in a variety of surface textures and weights up to 400 pounds. Heavier papers do not necessarily indicate higher quality, so careful selection of the appropriate paper requires research and experience.
Paper textures range from very rough to very soft and smooth, with many varieties between these two extremes. Paper with a texture uses a process called Cold Press . Smooth papers use a process involving heat to create a smooth paper surface and this is called Hot Press .
https://blog.schoolspecialty.com/hot-cold-press-paper-101/
For ink drawings a hard, smooth surface works well. Bristol paper of sufficient weight (thickness) match well with pen and ink techniques since the surface is smooth, the paper is available in archival form as well as many sizes. Another choice when inking is a coated, thick support termed Scratchboard . Scratchboard can be purchased with a white or black coating. If purchases with the white coating, it is prudent to "burnish" the white coating smooth using an eraser over the areas where you will ink. This allows the flexible nib pen to flow smoothly over the surface when loaded with ink. Areas of the white Scratchboard can be painted with black ink and scratched out as well. Sometimes, combining the ink lines and scratched out lines creates a nice combination in an ink drawing. (add shunt drawing) (or Dotson drawing)
Pencil Sharpeners
A great little sharpener is the German-made metal one pictured here. Sharpeners can be electric or manual. Charcoal/carbon pencils should be sharpened with a blade (box cutter) and will continue to break if used in an electric or wall-mounted pencil sharpener.
Erasers
Erasers are made of a variety of materials. The white plastic (rubber) erasers by StaedtlerMars are an all around versatile studio eraser. A kneaded eraser is unique and very useful as well. The kneaded eraser has some very specific uses in the drawing studio, especially when using very soft media like charcoal, Conté and pastels. There are other drawing erasers as well such as the Pink Pearl and the soap eraser. However, my recommendation is to stick with the white plastic and kneaded erasers as your studio erasers.
Blending Tools
Blending tools help create smooth values in your drawing. Paper products, such as paper towels, work well for blending. Fingers are generally not used because of the oils found on the skin surface may transfer to the paper. When this happens, the area in which you are drawing will appear darker in value than adjacent areas. This oil spot is difficult to remove or blend into the surrounding lighter area. So for finished, more precise drawings use a tool to blend rather than your fingers. For a larger, expressive drawing using your fingers may not impact the overall look of your drawing and may connect you more to the process of creating art. My opinion is to grab a large brush in that case, rather than getting into the habit of using fingers to blend.
Blending Stumps
Blending stumps are made of wrapped paper and come in a variety of sizes. Smaller ones are called Tortillions . Stumps allow for large area blending as well as very small, detail work. The tips of the pointed ends and diameter of the stumps are variable allowing for a variety of blending options.
Brushes
Brushes are not just for painting but find uses in drawing as well. Soft brushes are a great way to create incredible, subtle blending with graphite, charcoal and other dry media.
Drawing Boards
Drawing boards are transportable allowing you to work outside on a hard surface you can place on your lap, easel, table, bench, or on the ground.
Dry Media
A medium is the art tool and its accompanying technique (media is plural). Dry media used in drawing includes graphite pencils, charcoal and charcoal pencils, Conté crayons, pastels, and colored pencils.
Pencils
Lead pencils and lead-based paints are both toxic, and thankfully a thing of the past.
http://www.abc.net.au/science/articles/2004/09/09/1195234.htm
The pencils we use for drawing today are manufactured with varying hardness of graphite. The basic #2 pencil is in fact the middle of the hardness spectrum for graphite pencils and equivalent to an HB level of hardness. Pencils are graded by how soft or hard they are, again with HB situated in the middle. From HB, pencils becoming softer, and thus easier to make dark in value, are B, 2B, 3B, 4B and so on. You can really see a difference when working with a 6B (very soft) pencil versus an HB. From an HB, pencils progressively getting harder are H, 2H, 3H, 4H and so on. The hard pencils were often used in drafting since they hold a point, are precise and generally do not smear as easily. These more technical pencils were useful when using drafting tools like French Curves, Compasses, T-squares and rulers. These pencils have little use in the Drawing Studio since they produce a very light mark and can literally damage the paper because of their hard points. So for general studio drawing, using a range of HB through 6B is usually sufficient. Using a pencil designed as a "drawing pencil" has the advantage of using a thicker point or edge of the tip when drawing, whereas a mechanical pencil has a defined width to the tip and is a somewhat less flexible drawing tool. The softness of the lead and general feel of mechanical pencils is appealing and works well for smaller drawings, but not for larger studio drawings.
Graphite sticks are also a versatile drawing tool that allows the artist to use the thick edge or side to block in values. Although somewhat clumsy, with practice this can be an effective drawing too.
Charcoal and Conté
Charcoal comes in Compressed and Vine sticks and pencils. It is also graded for soft and hard qualities the same as pencils. Compressed charcoal is jet black; Vine Charcoal is lighter and must have a fixative sprayed over the Vine charcoal drawing to maintain the details since it is so delicate a medium. Compressed charcoal is more robust in nature and more suited to studio drawing techniques. Can easily be manipulated and blended lending itself to subtle shifts in value and delicate drawing transitions. It has been a staple of academic drawing for centuries and is arguably one of the oldest drawing media used by humans.
Conté is a favorite and also a drawing medium used by students and art masters alike. It comes in a variety of shades, colors and degrees of softness and hardness. It is not as easily manipulated as charcoal and is a bit more permanent when applied to paper. It tends to stain the paper and is difficult to erase. Also, once directly applied it has certain quality that disappears upon blending. So once it is blended, you make a commitment to work with that change on the character of the drawing. This will be covered in the Charcoal chapter.
Pastels and Colored Pencils
Pastels, chalk and colored pencils are used for drawing but will not be covered in depth in this beginning drawing course. Using these media adds color for the creation of the artwork which is best covered in 2-D Design and Color, Watercolor, and Painting courses.


<IP_ADDRESS>: Varying Rates of Speciation
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Learning Objectives
- Explain how the interaction of an organism’s population size in association with environmental changes can lead to different rates of speciation
Varying Rates of Speciation
Scientists around the world study speciation, documenting observations both of living organisms and those found in the fossil record. As their ideas take shape and as research reveals new details about how life evolves, they develop models to help explain rates of speciation. In terms of how quickly speciation occurs, two patterns are currently observed: the gradual speciation model and the punctuated equilibrium model.
In the gradual speciation model, species diverge gradually over time in small steps. In the punctuated equilibrium model, a new species changes quickly from the parent species and then remains largely unchanged for long periods of time afterward. This early change model is called punctuated equilibrium, because it begins with a punctuated or periodic change and then remains in balance afterward. While punctuated equilibrium suggests a faster tempo, it does not necessarily exclude gradualism.
The primary influencing factor on changes in speciation rate is environmental conditions. Under some conditions, selection occurs quickly or radically. Consider a species of snails that had been living with the same basic form for many thousands of years. Layers of their fossils would appear similar for a long time. When a change in the environment takes place, such as a drop in the water level, a small number of organisms are separated from the rest in a brief period of time, essentially forming one large and one tiny population. The tiny population faces new environmental conditions. Because its gene pool quickly became so small, any variation that surfaces and that aids in surviving the new conditions becomes the predominant form.
Contributions and Attributions
Contributions and Attributions
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44573/latest...ol11448/latest . License : CC BY: Attribution
- Hybrid speciation. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Hybrid_speciation . License : CC BY-SA: Attribution-ShareAlike
- Hybrid speciation. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Hybrid_speciation . License : CC BY-SA: Attribution-ShareAlike
- hybrid zone. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/hybrid%20zone . License : CC BY-SA: Attribution-ShareAlike
- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//biology/de...n/reconnection . License : CC BY-SA: Attribution-ShareAlike
- hybrid speciation. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/hybrid_speciation . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Reconnection and Rates of Speciation. January 16, 2015. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44573/latest/ . License : CC BY: Attribution
- OpenStax College, Biology. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44573/latest/?collection=col11448/latest . License : CC BY: Attribution
- gradualism. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/gradualism . License : CC BY-SA: Attribution-ShareAlike
- punctuated equilibrium. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/punctuated_equilibrium . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Reconnection and Rates of Speciation. January 16, 2015. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44573/latest/ . License : CC BY: Attribution
- OpenStax College, Reconnection and Rates of Speciation. October 16, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44573/latest/Figure_18_03_02.png . License : CC BY: Attribution
Key Points
- In the gradual speciation model, species diverge slowly over time in small steps while in the punctuated equilibrium model, a new species diverges rapidly from the parent species.
- The two key influencing factors on the change in speciation rate are the environmental conditions and the population size.
- Gradual speciation is most likely to occur in large populations that live in a stable environment, while the punctuation equilibrium model is more likely to occur in a small population with rapid environmental change.
Key Terms
- punctuated equilibrium : a theory of evolution holding that evolutionary change tends to be characterized by long periods of stability, with infrequent episodes of very fast development
- gradualism : in evolutionary biology, belief that evolution proceeds at a steady pace, without the sudden development of new species or biological features from one generation to the next


Identify the typical milestones of
motor development
Distinguish between gross and fine motor development
Explain influences on motor development in infancy and toddlerhood
Mateo and Jonas brought their eight-month-old daughter Bella to her pediatrician. Bella has been sitting independently for a couple of months and tries to pull herself up to a standing position, but she shows no interest in crawling. The pediatrician says there is no reason for concern. There are many important milestones in physical development, she says, but crawling is not one of them. This is quite surprising news to both dads. In fact, some babies never crawl: they go straight from sitting to walking. Others scoot on their bottoms, “commando” crawl on their bellies, or adopt a “crab crawl,” placing their weight on one leg while bending the other to propel themselves forward. The pediatrician says that not crawling will not prevent Bella from reaching other important physical milestones like standing and walking.
The first two years of life bring dramatic gains in the development of movement and motor skills. A newborn’s involuntary movements transform to the infant’s ability to exert some limited control over their muscles.
Newborn Reflexes
Babies are born with several reflexes (Table 3.5). Some, like coughing, sneezing, and blinking, stay with us, while others, like rooting, disappear.
 Some, such as the sucking reflex for nursing or bottle feeding, are for survival. Others are a stress response connected to the sympathetic nervous system. For example, the Moro reflex, also called the startle reflex, is activated when the infant is startled by a sudden movement or noise. The head automatically moves back, and the arms and legs flail out. Other reflexes do not have clear functions but are appealing to see, like the Babinski and plantar reflexes. If you glide your finger on the inner sole of the infant’s foot beneath their toes, the toes curl up, but if you stroke the outer sole, the toes spread out (Figure 3.12).
Some reflexes may be absent in newborns who have neurological impairments or who were born preterm, for example, while others may remain beyond the typical age of disappearance. Health-care professionals are likely to check for the appearance and disappearance of various reflexes to better understand how infants are developing and provide early intervention when needed (Swapna et al., 2020). This scan often begins with the Apgar test immediately after birth.
Milestones in Motor Development
Motor development milestones in the first two years of life are rapid and dramatic, and parents and caregivers often eagerly anticipate their child’s learning to sit, stand, and walk. Each achievement occurs during a given time span. The average age for walking is twelve months, for example, but infants can begin to walk as early as eight months or as late as seventeen to eighteen months. These milestones follow the predictable sequence of cephalocaudal and proximodistal development. For example, following the cephalocaudal pattern, infants are first able to hold their heads up, then sit, pull up, and finally walk. In the proximodistal pattern, infants can grab an object with their full hand before they can manipulate it with individual fingers.
Gross Motor Skills
Voluntary movements that rely on large muscle groups and typically activate the arms, legs, head, and torso are
gross motor skills
. Infants reach several major postural milestones in the first year (Table 3.6). Movement skills also typically build on one another. For example, after crawling, infants may pull themselves up and then gain the strength to walk. If a child is lagging on one or more milestones, the delay can be addressed through early intervention and monitored by a pediatrician. However, motor milestones also show cultural, experiential, and individual variability (Adolph & Hoch, 2019).
Muscle Control Milestone
Age Range Expected
Supports head
6–8 weeks old
Lifts head and chest when on tummy
2–5 months
Rolls from tummy to back
4–6 months
Sits alone (unsupported)
4–8 months
Stands with support
5–12 months
Stands without support
7–17 months
Walks with support
6–13 months
Walks without support
8–17 months
Table 3.6
Average Age Range for Muscle Control Milestones in the First Two Years
(sources: Adolph & Franchak, 2017; Noritz et al., 2013; WHO Multicentre Growth Reference Study Group & de Onis, 2006)
Link to Learning
During the first few months of life, infants are rapidly moving from one gross motor milestone to the next. This video shows
some of the motor milestones they reach
during the first three months.
Fine Motor Skills
Although gross motor skills typically appear first, infants and toddlers are also beginning to develop
fine motor skills
, coordinated movements performed by small muscles to manipulate and control objects or perform precise tasks like reaching for and grasping an object (Table 3.7). Finger movements readily come to mind, but the toes, eyes, face, tongue, and lips also execute fine motor movements.
Fine Motor Skill Milestone
Average Age Expected
Opens hands briefly
2 months
Holds a toy when put in their hand
4 months
Brings hand to mouth
4 months
Reaches to grab toy
6 months
Bangs objects together
9 months
Transfers objects between hands
9 months
Grasps with two fingers
12 months
Feeds self with fingers
18 months
Tries to use buttons or switches on a toy
24 months
Table 3.7
Average Age for Fine Motor Skills Emerging in the First Two Years of Life
(source: CDC, 2024)
Like gross motor skills, fine motor skills often develop in a predictable sequence. At about four months of age, the infant uses two hands to reach for an object and then with more experience uses only one arm. Grasping begins with hands and palms (no thumb), a skill called the ulnar grasp. At about six months, the palmar grasp reflex is replaced by voluntary grasping. Infants begin to use their thumbs at around nine months of age when they grasp with their forefinger and thumb (Figure 3.13), dramatically improving the ability to control and manipulate objects. These coordinated actions are quite an accomplishment in only a few months’ time.
The development of fine motor skills improves with practice, as you can observe if you watch an infant pick up objects and put them into a container repeatedly. With practice, infants will change their grasping patterns to fit the shape and size of the objects (Barrett et al., 2008). Fine motor skills, much like gross motor skills, also show variability based on family environment, culture, and opportunities to practice various movements (Hoch et al., 2024). For example, being in a home with more access to toys that call upon fine motor skills predicts improved fine motor skills.
Motor skills are also the result of interactions with other domains of development. For example, infants must plan and use their senses to move about in their environment. The ability to speak is also dependent on their being able to control the muscles in their mouth and larynx. In addition, although motor skills may appear to develop as individual skills, they build on each other, with each new one serving as the foundation for the next. As infants get older, individual motor skills combine into more complex skills, allowing infants to explore and learn about their surroundings.
One theory that has helped developmental psychologists understand the complexities of motor development is
dynamic systems theory
, which posits that developmental behaviors and milestones are the result of interactions between systems, including those within the individual. According to the original theorist, Esther Thelen, motor development is the result of interactions between internal developmental domains (e.g., cognition and the senses), maturation, and the environment (Fujihira & Taga, 2023; Thelen & Smith, 2007). For example, to walk, a child must have several coordinated and developed internal systems including balance, leg muscles, stepping movements, coordination, vision, and spatial awareness (Thelen & Ulrich, 1991).
From a dynamic systems perspective, every motor skill results from multiple maturational and environmental processes. For example, growth in the cerebellum, which contributes to balance and coordination, myelination of the nervous system, and synaptic pruning are critical for acquiring both fine and gross motor skills (Morizawa et al., 2022; Tiemeier et al., 2010). Gross motor skills are also influenced by
genetics
. Identical twins, who have nearly identical genetic makeup, are more similar in the timing of their motor milestones than fraternal twins, who share around half of the same genes (Ooki, 2006). However, even identical twins have unique experiences that can result in very different motor skill capabilities, as developmental psychologist and neurobiologist Myrtle McGraw observed in case study experiments in the 1930s (McGraw, 1935; Parnin, 2022).
Dynamic systems theory has since expanded into a dynamic systems approach, applying the theory to other domains of development over the lifespan, including cognition and language (Fogel, 2011; Spencer et al., 2011). You may notice that this approach incorporates many of the theories you learned about in Chapter 1 Lifespan Psychology and Developmental Theories, such as cognitive development and behavioral genetics.
While the dynamic systems approach improves our understanding of the influences of systems within the individual, ecological systems and sociocultural theories inform our view of the factors
outside
the individual that shape developmental outcomes. These can come from cultural and familial environments as well as from individual play behaviors and resources.
Although the sequence of motor milestones is expected to be similar for all children, the timing and abilities can vary widely, demonstrating that environmental and cultural factors play an important role (Adolph & Hoch, 2019). For example, in Tajikistan, infants are regularly swaddled and placed in
gahvora
cradles for long periods of time. Because of their temporary immobility, these infants reach motor milestones later (Karasik et al., 2018).
In Northern China, where infants lie on their backs with their body movement restricted by being buried in a bag of fine sand for most of the day for toileting practice, the onset of sitting, crawling, and walking is delayed by several months (Mei, 1994). In other cultures, such as in the United States, Argentina, South Korea, and Italy, infants spend a lot of time sitting with postural support from furniture (United States and Argentina) or their mother’s arms (South Korea and Italy) (Karasik et al., 2015). In contrast, infants in Western Kenya have more practice with both supported and independent sitting, which may allow them to sit independently somewhat earlier than in cultures where infants are provided with a lot of support while sitting (Karasik et al., 2015).
Crawling and
walking
also show great variability based on culture and parenting beliefs. In some cultures, infants skip
crawling
or walk before they crawl, based on factors like maternal handling, cultural beliefs, and whether parents encourage or discourage crawling (Adolph et al., 2018; Super, 1976). Parenting beliefs can also influence the way they handle their infants. In some cultures, caregivers perform exercises to encourage motor development that might be viewed as rough in another culture (Adolph et al., 2018).
In another example, research following the Back to Sleep campaign has shown that infants placed on their backs while sleeping reach some motor milestones significantly later than infants who slept on their stomachs (Brouwer et al., 2006; Johnson, 2004). In response, pediatricians began encouraging parents to give their infants “
tummy time
,” which is associated with earlier development of gross motor skills such as walking, standing, and sitting (Hewitt et al., 2020).
Play
Play uses motor skills and cognitive skills, both connected to brain development. For example, with time and practice, the infant who begins by grasping a spoon is soon able to use it as a tool for eating. Besides the physical ability required to do this, the child also uses cognitive skills to play. Using paintbrushes, crayons, balls, and blocks in
play
takes similar planning and coordination. By one year, the child can point to objects of interest, use more motor skills to explore environments, and interact more with others during playtime. At eighteen months, the child may seek interaction, enjoy patty-cake, and clap during play. Parents and caregivers can support motor development through play by providing age-appropriate toys and activities and encouraging exploration and movement in a safe, supportive environment (Table 3.8).
Play Activity
Motor Development
Age Range
Tummy time
Spending supervised time on their tummy can help infants strengthen neck and upper body muscles.
Infancy
Baby gym
Playing under a play gym with hanging toys encourages babies to reach and grasp.
Infancy
Mirror play
Playing with a baby-safe mirror encourages visual tracking and body awareness.
6 months–2 years
Crawling tunnel
Crawling through a toy tunnel promotes mobility and exploration.
6 months–1 year
Puzzles and building blocks
Interacting with age-appropriate puzzles and soft building blocks improves problem-solving skills, hand-eye coordination, and gross and fine motor skills.
6 months–2 years
Dance and music
Moving to music strengthens balance and coordination.
6 months–2 years
Arts and crafts
Finger painting, coloring, and cutting with safety scissors promote fine motor skills and creativity.
1–2 years
Obstacle courses
Engaging with a simple obstacle course with cushions, pillows, and tunnels encourages crawling, climbing, and balance.
1–2 years
Puppet shows
Interacting with puppets enhances visual tracking and fine motor skills.
1–2 years
Table 3.8
Play Activities and Related Motor Skills
Play starts in infancy and can inform several aspects of early development including motor skills, social interaction, self-exploration, and exploration of the environment. Play also helps caregivers and health-care providers observe how well development is progressing.
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Handbook of Child Psychology.
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Thelen, E., & Ulrich, B. D. (1991). Hidden skills: A dynamic systems analysis of treadmill stepping during the first year.
Monographs of the Society for Research in Child Development, 56
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Neuroimage
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Acta Paediatrica, 95
(S450), 66–75. doi.org/10.1111/j.1651-2227.2006.tb02377.x


26.5C: Uterus
The uterus is the largest and major organ of the female reproductive tract that is the site of fetal growth and is hormonally responsive.
- Describe the structure and function of the uterus
Key Points
- The body of the uterus is connected to the ovaries via the fallopian tubes, and opens into the vagina via the cervix.
-
Two Müllerian ducts
usually form initially in a female fetus, but in humans they completely fuse into a single uterus during gestation. - The uterus is essential in sexual response by directing blood flow to the pelvis and to the external genitalia, including the ovaries, vagina, labia, and clitoris.
- The reproductive function of the uterus is to accept a fertilized ovum which passes through the utero-tubal junction from the fallopian tube.
- The lining of the uterine cavity is called the endometrium.
Key Terms
- linea terminalis : Part of the pelvic brim, which is the edge of the pelvic inlet.
- adenomyosis : A condition characterized by the breaking through of the endometrium into the muscle wall of the uterus.
- uterus : An organ of the female reproductive system in which the young are conceived and develop until birth; the womb.
- endometrium : The mucous membrane that lines the uterus in mammals and in which fertilized eggs are implanted.
- fallopian tubes : The fallopian tubes, also known as oviducts, uterine tubes, and salpinges (singular salpinx) are two very fine tubes lined with ciliated epithelia, and lead from the ovaries of female mammals into the uterus via the utero-tubal junction.
The uterus or womb is a major female hormone -responsive reproductive sex organ of most mammals including humans. One end, the cervix, opens into the vagina, while the other is connected to one or both fallopian tubes, depending on the species. It is within the uterus that the fetus develops during gestation, usually developing completely in placental mammals such as humans.
Two Müllerian ducts usually form initially in a female fetus and, in humans, they completely fuse into a single uterus depending on the species. The uterus consists of a body and a cervix. The cervix protrudes into the vagina. The uterus is held in position within the pelvis by condensations of endopelvic fascia, which are called ligaments. These ligaments include the pubocervical, transverse, cervical, cardinal, and uterosacral ligaments. It is covered by a sheet-like fold of peritoneum, the broad ligament.
The uterus is essential in sexual response by directing blood flow to the pelvis and to the external genitalia, including the ovaries, vagina, labia, and clitoris. The reproductive function of the uterus is to accept a fertilized ovum which passes through the utero-tubal junction from the fallopian tube. It implants into the endometrium, and derives nourishment from blood vessels which develop exclusively for this purpose.
Uterus : Vessels of the uterus and its appendages, rear view.
The fertilized ovum becomes an embryo, attaches to a wall of the uterus, creates a placenta, and develops into a fetus (gestates) until childbirth. Due to anatomical barriers such as the pelvis, the uterus is pushed partially into the abdomen due to its expansion during pregnancy. Even during pregnancy, the mass of a human uterus amounts to only about a kilogram (2.2 pounds).
The uterus is located inside the pelvis immediately dorsal (and usually somewhat rostral) to the urinary bladder and ventral to the rectum. The human uterus is pear-shaped and about three inches (7.6 cm) long. The uterus can be divided anatomically into four segments: The fundus, corpus, cervix and the internal os.
Ovary : A pictorial illustration of the female reproductive system.
The uterus is in the middle of the pelvic cavity in frontal plane (due to ligamentum latum uteri). The fundus does not surpass the linea terminalis. The fundus of the uterus is the top, rounded portion, opposite from the cervix. The vaginal part of the cervix does not extend below interspinal line. The uterus is mobile and moves under the pressure of the full bladder or full rectum anteriorly, whereas if both are full it moves upwards. Increased intra-abdominal pressure pushes it downwards. The mobility is conferred to it by musculo-fibrous apparatus that consists of a suspensory and sustentacular part. Under normal circumstances the suspensory part keeps the uterus in anteflexion and anteversion (in 90% of women) and keeps it “floating” in the pelvis. In cases where the uterus is “tipped,” also known as retroverted uterus, women may have symptoms of pain during sexual intercourse, pelvic pain during menstruation, minor incontinence, urinary tract infections, difficulty conceiving, and difficulty using tampons. A pelvic examination by a doctor can determine if a uterus is tipped.
The lining of the uterine cavity is called the endometrium. It consists of the functional endometrium and the basal endometrium from which the former arises. Damage to the basal endometrium results in adhesion formation and/or fibrosis (Asherman’s syndrome). In all placental mammals, including humans, the endometrium builds a lining periodically which is shed or reabsorbed if no pregnancy occurs. Shedding of the functional endometrial lining is responsible for menstrual bleeding (known colloquially as a “period” in humans, with a cycle of approximately 28 days, +/- 7 days of flow and +/- 21 days of progression) throughout the fertile years of a female and for some time beyond.
Depending on the species and attributes of physical and psychological health, weight, environmental factors of circadian rhythm, photoperiodism (the physiological reaction of organisms to the length of day or night), the effect of menstrual cycles to the reproductive function of the uterus is subject to hormone production, cell regeneration, and other biological activities. The menstrual cycles may vary from a few days to six months, but can vary widely even in the same individual, often stopping for several cycles before resuming.
The uterus mostly consists of smooth muscle, known as myometrium. The innermost layer of myometrium is known as the junctional zone, which becomes thickened in adenomyosis. The parametrium is the loose connective tissue around the uterus. The perimetrium is the peritoneum covering of the fundus and ventral and dorsal aspects of the uterus. The uterus is primarily supported by the pelvic diaphragm, perineal body, and the urogenital diaphragm. Secondarily, it is supported by ligaments and the peritoneum (broad ligament of uterus).


6.E: Thermochemistry- Homework
Turn in your answers for the following questions - show your work
- An electric stove has a 1200 watt heating element. If it takes 5 minutes for a teakettle to come to a boil, how much energy (in joules) is required?
- A piece of titanium metal (mass 452.398 g) is placed in boiling water (100.00 °C). After 20 minutes it is removed from the boiling water and placed in a 1.000 liter container of water at 20.00 °C. The temperature of the water increases to 24.28 °C. What is the specific heat of titanium? How does this compare to the value given in the CRC Handbook of Chemistry and Physics?
- Next the same piece of titanium is heated in an acetylene flame (like that used for welding) to an unknown temperature. When the piece of titanium is placed in a 10.000 liter container of water at 20.00 °C the final temperature is now 30.72 °C. What is the temperature of the flame (assuming that the piece of titanium was at equilibrium)? At what temperature does titanium melt?
-
Balance the following reactions, calculate [delta]H
rxn
, calculate the energy released (or required) for the combustion of 10.00 grams of fuel (H
2
, CH
4
(methane), or C
4
H
10
(butane)) in each reaction, and compare the energetics for the reactions (which is the best fuel).
- H 2 (g) + O 2 (g) --> H 2 O (g)
- CH 4 (g) + O 2 (g) --> CO 2 (g) + H 2 O (g)
- C 4 H 10 (g) + O 2 (g) --> CO 2 (g) + H 2 O (g)
The Following Questions are for your practice - Do Not Turn In. They include answers so you can check your work
Q5.1.6
How much heat, in joules and in calories, must be added to a 75.0–g iron block with a specific heat of 0.449 J/g °C to increase its temperature from 25 °C to its melting temperature of 1535 °C?
S5.1.6
\(q=mCΔ°T\)
\(q=(75.0g)\times(\dfrac{0.449\:J}{g\:°C})\times(1,510°K) = 50,800J\)
50,800J ; 12,200cal
Q5.1.8
How much would the temperature of 275 g of water increase if 36.5 kJ of heat were added?
S5.1.8
ΔT ° = 31.7° C
Calorimetry
Q5.2.6
How much will the temperature of a cup (180 g) of coffee at 95 °C be reduced when a 45 g silver spoon (specific heat 0.24 J/g °C) at 25 °C is placed in the coffee and the two are allowed to reach the same temperature? Assume that the coffee has the same density and specific heat as water.
S5.2.6
The temperature of the coffee will drop 1 degree.
Q5.2.9
If a reaction produces 1.506 kJ of heat, which is trapped in 30.0 g of water initially at 26.5 °C in a calorimeter like that in Figure , what is the resulting temperature of the water?
S5.2.9
38.5 °C
Enthalpy
Q5.3.10
When 2.50 g of methane burns in oxygen, 125 kJ of heat is produced. What is the enthalpy of combustion per mole of methane under these conditions?
S5.3.10
802 kJ mol −1
Q5.3.28
Calculate the standard molar enthalpy of formation of NO( g ) from the following data:
\(\ce{N2}(g)+\ce{2O2}⟶\ce{2NO2}(g)\hspace{20px}ΔH^\circ_{298}=\mathrm{66.4\:kJ}\)
\(\ce{2NO}(g)+\ce{O2}⟶\ce{2NO2}(g)\hspace{20px}ΔH^\circ_{298}=\mathrm{−114.1\:kJ}\)
S5.3.28
90.3 mol −1 of NO
Q5.3.32
The decomposition of hydrogen peroxide, H 2 O 2 , has been used to provide thrust in the control jets of various space vehicles. Using the data in Appendix G , determine how much heat is produced by the decomposition of exactly 1 mole of H 2 O 2 under standard conditions.
\(\ce{2H2O2}(l)⟶\ce{2H2O}(g)+\ce{O2}(g)\)
S5.3.32
−54.04 kJ mol −1
Q5.3.45
Propane, C 3 H 8 , is a hydrocarbon that is commonly used as a fuel.
- Write a balanced equation for the complete combustion of propane gas.
- Assuming that all of the heat released in burning 25.0 grams of propane is transferred to 4.00 kilograms of water, calculate the increase in temperature of the water.
S5.3.45
- \(\ce{C3H8}(g)+\ce{5O2}(g)⟶\ce{3CO2}(g)+\ce{4H2O}(l)\);
- 75.4 °C


7.1.4C: Illegitimate Opportunity Structures - Social Class and Crime
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Learning Objectives
Explain how illegitimate opportunity structures function in different subcultures
American sociologists Richard Cowan and Lloyd Ohlin extended Robert K. Merton’s social strain theory to directly address juvenile delinquency and social class. If you recall, social strain theory develops a typology of deviance in which an individual can deviate on two planes. An individual can be deviant by refusing to accept social norms or an individual can deviate by accepting social norms but using deviant means to achieve their realization. In the context of the U.S., in which prosperity is a social value, one could deviate by rejecting the notion of wealth. Alternatively, one could deviate by aspiring to a wealthy lifestyle but earning one’s living as a pickpocket.
In 1960, Cowan and Ohlin published Delinquency and Opportunity: A Theory of Delinquent Gangs . In this work, they noted that the individuals who achieved social norms by deviant means frequently operated from within institutions that, similarly to those operating in normative institutions, had rules of behavior. A key to understanding Cowan and Ohlin’s theory is the notion of subculture. A subculture is a group of people with a culture that differentiates them from the larger culture to which they belong. Subcultures create a stacking or layering effect within a larger cultural context. While a pickpocket may deviate from American social norms, he adheres to social norms of a smaller group of individuals who identify as American pickpockets. Cowan and Ohlin asserted that subcultures have rules of their own. Illegitimate opportunity structures are the rules that operate within deviant subcultures. Cowan and Ohlin emphasized how the structures of these deviant subcultures paralleled the rules and operations of more socially acceptable institutions.
Cowan and Ohlin used juvenile delinquency as a case study to explore this theory of illegitimate opportunity structures. In a criminal subculture, youth learn to use crime for material gain. This subculture usually forms in areas where there is an established organization of adult crime that provides an illegitimate opportunity structure for youths to learn how to behave criminally for material success. In a conflict subculture, youth learn to form gangs as a way to express frustration about the lack of normative opportunity structures in their neighborhood. New initiates into the gang will learn how to engage in conflict or gang activities to express frustrations by watching gang leadership. Thus, gangs become a subculture of their own, in contradistinction to the normative, peaceful model of youth behavior. Finally, in a retreatist subculture youth learn to reject both legitimate and illegitimate opportunity structures. These individuals are thought to be “double failures” in that they engage in conduct that is neither normative and accepted by society at large nor deviant but accepted by a subculture.
Criminal and conflict subcultures demonstrate that individuals can reject the normative means of the culture at large and still find a place within a smaller deviant subculture. The retreatist subculture is the exception that proves the rule of illegitimate opportunity structures. The extreme deviance and isolation of individuals affiliated with a retreatist subculture demonstrate that others who engage in deviant behavior are able to find a subculture to which to subscribe.
Key Points
- American sociologists Richard Cowan and Lloyd Ohlin extended Robert K. Merton’s social strain theory to directly address juvenile delinquency and social class.
- A subculture is a group of people with a culture that differentiates them from the larger culture to which they belong.
- In a criminal subculture, youth learn to use crime for material gain.
- In a conflict subculture, youth learn to form gangs as a way to express frustration about the lack of normative opportunity structures in their neighborhood.
- In a retreatist subculture youth learn to reject both legitimate and illegitimate opportunity structures.
- In a retreatist subculture youth learn to reject both legitimate and illegitimate opportunity structures
Key Terms
- Conflict subculture : In a conflict subculture, youth learn to form gangs as a way to express frustration about the lack of normative opportunity structures in their neighborhood.
- subculture : A portion of a culture distinguished from the larger society around it by its customs or other features.
- Illegitimate opportunity structure : In criminology, subcultural theory emerged from the work of the Chicago School on gangs and developed through the symbolic interactionism school into a set of theories arguing that certain groups or subcultures in society have values and attitudes that are conducive to crime and violence. The primary focus is on juvenile delinquency because theorists believe that if this pattern of offending can be understood and controlled, it will break the transition from teenage offender into habitual criminal.
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13.1: Prelude to F Distribution and One-Way ANOVA
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CHAPTER OBJECTIVES
By the end of this chapter, the student should be able to:
Many statistical applications in psychology, social science, business administration, and the natural sciences involve several groups. For example, an environmentalist is interested in knowing if the average amount of pollution varies in several bodies of water. A sociologist is interested in knowing if the amount of income a person earns varies according to his or her upbringing. A consumer looking for a new car might compare the average gas mileage of several models.
Figure 13.1.1. One-way ANOVA is used to measure information from several groups.For hypothesis tests comparing averages between more than two groups, statisticians have developed a method called "Analysis of Variance" (abbreviated \(ANOVA\)). In this chapter, you will study the simplest form of \(ANOVA\) called single factor or one-way \(ANOVA\). You will also study the \(F\) distribution, used for one-way \(ANOVA\), and the test of two variances. This is just a very brief overview of one-way \(ANOVA\). You will study this topic in much greater detail in future statistics courses. One-Way \(ANOVA\), as it is presented here, relies heavily on a calculator or computer.
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by
OpenStax College
is licensed under a Creative Commons Attribution License 4.0 license. Download for free at
<EMAIL_ADDRESS>.


17.6: Assignment- HR’s Sustainability Role
Scenario
Sustainability initiatives have the potential to yield a number of benefits that build business competitiveness, employee engagement and satisfaction, and community quality of life. What role does HR play in advancing sustainability?
Your Task
In this rotation, you are reporting to the firm’s CSR & Sustainability lead. The team is developing a number of HR materials to serve as reference pieces for clients that are interested in increasing their sustainability. You have been assigned to develop an action plan to address this bullet point from The Business Guide to Sustainability:
- Align HR systems to embed sustainability in the culture: Incorporate sustainability into job descriptions, selection and evaluation criteria, employee orientation and training, evaluation and compensation.
Specifically, your task is to develop a 1-page list of specific, actionable recommendations for incorporating sustainability into these materials.
Grading Rubric
| Criteria | Inadequate (40%) | Minimal (60%) | Adequate (80%) | Exemplary (100%) | Total Points |
|---|---|---|---|---|---|
| Organization and format |
2 pts
Writing lacks logical organization. It may show some coherence but ideas lack unity. Serious errors and generally is an unorganized format and information. |
3 pts
Writing is coherent and logically organized, using a format suitable for the material presented. Some points may be contextually misplaced and/or stray from the topic. Transitions may be evident but not used throughout the essay. Organization and format used may detract from understanding the material presented. |
4 pts
Writing is coherent and logically organized, using a format suitable for the material presented. Transitions between ideas and paragraphs create coherence. Overall unity of ideas is supported by the format and organization of the material presented. |
5 pts
Writing shows high degree of attention to details and presentation of points. Format used enhances understanding of material presented. Unity clearly leads the reader to the writer’s conclusion and the format and information could be used independently. |
5 pts |
| Content |
8 pts
Some but not all required questions are addressed. Content and/or terminology is not properly used or referenced. Little or no original thought is present in the writing. Concepts presented are merely restated from the source, or ideas presented do not follow the logic and reasoning presented throughout the writing. |
12 pts
All required questions are addressed but may not be addressed with thoughtful consideration and/or may not reflect proper use of content terminology or additional original thought. Additional concepts may not be present and/or may not be properly cited sources. |
16 pts
All required questions are addressed with thoughtful consideration reflecting both proper use of content terminology and additional original thought. Some additional concepts may be presented from other properly cited sources, or originated by the author following logic and reasoning they’ve clearly presented throughout the writing. |
20 pts
All required questions are addressed with thoughtful in-depth consideration reflecting both proper use of content terminology and additional original thought. Additional concepts are clearly presented from properly cited sources, or originated by the author following logic and reasoning they’ve clearly presented throughout the writing. |
20 pts |
| Development—Critical Thinking |
8 pts
Shows some thinking and reasoning but most ideas are underdeveloped, unoriginal, and/or do not address the questions asked. Conclusions drawn may be unsupported, illogical or merely the author’s opinion with no supporting evidence presented. |
12 pts
Content indicates thinking and reasoning applied with original thought on a few ideas, but may repeat information provided and/ or does not address all of the questions asked. The author presents no original ideas, or ideas do not follow clear logic and reasoning. The evidence presented may not support conclusions drawn. |
16 pts
Content indicates original thinking, cohesive conclusions, and developed ideas with sufficient and firm evidence. Clearly addresses all of the questions or requirements asked. The evidence presented supports conclusions drawn. |
20 pts
Content indicates synthesis of ideas, in-depth analysis and evidence beyond the questions or requirements asked. Original thought supports the topic, and is clearly a well-constructed response to the questions asked. The evidence presented makes a compelling case for any conclusions drawn. |
20 pts |
| Grammar, Mechanics, Style |
2 pts
Writing contains many spelling, punctuation, and grammatical errors, making it difficult for the reader to follow ideas clearly. There may be sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices disrupts the content. Additional information may be presented but in an unsuitable style, detracting from its understanding. |
3 pts
Some spelling, punctuation, and grammatical errors are present, interrupting the reader from following the ideas presented clearly. There may be sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices may detract from the content. Additional information may be presented, but in a style of writing that does not support understanding of the content. |
4 pts
Writing is free of most spelling, punctuation, and grammatical errors, allowing the reader to follow ideas clearly. There are no sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices enhance the content. Additional information is presented in a cohesive style that supports understanding of the content. |
5 pts
Writing is free of all spelling, punctuation, and grammatical errors and written in a style that enhances the reader’s ability to follow ideas clearly. There are no sentence fragments and run-ons. The style of writing, tone, and use of rhetorical devices enhance the content. Additional information is presented to encourage and enhance understanding of the content. |
5 pts |
| Total: | 50 pts |


5.6: The Law of Quadratic Reciprocity
Given that \(p\) and \(q\) are odd primes. Suppose we know whether \(q\) is a quadratic residue of \(p\) or not. The question that this section will answer is whether \(p\) will be a quadratic residue of \(q\) or not. Before we state the law of quadratic reciprocity, we will present a Lemma of Eisenstein which will be used in the proof of the law of reciprocity. The following lemma will relate Legendre symbol to the counting lattice points in the triangle.
Lemma of Eisenstein
If \(p\neq 2\) is a prime and \(a\) is an odd integer such that \(p\nmid a\), then
\[\left(\frac{a}{p}\right)=(-1)^{\sum_{i=1}^{(p-1)/2}[ia/p]}.\]
Consider the least positive residues of the integers \(a, 2a,...,((p-1)/2)a\); let \(m_1,m_2,...,m_s\) be integers of this set such that \(m_i>p/2\) for all \(i\) and let \(n_1,n_2,...,n_t\) be those integers where \(n_i<p/2\). Using the division algorithm, we see that \[ia=p[ia/p]+r\] where \(r\) is one of the \(m_i\) or \(n_i\). By adding the \((p-1)/2\) equations, we obtain
\[\label{qr1} \sum_{i=1}^{(p-1)/2}ia=\sum_{i=1}^{(p-1)/2}p[ia/p]+\sum_{i=1}^sm_i+\sum_{i=1}^tn_i.\]
As in the proof of Gauss’s Lemma, we see that \[p-m_1,p-m_2,...,p-m_s,p-n_1,p-n_2,...,p-n_t\] are precisely the integers \(1,2,...,(p-1)/2\), in the same order. Now we obtain
\[\label{qr2} \sum_{i=1}^{(p-1)/2}i=\sum_{i=1}^s(p-m_i)+\sum_{i=1}^tn_i=ps-\sum_{i=1}^sm_i+\sum_{i=1}^tn_i.\] We subtract \((\ref{qr2})\) from \((\ref{qr1})\) to get \[\sum_{i=1}^{(p-1)/2}ia-\sum_{i=1}^{(p-1)/2}i=\sum_{i=1}^{(p-1)/2}p[ia/p]-ps+2\sum_{i=1}^sm_i.\]
Now since we are taking the following as exponents for \(-1\), it suffice to look at them modulo 2. Thus
\[0\equiv \sum_{i=1}^{(p-1)/2}[ia/p]-s(mod \ 2).\] \[\sum_{i=1}^{(p-1)/2}[ia/p]\equiv s(mod \ 2)\] Using Gauss’s lemma, we get
\[\left(\frac{a}{p}\right)=(-1)^s=(-1)^{\sum_{i=1}^{(p-1)/2}[ia/p]}.\]
The Law of Quadratic Reciprocity
Let \(p\) and \(q\) be distinct odd primes. Then
\[\left(\frac{p}{q}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}.\frac{q-1}{2}}\]
We consider now the pairs of integers also known as lattice points \((x,y)\) with \[1\leq x\leq (p-1)/2 \mbox{and} \ \ 1\leq y\leq (q-1)/2.\] The number of such pairs is \(\frac{p-1}{2}.\frac{q-1}{2}\). We divide these pairs into two groups depending on the sizes of \(qx\) and \(py\). Note that \(qx\neq py\) for all pairs because \(p\) and \(q\) are distinct primes.
We now count the pairs of integers \((x,y)\) with \[1\leq x\leq (p-1)/2, \ \ 1\leq y\leq (q-1)/2 \mbox{and} \ \ qx>py.\] Note that these pairs are precisely those where
\[1\leq x\leq (p-1)/2 \mbox{and} \ \ 1\leq y\leq qx/p.\]
For each fixed value of \(x\) with \(1\leq x\leq (p-1)/2\), there are \([qx/p]\) integers satisfying \(1\leq y\leq qx/p\). Consequently, the total number of pairs with are
\[1\leq x\leq (p-1)/2, \ \ 1\leq y\leq qx/p, \mbox{and} \ \ qx>py\] is \[\sum_{i=1}^{(p-1)/2}[qi/p].\]
Consider now the pair of integers \((x,y)\) with
\[1\leq x\leq (p-1)/2, \ \ 1\leq y\leq (q-1)/2, \mbox{and} \ \ qx<py.\]
Similarly, we find that the total number of such pairs of integers is
\[\sum_{i=1}^{(q-1)/2}[pi/q].\]
Adding the numbers of pairs in these classes, we see that \[\sum_{i=1}^{(p-1)/2}[qi/p]+ \sum_{i=1}^{(q-1)/2}[pi/q]=\frac{p-1}{2}.\frac{q-1}{2},\] and hence using Lemma 14, we get that
\[\left(\frac{p}{q}\right)\left(\frac{p}{q}\right)=(-1)^{\frac{p-1}{2}.\frac{q-1}{2}}\]
Exercises
- Evaluate \(\left(\frac{3}{53}\right)\) .
- Evaluate \(\left(\frac{31}{641}\right)\) .
- Using the law of quadratic reciprocity, show that if \(p\) is an odd prime, then \[\left(\frac{3}{p}\right)=\left\{\begin{array}{lcr} \ 1 &{\mbox{if}\ p\equiv \pm1(mod \ 12)} \\ \ -1 &{\mbox{if}\ p\equiv \pm 5(mod \ 12)}. \\ \end{array}\right .\]
- Show that if \(p\) is an odd prime, then \[\left(\frac{-3}{p}\right)=\left\{\begin{array}{lcr} \ 1 &{\mbox{if}\ p\equiv 1(mod \ 6)} \\ \ -1 &{\mbox{if}\ p\equiv -1 (mod \ 6)}. \\ \end{array}\right .\]
- Find a congruence describing all primes for which 5 is a quadratic residue.
Contributors and Attributions
-
Dr. Wissam Raji, Ph.D., of the American University in Beirut. His work was selected by the Saylor Foundation’s Open Textbook Challenge for public release under a Creative Commons Attribution ( CC BY ) license.


1.5: The Equivalence Principle (Part 1)
Proportionality of Inertial and Gravitational Mass
What physical interpretation should we give to the “lines” described in ordered geometry? Galileo described an experiment (which he may or may not have actually performed) in which he simultaneously dropped a cannonball and a musket ball from a tall tower. The two objects hit the ground simultaneously, disproving Aristotle’s assertion that objects fell at a speed proportional to their weights. On a graph of spacetime with x and t axes, the curves traced by the two objects, called their world-lines , are identical parabolas. (The paths of the balls through x − y − z space are straight, not curved.) One way of explaining this observation is that what we call “mass” is really two separate things, which happen to be equal. Inertial mass , which appears in Newton’s a = \(\frac{F}{m}\), describes how difficult it is to accelerate an object. Gravitational mass describes the strength with which gravity acts. The cannonball has a hundred times more gravitational mass than the musket ball, so the force of gravity acting on it is a hundred times greater. But its inertial mass is also precisely a hundred times greater, so the two effects cancel out, and it falls with the same acceleration. This is a special property of the gravitational force. Electrical forces, for example, do not behave this way. The force that an object experiences in an electric field is proportional to its charge, which is unrelated to its inertial mass, so different charges placed in the same electric field will in general have different motions.
Geometrical Treatment of Gravity
Einstein realized that this special property of the gravitational force made it possible to describe gravity in purely geometrical terms. We define the world-lines of small objects 9 acted on by gravity to be the lines described by the axioms of the geometry. Since we normally think of the “lines” described by Euclidean geometry and its kin as straight lines, this amounts to a redefinition of what it means for a line to be straight. By analogy, imagine stretching a piece of string taut across a globe, as we might do in order to plan an airplane flight or aim a directional radio antenna. The string may not appear straight as viewed from the three-dimensional Euclidean space in which the globe is embedded, but it is as straight as possible in the sense that it is the path followed by a radio wave, 10 or by an airplane pilot who keeps her wings level and her rudder straight. The world-“line” of an object acted on by nongravitational forces is not considered to be a straight “line” in the sense of O1-O4. When necessary, one eliminates this ambiguity in the overloaded term “line” by referring to the lines of O1-O4 as geodesics. The world-line of a low-mass object acted on only by gravity is one type of geodesic. 11
Notes
9 The reason for the restriction to small objects is essentially gravitational radiation. The object should also be electrically neutral, and neither the object nor the surrounding spacetime should contain any exotic forms of negative energy. This is discussed in more detail later. See also problem 1.
10 Radio waves in the HF band tend to be trapped between the ground and the ionosphere, causing them to curve over the horizon, allowing long-distance communication.
11 For more justification of this statement, see section 9, problem 1.
We can now see the deep physical importance of statement O1, that two events determine a line. To predict the trajectory of a golf ball, we need to have some initial data. For example, we could measure event A when the ball breaks contact with the club, and event B an infinitesimal time after A. 12 This pair of observations can be thought of as fixing the ball’s initial position and velocity, which should be enough to predict a unique world-line for the ball, since relativity is a deterministic theory. With this interpretation, we can also see why it is not necessarily a disaster for the theory if O1 fails sometimes. For example, event A could mark the launching of two satellites into circular orbits from the same place on the Earth, heading in opposite directions, and B could be their subsequent collision on the opposite side of the planet. Although this violates O1, it doesn’t violate determinism. Determinism only requires the validity of O1 for events infinitesimally close together. Even for randomly chosen events far apart, the probability that they will violate O1 is zero.
Note
Regarding infinitesimals, see Example 3.
Eötvös Experiments
Einstein’s entire system breaks down if there is any violation, no matter how small, of the proportionality between inertial and gravitational mass, and it therefore becomes very interesting to search experimentally for such a violation. For example, we might wonder whether neutrons and protons had slightly different ratios of gravitational and inertial mass, which in a Galileo-style experiment would cause a small difference between the acceleration of a lead weight, with a large neutron-to-proton ratio, and a wooden one, which consists of light elements with nearly equal numbers of neutrons and protons. The first high-precision experiments of this type were performed by Eötvös around the turn of the twentieth century, and they verified the equivalence of inertial and gravitational mass to within about one part in 10 8 . These are generically referred to as Eötvös experiments.
Figure 1.5.5 below shows a strategy for doing Eötvös experiments that allowed a test to about one part in 10 12 . The top panel is a simplified version. The platform is balanced, so the gravitational masses of the two objects are observed to be equal. The objects are made of different substances. If the equivalence of inertial and gravitational mass fails to hold for these two substances, then the force of gravity on each mass will not be exact proportion to its inertia, and the platform will experience a slight torque as the earth spins.
Figure \(\PageIndex{5}\): An Eötvös experiment. Top: simplified version. Bottom: realistic version by Braginskii and Panov. (Drawing after Braginskii and Panov.)
The bottom panel shows a more realistic drawing of an experiment by Braginskii and Panov. 13 The whole thing was encased in a tall vacuum tube, which was placed in a sealed basement whose temperature was controlled to within 0.02 °C. The total mass of the platinum and aluminum test masses, plus the tungsten wire and the balance arms, was only 4.4 g. To detect tiny motions, a laser beam was bounced off of a mirror attached to the wire. There was so little friction that the balance would have taken on the order of several years to calm down completely after being put in place; to stop these vibrations, static electrical forces were applied through the two circular plates to provide very gentle twists on the ellipsoidal mass between them.
In the 45 years since Braginskii and Panov’s work, improvements have been made in more direct experimental tests of the equivalence principle, in which the test masses simply free-fall. The best earthbound experiment of this type 14 has given a bound of 10 −9 , while a new experiment in orbit 15 has tightened this to 10 −14 .
Equivalence of Gravitational Fields and Accelerations
One consequence of the Eötvös experiments’ null results is that it is not possible to tell the difference between an acceleration and a gravitational field. At certain times during Betty’s field trip, she feels herself pressed against her seat, and she interprets this as evidence that she’s in a space vessel that is undergoing violent accelerations and decelerations. But it’s equally possible that Tutorbot has simply arranged for her capsule to be hung from a rope and dangled into the gravitational field of a planet. Suppose that the first explanation is correct. The capsule is initially at rest in outer space, where there is no gravity. Betty can release a pencil and a lead ball in the air inside the cabin, and they will stay in place. The capsule then accelerates, and to Betty, who has adopted a frame of reference tied to its deck, ceiling and walls, it appears that the pencil and the ball fall to the deck. They are guaranteed to stay side by side until they hit the deckplates, because in fact they aren’t accelerating; they simply appear to accelerate, when in reality it’s the deckplates that are coming up and hitting them. But now consider the second explanation, that the capsule has been dipped into a gravitational field. The ball and the pencil will still fall side by side to the floor, because they have the same ratio of gravitational to inertial mass.
The Equivalence Principle
This leads to one way of stating a central principle of relativity known as the equivalence principle : Accelerations and gravitational fields are equivalent. There is no experiment that can distinguish one from the other. 16
Note
This statement of the equivalence principle is summarized, along with some other forms of it to be encountered later.
To see what a radical departure this is, we need to compare with the completely different picture presented by Newtonian physics and special relativity. Newton’s law of inertia states that “Every object perseveres in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed thereon.” 17 Newton’s intention here was to clearly state a contradiction of Aristotelian physics, in which objects were supposed to naturally stop moving and come to rest in the absence of a force. For Aristotle, “at rest” meant at rest relative to the Earth, which represented a special frame of reference. But if motion doesn’t naturally stop of its own accord, then there is no longer any way to single out one frame of reference, such as the one tied to the Earth, as being special. An equally good frame of reference is a car driving in a straight line down the interstate at constant speed. The earth and the car both represent valid inertial frames of reference, in which Newton’s law of inertia is valid. On the other hand, there are other, noninertial frames of reference, in which the law of inertia is violated. For example, if the car decelerates suddenly, then it appears to the people in the car as if their bodies are being jerked forward, even though there is no physical object that could be exerting any type of forward force on them. This distinction between inertial and noninertial frames of reference was carried over by Einstein into his theory of special relativity, published in 1905.
Note
paraphrased from a translation by Motte, 1729
But by the time he published the general theory in 1915, Einstein had realized that this distinction between inertial and noninertial frames of reference was fundamentally suspect. How do we know that a particular frame of reference is inertial? One way is to verify that its motion relative to some other inertial frame, such as the Earth’s, is in a straight line and at constant speed. But how does the whole thing get started? We need to bootstrap the process with at least one frame of reference to act as our standard. We can look for a frame in which the law of inertia is valid, but now we run into another difficulty. To verify that the law of inertia holds, we have to check that an observer tied to that frame doesn’t see objects accelerating for no reason. The trouble here is that by the equivalence principle, there is no way to determine whether the object is accelerating “for no reason” or because of a gravitational force. Betty, for example, cannot tell by any local measurement (i.e., any measurement carried out within the capsule) whether she is in an inertial or a noninertial frame.
We could hope to resolve the ambiguity by making non-local measurements instead. For example, if Betty had been allowed to look out a porthole, she could have tried to tell whether her capsule was accelerating relative to the stars. Even this possibility ends up not being satisfactory. The stars in our galaxy are moving in circular orbits around the galaxy. On an even larger scale, the universe is expanding in the aftermath of the Big Bang. It spent about the first half of its history decelerating due to gravitational attraction, but the expansion is now observed to be accelerating, apparently due to a poorly understood phenomenon referred to by the catch-all term “dark energy.” In general, there is no distant background of physical objects in the universe that is not accelerating.
Lorentz Frames
The conclusion is that we need to abandon the entire distinction between Newton-style inertial and noninertial frames of reference. The best that we can do is to single out certain frames of reference defined by the motion of objects that are not subject to any nongravitational forces. A falling rock defines such a frame of reference. In this frame, the rock is at rest, and the ground is accelerating. The rock’s world-line is a straight line of constant x = 0 and varying t. Such a free-falling frame of reference is called a Lorentz frame. The frame of reference defined by a rock sitting on a table is an inertial frame of reference according to the Newtonian view, but it is not a Lorentz frame.
In Newtonian physics, inertial frames are preferable because they make motion simple: objects with no forces acting on them move along straight world-lines. Similarly, Lorentz frames occupy a privileged position in general relativity because they make motion simple: objects move along “straight” world-lines if they have no nongravitational forces acting on them.
Example 1: The artificial horizon
The pilot of an airplane cannot always easily tell which way is up. The horizon may not be level simply because the ground has an actual slope, and in any case the horizon may not be visible if the weather is foggy. One might imagine that the problem could be solved simply by hanging a pendulum and observing which way it pointed, but by the equivalence principle the pendulum cannot tell the difference between a gravitational field and an acceleration of the aircraft relative to the ground — nor can any other accelerometer, such as the pilot’s inner ear.
For example, when the plane is turning to the right, accelerometers will be tricked into believing that “down” is down and to the left. To get around this problem, airplanes use a device called an artificial horizon, which is essentially a gyroscope. The gyroscope has to be initialized when the plane is known to be oriented in a horizontal plane. No gyroscope is perfect, so over time it will drift. For this reason the instrument also contains an accelerometer, and the gyroscope is automatically restored to agreement with the accelerometer, with a time-constant of several minutes. If the plane is flown in circles for several minutes, the artificial horizon will be fooled into indicating that the wrong direction is vertical.
Example 2: no antigravity
This whole chain of reasoning was predicated on the null results of Eötvös experiments. In the Rocky and Bullwinkle cartoons, there is a non-Eötvösian substance called upsidasium, which falls up instead of down. Its ratio of gravitational to inertial mass is apparently negative. If such a substance could be found, it would falsify the equivalence principle. Example 10.
Operational Definition of a Lorentz Frame
We can define a Lorentz frame in operational terms using an idealized variation (Figure \(\PageIndex{9}\)) on a device actually built by Harold Waage at Princeton as a lecture demonstration to be used by his partner in crime John Wheeler. Build a sealed chamber whose contents are isolated from all nongravitational forces. Of the four known forces of nature, the three we need to exclude are the strong nuclear force, the weak nuclear force, and the electromagnetic force. The strong nuclear force has a range of only about 1 fm (10 −15 m), so to exclude it we merely need to make the chamber thicker than that, and also surround it with enough paraffin wax to keep out any neutrons that happen to be flying by. The weak nuclear force also has a short range, and although shielding against neutrinos is a practical impossibility, their influence on the apparatus inside will be negligible. To shield against electromagnetic forces, we surround the chamber with a Faraday cage and a solid sheet of mu-metal. Finally, we make sure that the chamber is not being touched by any surrounding matter, so that short-range residual electrical forces (sticky forces, chemical bonds, etc.) are excluded. That is, the chamber cannot be supported; it is free-falling.
Crucially, the shielding does not exclude gravitational forces. There is in fact no known way of shielding against gravitational effects such as the attraction of other masses or the propagation of gravitational waves. Because the shielding is spherical, it exerts no gravitational force of its own on the apparatus inside.
Inside, an observer carries out an initial calibration by firing bullets along three Cartesian axes and tracing their paths, which she defines to be linear.
We’ve gone to elaborate lengths to show that we can really determine, without reference to any external reference frame, that the chamber is not being acted on by any nongravitational forces, so that we know it is free-falling. In addition, we also want the observer to be able to tell whether the chamber is rotating. She could look out through a porthole at the stars, but that would be missing the whole point, which is to show that without reference to any other object, we can determine whether a particular frame is a Lorentz frame. One way to do this would be to watch for precession of a gyroscope. Or, without having to resort to additional apparatus, the observer can check whether the paths traced by the bullets change when she changes the muzzle velocity. If they do, then she infers that there are velocity-dependent Coriolis forces, so she must be rotating. She can then use flywheels to get rid of the rotation, and redo the calibration.
After the initial calibration, she can always tell whether or not she is in a Lorentz frame. She simply has to fire the bullets, and see whether or not they follow the precalibrated paths. For example, she can detect that the frame has become non-Lorentzian if the chamber is rotated, allowed to rest on the ground, or accelerated by a rocket engine.
It may seem that the detailed construction of this elaborate thought-experiment does nothing more than confirm something obvious. It is worth pointing out, then, that we don’t really know whether it works or not. It works in general relativity, but there are other theories of gravity, such as Brans-Dicke gravity, that are also consistent with all known observations, but in which the apparatus in Figure 1.5.8 doesn’t work. Two of the assumptions made above fail in this theory: gravitational shielding effects exist, and Coriolis effects become undetectable if there is not enough other matter nearby.
Locality of Lorentz Frames
It would be convenient if we could define a single Lorentz frame that would cover the entire universe, but we can’t. In Figure 1.5.9, two girls simultaneously drop down from tree branches — one in Los Angeles and one in Mumbai. The girl free-falling in Los Angeles defines a Lorentz frame, and in that frame, other objects falling nearby will also have straight world-lines. But in the LA girl’s frame of reference, the girl falling in Mumbai does not have a straight world-line: she is accelerating up toward the LA girl with an acceleration of about 2g.
A second way of stating the equivalence principle is that it is always possible to define a local Lorentz frame in a particular neighborhood of spacetime. 18 It is not possible to do so on a universal basis.
Note
This statement of the equivalence principle is summarized, along with some other forms of it, in Appendix C.
The locality of Lorentz frames can be understood in the analogy of the string stretched across the globe. We don’t notice the curvature of the Earth’s surface in everyday life because the radius of curvature is thousands of kilometers. On a map of LA, we don’t notice any curvature, nor do we detect it on a map of Mumbai, but it is not possible to make a flat map that includes both LA and Mumbai without seeing severe distortions.
Terminology
The meanings of words evolve over time, and since relativity is now a century old, there has been some confusing semantic drift in its nomenclature. This applies both to “inertial frame” and to “special relativity.”
Early formulations of general relativity never refer to “inertial frames,” “Lorentz frames,” or anything else of that flavor. The very first topic in Einstein’s original systematic presentation of the theory 19 is an example (Figure 1.5.10) involving two planets, the purpose of which is to convince the reader that all frames of reference are created equal, and that any attempt to make some of them into second-class citizens is invidious. Other treatments of general relativity from the same era follow Einstein’s lead. 20 The trouble is that this example is more a statement of Einstein’s aspirations for his theory than an accurate depiction of the physics that it actually implies. General relativity really does allow an unambiguous distinction to be made between Lorentz frames and non-Lorentz frames, as described earlier. Einstein’s statement should have been weaker: the laws of physics (such as the Einstein field equation) are the same in all frames (Lorentz or non-Lorentz). This is different from the situation in Newtonian mechanics and special relativity, where the laws of physics take on their simplest form only in Newton-inertial frames.
Note
Two that I believe were relatively influential are Born’s 1920 Einstein’s Theory of Relativity and Eddington’s 1924 The Mathematical Theory of Relativity . Born follows Einstein’s “Foundation” paper slavishly. Eddington seems only to mention inertial frames in a few places where the context is Newtonian.
Because Einstein didn’t want to make distinctions between frames, we ended up being saddled with inconvenient terminology for them. The least verbally awkward choice is to hijack the term “inertial,” redefining it from its Newtonian meaning. We then say that the Earth’s surface is not an inertial frame, in the context of general relativity, whereas in the Newtonian context it is an inertial frame to a very good approximation. This usage is fairly standard, 21 but would have made Newton confused and Einstein unhappy. If we follow this usage, then we may sometimes have to say “Newtonianinertial” or “Einstein-inertial.” A more awkward, but also more precise, term is “Lorentz frame,” as used in this book; this seems to be widely understood. 22
The distinction between special and general relativity has undergone a similar shift over the decades. Einstein originally defined the distinction in terms of the admissibility of accelerated frames of reference. This, however, puts us in the absurd position of saying that special relativity, which is supposed to be a generalization of Newtonian mechanics, cannot handle accelerated frames of reference in the same way that Newtonian mechanics can. In fact both Newtonian mechanics and special relativity treat Newtonian-noninertial frames of reference in the same way: by modifying the laws of physics so that they do not take on their most simple form (e.g., violating Newton’s third law), while retaining the ability to change coordinates back to a preferred frame in which the simpler laws apply. It was realized fairly early on 23 that the important distinction was between special relativity as a theory of flat spacetime, and general relativity as a theory that described gravity in terms of curved spacetime. All relativists writing since about 1950 seem to be in agreement on this more modern redefinition of the terms. 24
In an accelerating frame, the equivalence principle tells us that measurements will come out the same as if there were a gravitational field. But if the spacetime is flat, describing it in an accelerating frame doesn’t make it curved. (Curvature is a physical property of spacetime, and cannot be changed from zero to nonzero simply by a choice of coordinates.) Thus relativity allows us to have gravitational fields in flat space — but only for certain special configurations like this one. Special relativity is capable of operating just fine in this context. For example, Chung et al. 25 did a high-precision test of special relativity in 2009 using a matter interferometer in a vertical plane, specifically in order to test whether there was any violation of special relativity in a uniform gravitational field. Their experiment is interpreted purely as a test of special relativity, not general relativity.
Chiao's Paradox
The remainder of this subsection deals with the subtle question of whether and how the equivalence principle can be applied to charged particles. You may wish to skip it on a first reading. The short answer is that using the equivalence principle to make conclusions about charged particles is like the attempts by slaveholders and abolitionists in the 19th century U.S. to support their positions based on the Bible: you can probably prove whichever conclusion was the one you set out to prove.
The equivalence principle is not a single, simple, mathematically well defined statement. 26 As an example of an ambiguity that is still somewhat controversial, 90 years after Einstein first proposed the principle, consider the question of whether or not it applies to charged particles. Raymond Chiao 27 proposes the following thought experiment, which I’ll refer to as Chiao’s paradox. Let a neutral particle and a charged particle be set, side by side, in orbit around the earth. Assume (unrealistically) that the space around the earth has no electric or magnetic field. If the equivalence principle applies regardless of charge, then these two particles must go on orbiting amicably, side by side. But then we have a violation of conservation of energy, since the charged particle, which is accelerating, will radiate electromagnetic waves (with very low frequency and amplitude). It seems as though the particle’s orbit must decay.
Note
A good recent discussion of this is “Theory of gravitation theories: a noprogress report,” Sotiriou, Faraoni, and Liberati, http://arxiv.org/abs/0707.2748
The resolution of the paradox, as demonstrated by hairy calculations 28 is interesting because it exemplifies the local nature of the equivalence principle. When a charged particle moves through a gravitational field, in general it is possible for the particle to experience a reaction from its own electromagnetic fields. This might seem impossible, since an observer in a frame momentarily at rest with respect to the particle sees the radiation fly off in all directions at the speed of light. But there are in fact several different mechanisms by which a charged particle can be reunited with its long-lost electromagnetic offspring. An example (not directly related to Chiao’s scenario) is the following.
Note
The first detailed calculation appears to have been by Cécile and Bryce DeWitt, “Falling Charges,” Physics 1 (1964) 3. This paper is unfortunately very difficult to obtain now. A more recent treatment by Grøn and Næss is accessible at arxiv.org/abs/0806.0464v1. A full exposition of the techniques is given by Poisson, “The Motion of Point Particles in Curved Spacetime,” www.livingreviews.org/lrr-2004-6.
Bring a laser very close to a black hole, but not so close that it has strayed inside the event horizon, which is the spherical point of no return from within which nothing can escape. Example 14 gives a plausibility argument based on Newtonian physics that the radius 29 of the event horizon should be something like \(r_{H} = \frac{GM}{c^{2}}\), and section 6.3 derives the relativistically correct factor of 2 in front, so that \(r_{H} = \frac{2GM}{c^{2}}\). It turns out that at r = \(\left( \frac{3}{2}\right) r_{H}\), a ray of light can have a circular orbit around the black hole. Since this is greater than r H , we can, at least in theory, hold the laser stationary at this value of r using a powerful rocket engine. If we point the laser in the azimuthal direction, its own beam will come back and hit it.
Note
Because relativity describes gravitational fields in terms of curvature of spacetime, the Euclidean relationship between the radius and circumference of a circle fails here. The r coordinate should be understood here not as the radius measured from the center but as the circumference divided by 2\(\pi\).
Since matter can experience a back-reaction from its own electromagnetic radiation, it becomes plausible how the paradox can be resolved. The equivalence principle holds locally , i.e., within a small patch of space and time. If Chiao’s charged and neutral particle are released side by side, then they will obey the equivalence principle for at least a certain amount of time — and “for at least a certain amount of time” is all we should expect, since the principle is local. But after a while, the charged particle will start to experience a back-reaction from its own electromagnetic fields, and this causes its orbit to decay, satisfying conservation of energy. Since Chiao’s particles are orbiting the earth, and the earth is not a black hole, the mechanism clearly can’t be as simple as the one described above, but Grøn and Næss show that there are similar mechanisms that can apply here, e.g., scattering of light waves by the nonuniform gravitational field.
It is worth keeping in mind the DeWitts’ caution that “The questions answered by this investigation are of conceptual interest only, since the forces involved are far too small to be detected experimentally” (see problem 8).
References
13 V.B. Braginskii and V.I. Panov, Soviet Physics JETP 34, 463 (1972).
14 Carusotto et al., “Limits on the violation of g-universality with a Galileotype experiment,” Phys Lett A183 (1993) 355. Freely available online at researchgate.net.
15 Touboul et al., “The MICROSCOPE mission: first results of a space test of the Equivalence Principle,” arxiv.org/abs/1712.01176
19 Einstein, “The Foundation of the General Theory of Relativity,” 1916. An excerpt is given later.
21 Misner, Thorne, and Wheeler, Gravitation , 1973, p. 18
22 ibid , p. 19
23 Eddington, op. cit.
24 Misner, Thorne, and Wheeler, op. cit. , pp.163-164. Penrose, The Road to Reality , 2004, p. 422. Taylor and Wheeler, Spacetime Physics , 1992, p. 132. Schutz, A First Course in General Relativity , 2009, pp. 3, 141. Hobson, General Relativity: An Introduction for Physicists , 2005, sec. 1.14.
25 arxiv.org/abs/0905.1929
27 arxiv.org/abs/quant-ph/0601193v7