10.5: Steady-Flow Devices
A steady-flow device is any device that will have a continuous flow of material through it. Some examples of steady-flow devices include pipes, nozzles, diffusers, and pumps. Generally, the material flowing through the device is a gas or liquid, and if the device in any way changes the velocity of the fluid then that fluid will exert a force on the steady flow device in return.
In order to determine the forces at play on a steady-flow device, we will start with our impulse-momentum equation.
\[ \vec{J} = m \vec{v}_f - m \vec{v}_i \]
Because this is a continuous process, it doesn't really make sense to have initial and final velocities. Instead, we will have inlet and outlet velocities. Also, the mass will need to be changed to the mass flow rate (the mass entering or leaving the device per unit time) to deal with the continuous-flow nature of the system.
Dividing our initial impulse-momentum equation by time on both sides will give us the desired mass flow rate on the right, while the time on the left will cancel out the time component of the impulse.
\[ \frac{\vec{F} * t}{t} = \frac{m_{out}}{t} \vec{v}_{out} - \frac{m_{in}}{t} \vec{v}_{in} \]
Simplifying this equation, we will arrive at our final equation, which relates the force our steady-flow device exerts on the fluid to the mass flow rates and velocities at the inlet and outlet. The force the fluid exerts on the device would simply be equal and opposite to the force below.
\[ \vec{F} = \dot{m}_{out} \vec{v}_{out} - \dot{m}_{in} \vec{v}_{in} \]
One final note is that these equations are vector equations. If the device is changing the direction of the flow of a fluid you will need to break the force and velocities into \(x\) and \(y\) components and split the above equation into \(x\) and \(y\) components.
Finding Mass Flow Rate:
If the mass flow rate in or out of your device is not given directly, you may need to find those values. First we can use a simple identity: we know the mass flow rate will be equal to the density of the fluid \((\rho)\) times the volumetric flow rate. Furthermore, the volumetric flow rate can be related to the geometry of the device, in that it will be equal to the average velocity of the fluid at the inlet or outlet times the cross-sectional area at the inlet or outlet. Putting this all together, we arrive at the following formula.
\[ \dot{m} = \rho \dot{V} = \rho \vec{v} A \]
Example \(\PageIndex{1}\)
A firefighter supports a hose as shown below. The hose has a volumetric flow rate of 60 gal/min and the nozzle reduces in diameter from 4 cm to 2 cm. What force will the firefighter have to exert, in Newtons, to keep the hose in place?
- Solution
Example \(\PageIndex{2}\)
A 90-degree elbow joint redirects the flow along a pipe of diameter 3 cm. If water (density=1000 kg/m 3 ) is traveling through the pipe with an average speed of 5 m/s, what is the magnitude and direction of the force the water exerts on the elbow joint?
- Solution


6.2: Centripetal Acceleration
Learning Objectives
By the end of this section, you will be able to:
- Establish the expression for centripetal acceleration.
- Explain the centrifuge.
We know from kinematics that acceleration is a change in velocity, either in its magnitude or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the magnitude of the velocity might be constant. You experience this acceleration yourself when you turn a corner in your car. (If you hold the wheel steady during a turn and move at constant speed, you are in uniform circular motion.) What you notice is a sideways acceleration because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration will become. In this section we examine the direction and magnitude of that acceleration.
Figure \(\PageIndex{1}\) shows an object moving in a circular path at constant speed. The direction of the instantaneous velocity is shown at two points along the path. Acceleration is in the direction of the change in velocity, which points directly toward the center of rotation (the center of the circular path). This pointing is shown with the vector diagram in the figure. We call the acceleration of an object moving in uniform circular motion (resulting from a net external force) the centripetal acceleration \(a_c\); centripetal means “toward the center” or “center seeking.”
The direction of centripetal acceleration is toward the center of curvature, but what is its magnitude? Note that the triangle formed by the velocity vectors and the one formed by the radii \(r\) and \(\Delta s\) are similar. Both the triangles ABC and PQR are isosceles triangles (two equal sides). The two equal sides of the velocity vector triangle are the speeds \( v_1 =v_2 = v \) Using the properties of two similar triangles, we obtain
\[ \dfrac{\Delta v}{v} = \dfrac{\Delta s}{r}. \]
Acceleration is \(\Delta v/\Delta t \) and so we first solve this expression for \(\delta v \):
\[\delta v = \dfrac{v}{r} \Delta s. \]
Then we divide this by \(\Delta t \), yielding
\[\dfrac{\Delta v}{\Delta t} = \dfrac{v}{r} \times \dfrac{\Delta s}{\Delta t}. \]
Finally, noting that \( \Delta v/\Delta t = a_c \) and that \(\delta s/\Delta t = v \) the linear or tangential speed, we see that the magnitude of the centripetal acceleration is
\[ a_c = \dfrac{v^2}{r}, \]
which is the acceleration of an object in a circle of radius \(r\) at a speed \(v\).
So, centripetal acceleration is greater at high speeds and in sharp curves (smaller radius), as you have noticed when driving a car. But it is a bit surprising that \(a_c\) is proportional to speed squared, implying, for example, that it is four times as hard to take a curve at 100 km/h than at 50 km/h. A sharp corner has a small radius, so that \(a_c\) is greater for tighter turns, as you have probably noticed.It is also useful to express \(a_c\) in terms of angular velocity. Substituting \( v = r\omega \) into the above expression, we find \( a_c = (r \omega^2)/r = r \omega^2 \). We can express the magnitude of centripetal acceleration using either of two equations:
\[ a_c = \dfrac{v^2}{r}; \, a_c = r \omega^2 \]
Recall that the direction of \(a_c\) is toward the center. You may use whichever expression is more convenient, as illustrated in examples below.
A centrifuge (Figure \(\PageIndex{2b}\)) is a rotating device used to separate specimens of different densities. High centripetal acceleration significantly decreases the time it takes for separation to occur, and makes separation possible with small samples. Centrifuges are used in a variety of applications in science and medicine, including the separation of single cell suspensions such as bacteria, viruses, and blood cells from a liquid medium and the separation of macromolecules, such as DNA and protein, from a solution. Centrifuges are often rated in terms of their centripetal acceleration relative to acceleration due to gravity \((g)\) maximum centripetal acceleration of several hundred thousand \(g\) is possible in a vacuum. Human centrifuges, extremely large centrifuges, have been used to test the tolerance of astronauts to the effects of accelerations larger than that of Earth’s gravity.
Example \(\PageIndex{1}\): Centripetal Acceleration vs. Gravity?
What is the magnitude of the centripetal acceleration of a car following a curve of radius 500 m at a speed of 25.0 m/s (about 90 km/h)? Compare the acceleration with that due to gravity for this fairly gentle curve taken at highway speed. See Figure \(\PageIndex{2a}\).
Strategy
Because \(v\) and \(r\) are given, the first expression in \( a_c = \frac{v^2}{r}: \, a_c = r \omega^2 \) is the most convenient to use.
Solution
Entering the given values of \( v = 25.0 \, m/s and r = 500 \, m \) into the first expression for \(a_c\) gives
\[a_c = \dfrac{v^2}{r} = \dfrac{(25.0 m/s)^2}{500 \, m} = 1.25 \, m/s^2. \nonumber \]
Discussion
To compare this with the acceleration due to gravity \( (g = 9.80 \, m/s^2) \), we take the ratio of \(a_c/g = (1.25 \, m/s^2)/(9.80 \, m/s^2) = 0.128. \) Thus, \(a_c = 0.128 g \) and is noticeable especially if you were not wearing a seat belt.
Example \(\PageIndex{2}\): How Big is the Centripetal Acceleration in an Ultracentrifuge?
Calculate the centripetal acceleration of a point 7.50 cm from the axis of an ultracentrifuge spinning at \(7.4 \times 10^7 \, rev/min.\) Determine the ratio of this acceleration to that due to gravity. See Figure Figure \(\PageIndex{2b}\).
Strategy
The term rev/min stands for revolutions per minute. By converting this to radians per second, we obtain the angular velocity \(\omega\). Because \(r\) is given, we can use the second expression in the equation \(a_c = \frac{v^2}{r}; \, a_c = r\omega^2 \) to calculate the centripetal acceleration.
Solution
To convert \(7.40 \times 10^4 \, rev/min \) to radians per second, we use the facts that one revolution is \(2\pi \, rad\) and one minute is 60.0 s. Thus,
\[\omega = 7.40 \times 10^4 \dfrac{rev}{min} \times \dfrac{2 \pi \, rad}{1 \, rev} \times \dfrac{1 \, min}{60 \, sec} = 7745 \, rad/sec. \nonumber \]
Now the centripetal acceleration is given by the second expression in \( a_c = \frac{v^2}{r}; \, a_c = r\omega^2\) as
\[ a_c = r\omega^2. \nonumber\]
Converting 7.50 cm to meters and substituting known values gives
\[a_c = (0.0750 \, m)(7854 \, rad/sec)^2 = 4.50 \times 10^6 \, m/s^2. \nonumber\]
Note that the unitless radians are discarded in order to get the correct units for centripetal acceleration. Taking the ratio of \(a_c\) to \(g\) yields
\[\dfrac{a_c}{g} = \dfrac{4.63 \times 10^6}{9.80} = 4.59 \times 10^5. \nonumber\]
Discussion
This last result means that the centripetal acceleration is 472,000 times as strong as \(g\). It is no wonder that such high \(\omega\) centrifuges are called ultracentrifuges. The extremely large accelerations involved greatly decrease the time needed to cause the sedimentation of blood cells or other materials.
Of course, a net external force is needed to cause any acceleration, just as Newton proposed in his second law of motion. So a net external force is needed to cause a centripetal acceleration. In the section on Centripetal Force , we will consider the forces involved in circular motion.
PHET EXPLORATIONS: LADYBUG MOTION 2D
Learn about position, velocity and acceleration vectors. Move the ladybug by setting the position, velocity or acceleration, and see how the vectors change. Choose linear, circular or elliptical motion, and record and playback the motion to analyze the behavior.
Summary
- Centripetal acceleration \(a_c\) is the acceleration experienced while in uniform circular motion. It always points toward the center of rotation. It is perpendicular to the linear velocity \(v\) and has the magnitude \[a_c = \dfrac{v^2}{r}; \, a_c = r\omega^2. \nonumber \]
- The unit of centripetal acceleration is \(m/s^2.\)
Glossary
- centripetal acceleration
- the acceleration of an object moving in a circle, directed toward the center
- ultracentrifuge
- a centrifuge optimized for spinning a rotor at very high speeds


1.11: Worksheet Answers
1. What is the most abundant molecule in the cell?
- Water (H 2 O). Roughly 70% of a cell consists of water.
2. Name one inorganic molecule in the cell.
- One of: Sodium chloride (NaCl), potassium chloride (KCl), water (H 2 O), oxygen (O 2 ) etc.
3. What is the difference between inorganic and organic molecules?
- All organic molecules contain carbon (C) while inorganic molecules can contain any atom except carbon.
4. Name the 4 different types of organic molecules to be found in the cell.
- 1. Carbohydrates
- 2. Fats
- 3. Proteins
- 4. Nucleic acids
5. What is the characteristic of carbon atoms that makes them so versatile?
- They can bond with 4 other atoms allowing them to make rings and chains and thus form large, complex molecules.
6. Name the 3 atoms that are always found in carbohydrates.
- Carbon (C)
- Hydrogen (H)
- Oxygen (O)
7. Fats and carbohydrates both contain the same atoms. What distinguishes carbohydrates from fats?
- In carbohydrates H and O are in the ratio 2:1. There is no such relationship in fats.
8. Give the 3 main functions of carbohydrates.
- 1. As an energy store e.g. as glycogen stored in the liver.
- 2. Providing energy in the cell
- 3. Making structures of the body e.g. in cartilage, bone
- Plus other functions
9. Which of the following is NOT one of the main types of carbohydrate.
-
- Monosaccharide / Polysaccharide / Sucrose / Disaccharide. Sucrose is a disaccharide.
10. Monosaccharides: (Indicate if true (T) or false (F))
-
- are sweet to taste T
- are insoluble in water F
- provide energy T
- form important structures in the cell or animal F
- are not easily digested in the gut of mammals F
- are found in grains and potatoes F
- are found in fruits and energy drinks T
- include sucrose, maltose and lactose F
- can be absorbed into the blood directly from the gut without the need for digestion T
- include glucose T
- are formed when disaccharides are split T
11. Disaccharides:
-
- are sweet to taste . T
- are soluble in water . T
- provide energy. T
- form important structures in the cell or animal. F
- are not easily digested in the gut. F
- are found in grains and potatoes. F
- are found in milk, malt and your sugar bowl . T
- include sucrose, maltose and lactose . T
- can be absorbed into the blood directly from the gut without the need for digestion. F
- include glucose. F
- are formed when 2 monosaccharides are joined . T
12. Polysaccharides:
-
- are sweet to taste. F
- are soluble water. F. Some are but most are not
- provide an energy store . T. Starch is the energy store in plants while glycogen is in animals. cellulose (plant cell walls) is digested by herbivores and provides them with their energy.
- form important structures in the cell or animal's body . T. e.g. cellulose plant cell walls, lignin is wood and chitin forms the outer skeleton of insects.
- can be absorbed into the blood directly from the gut without the need for digestion. F
- are found in grains and potatoes . T
- include sucrose, maltose and lactose. F
- do not need to be digested in the gut. F
- include glucose. F
- are formed when monosaccharides are joined in long strings . T
- include cellulose, starch and glycogen . T
13. Fats:
-
- are soluble in water. F
- are only soluble in solvents like ethanol . T
- contain hydrogen and oxygen in the ratio 2:1. F
- include triglycerides, phospholipids, and some vitamins . T
- provide the body’s most concentrated source of energy . T. Fats provide approximately 2 1/2 times more energy than the equivalent weight of carbohydrate.
- are formed by the joining of several glucose molecules. F
- are found in nuts, seeds and dairy produce . T
14. Give 2 differences between saturated and unsaturated fats.
- Saturated fats contain the maximum number of hydrogen atoms, come from animals, and tend to be solid at room temp. Unsaturated fats can incorporate more hydrogen into their structures, come from plants and fish (and poultry) and tend to be liquid at room temperature.
15. Proteins:
-
- are formed from carbon, hydrogen and oxygen molecules. F
- all contain nitrogen . T
- are made up from many fatty acids. F
- are made up from many amino acids . T
- are very small molecules. F. Proteins form some of the largest and most complex molecules in the body.
- are very stable molecules and can withstand heating, etc. F. Proteins are very sensitive to heat, acid/base balance (pH) and solvents which can destroy their molecular shape (denaturation) causing them to lose their function in the cell or body.
- are digested in the gut by the enzymes pepsin, trypsin, peptidases and proteases . T
- are digested in the gut by amylases.
- have a major role in the cell is as enzymes . T
- have a major role in the cell is to provide energy. F. Note that proteins can provide energy in some circumstances (e.g. during starvation) and in some animals (e.g.the cat) but generally they are not used for energy.
- are found in cream, potatoes and fruit. F
- are found in meat, nuts and cheese . T
16. These statements are true of amino acids:
-
- There are 20 different amino acids . T
- All of these can be found in plants. F. Only something like 10 of them (non essential amino acids) are found in sufficient quantities in all plants to provide a balanced diet.
- The amino acids not found in plants can be made by the body. F. These essential amino acids can not be made in the body. They must be taken into the body in the diet.
- Only meat and dairy produce contain all the essential amino acids . T
- The bond joining amino acids together is called a protein bond. F. It is called a peptide bond.
17. Which is the odd one out?
-
- proteins / fats / glucose / carbohydrates / nucleic acids. Glucose is a type of carbohydrate. All the others are major types of organic molecule.
- glucose / starch / amino acids / fatty acids. Starch is formed from many glucose molecules while the others in the group are all the smallest subunits of carbohydrates, proteins and fats respectively.
- starch / cellulose / glycogen / glucose . Glucose is the only molecule in the group that is not a polysaccharide.
- carbon / iron / hydrogen / oxygen. All except iron are found in all carbohydrates .
- enzymes / plasma membrane / antibodies / plant cell wall (cellulose ). All except cellulose are made of protein.


14.5: African Americans
California has 2.3 million African Americans as of 2010, the largest population of black or African Americans of the Western U.S states, and the 5 th largest black population in the United States. Cities that have the largest share of African Americans and have historically been black cultural centers include (11 largest in the state): Compton, Inglewood, Long Beach, Los Angeles, Oakland, Richmond, Riverside, Sacramento, San Bernardino, San Diego, and Vallejo.
Most African Americans in California have origins from Southern states like Louisiana, Texas, Arkansas, and Oklahoma.
There are many other cities and towns in the state with sizeable African American populations. These include:
- Northern California : Berkeley, Dublin, East Palo Alto, Emeryville, Fairfield, Hayward, Marin City, San Leandro, and Suisun City.
- Central California : Bakersfield, Fresno, Kings-Tulare counties, Merced, Modesto, and Stockton.
- Southern California : Adelanto, Altadena, Apple Valley, California City, Carson, Corona, Desert Hot Springs, Fontana, Gardena, Hawthorne, Lakewood, Lancaster, Lynwood, Moreno Valley, National City, Oceanside, Palmdale, Palm Springs, Pasadena, Perris, Pomona, Rialto, Twentynine Palms, and Victorville.
African Americans are approximately 7% of the state population. The state percentage has dropped in the 1990s and 2000s, though the state's overall number of African Americans has increased in that time. The black population in East and West Oakland and South-Central Los Angeles - places where they held the majority for decades - has greatly decreased as the black middle class has relocated to nearby suburbs, including those in the Inland Empire and Antelope Valley in Southern California and the Sacramento metropolitan area in Northern California. Many African Americans have also moved to the South, where their grandparents may have come from in the "Great Migration" of the mid-20th century.
African Americans have made important contributions to the state's hip-hop and R&B music culture. African American musical artists born and/or raised in California include: Snoop Dogg, Dr. Dre, Ice Cube, E-40, Nate Dogg, En Vogue, Tony! Toni! Tone!, Too Short, Eazy-E, N.W.A, Keyshia Cole, Digital Underground, JJ Fad, Barry White, The Pointer Sisters, Saweetie, Tyga, Doja Cat, Blueface, and Kendrick Lamar.
California has several West Indian (Afro-Caribbean American) and African immigrants from countries such as Cape Verde, Eritrea, Ethiopia, Ghana, Kenya, Nigeria, Senegal, Somalia, South Africa (including white South Africans), Tanzania, and Yemen across from the Horn of Africa. Immigrants from these countries have established communities in several cities in the state, such as Little Ethiopia in West Los Angeles. In addition to the Ethiopian community in Los Angeles, there is a significant Ethiopian population in Oakland/Berkeley. Nuer refugees from South Sudan have migrated to the Sacramento area. There are Cape Verdean communities in Solano County, the Santa Clara Valley, and the San Diego area. As of 2019, there are 13,061 Jamaicans in the state.
18 th Century -Slavery in California
People of African descent first appeared in California from Mexico due to the Spanish Conquest. Spanish soldiers, priests, and settlers brought black slaves and free blacks into the state in the 18th-century. The settlers and escort soldiers who founded the towns of San José de Guadalupe (San Jose), Yerba Buena (San Francisco), Monterey, San Diego, and La Reina de Los Ángeles (Los Angeles) were primarily mestizo and of mixed Negro and Native American ancestry from the province of Sonora y Sinaloa in Mexico. There were also many mulattoes (part black, part Spanish) in Alta California.
19th Century - Resilience & Leadership
Influential people of African ancestry were among the earliest California settlers and landowners. Pío Pico was a Californio politician, ranchero, and entrepreneur of mixed race with African ancestry, he had served as the last governor of Alta California under Mexican rule (from 1845 until 1846). Juana Briones de Miranda was a Californio businesswoman of mixed race with African ancestry, she is considered the "Founding Mother of San Francisco", as an early settler of Yerba Buena (now San Francisco). William Leidesdorff was black and multi-racial, he was one another founder of San Francisco.
After the discovery of gold in California on January 24, 1848, African Americans in search of wealth, and freedom arrived in the state during the California Gold Rush seeking their own gold discoveries. Additionally, white Southerners brought black slaves into the California mines starting in 1849, and were primarily migrating from Texas, Mississippi, Missouri, and Arkansas. The Sweet Vengeance Mine was a gold mine in Browns Valley, discovered by African American miners during the Gold Rush. Moses Rodgers was considered one of the best miners in the state.
It is important to note here that when California became a state, it was a free, non-slavery state, by the Compromise of 1850.
Some of the oldest African American churches in California are the Saint Andrews African Methodist Episcopal Church of Sacramento (founded in 1850, formerly known as Bethel African Methodist Episcopal Church), the Third Baptist Church in San Francisco (founded in 1852), Bethel African Methodist Episcopal Church (Bethel AME Church) in San Francisco (founded in 1852), African Methodist Episcopal Zion Church (AME Zion Church) in San Francisco (founded in 1852), and the First African Methodist Episcopal Church of Los Angeles (founded in 1872). In the 1870s, Rev. Peter William Cassey helped form two new Black Episcopalian churches in San Francisco; "Christ's Mission Church" (or Christ Mission Church), and he worked closely with the congregation from what later became St. Cyprian’s Church, however neither group had a building at that time.
Many of the earliest African Americans in the state held the California State Convention of Colored Citizens, a series of colored convention events active from 1855 to 1902. At the conventions they had elected delegates from the various counties and would discuss topics like slavery, public education, and voting rights.
Archy Lee had been formerly enslaved African-American and he was part of a series of notable 19th-century court cases that helped defined civil rights in the state by 1858. Edward Duplex was the first Black mayor in California, elected to office in Wheatland in 1888.
The first census recorded of African Americans in California appeared in 1850 with 962 people, and in 1860 with 4,086 people. Then, in 1910 the number rose to 22,000.
Case Study - Biddy Mason (c. 1818-1891)
Biddy Mason stands as a remarkable figure in California's history. Born into slavery in Georgia, she arrived in Los Angeles in the 1850s after a forced journey across the country with her enslaver Robert Smith. Despite facing the ongoing legal ambiguity of slavery in California, Mason successfully challenged Smith's claim of ownership in court, securing her freedom in a landmark 1856 case.
This legal victory not only secured her own liberty but also established a precedent for other enslaved people in California. Mason leveraged her newfound freedom to become a successful businesswoman, acquiring significant property holdings in what is now downtown Los Angeles. Her entrepreneurial spirit and financial acumen allowed her to transition from a life of forced labor to one of economic independence. In fact, at that time, Mason was one of the richest women in Los Angeles.
However, Mason's legacy extends far beyond personal achievement. Recognizing the importance of community and faith, she co-founded the First African Methodist Episcopal Church (FAME) in 1872, the oldest Black church in Los Angeles. This institution served not only as a spiritual center but also as a social hub for the burgeoning Black community, fostering a sense of belonging and support.
Beyond her own congregation, Mason became renowned for her philanthropy. She actively supported those in need, providing food, shelter, and financial assistance to the less fortunate. Her generosity earned her the affectionate title of "Aunt Biddy" within the Los Angeles community.
20 th Century - The Second Great Migration
In the 1920s, San Francisco's Terrific Street was an entertainment district that was home to numerous black and tan clubs. These clubs were interracial and often highlighted African American culture.
African Americans migrated to California in large numbers during the Second Great Migration, which took place between 1940 and 1970. Many of these migrants were seeking to escape Jim Crow laws in the South, while others were drawn to California's job opportunities in the defense industry and shipyards.
By the end of World War II, the African American population in California had grown significantly. In 1910, there were only 21,645 African Americans living in the state, but by 1970, that number had increased to over 1.4 million.
In the late 1940s and early 1960s, a new style of jazz called West Coast jazz emerged in California. This genre was largely developed by African American musicians, and it helped to put the state on the map as a center for jazz. Some of the most important figures in West Coast jazz include John Coltrane, Miles Davis, and Etta James.
In 1991, Rodney King, an African American, was beaten by three Los Angeles Police Department (LAPD) officers during his arrest. The beating was caught on videotape, and it sparked outrage across the country. The acquittal of the officers led to the 1992 Los Angeles riots, which were some of the most destructive in U.S. history.
The African American experience in California has been a complex one, as is has been both a haven and a source of discrimination for African Americans. However, the state has also been home to some of the most important figures in African American history and culture.
21 st Century - The New Migration
In the 2010s, California was a net loser of black migration for the first time in three decades. Most exiting California blacks are returning to Texas and the Atlanta metropolitan area. In 2018, there were Black neighborhoods and cities with Black populations surpassing 15% in Southern California like in Compton, South Los Angeles, and Inglewood, and in Northern California like Stockton, Oakland, and Vallejo. Oakland has been noted for being a center of Northern California's black population, with it being at least 25% black as of 2020. Many African Americans who settled in California, likewise in Oakland, worked on the railroad in Oakland and East Bay areas in the early-to-mid 1900s.
The Black Lives Matter movement has rippled through California, leaving its mark on social, educational, and political spheres. One significant impact is the renaming of streets, schools, and landmarks that previously honored slaveholders or figures associated with racism. This symbolic act aims to create a more inclusive public landscape that reflects the state's diverse history.
The movement has also spurred a shift in education, with a growing emphasis on incorporating Black history and culture into school curriculums and public institutions. This fosters a deeper understanding of the African American experience and its contributions to California's social fabric.
Beyond education, Black Lives Matter has significantly raised awareness of racial injustice, police brutality, and systemic racism faced by Black communities. This heightened awareness is a crucial step towards addressing these critical issues. The movement's influence extends to policy changes, potentially leading to reforms in areas like law enforcement practices and criminal justice.


5.1: Stand Structure
5.1 Stand Structure
One of the questions we often ask about forests is simply, “What’s out there?” What species are growing there? What does the forest look like visually? Are the trees young, old, or mixed? What is the proportion of small to large trees? Is there a tree canopy with a shrub layer underneath? Is the understory mainly herbaceous plants? Are there several layers of trees? Are snags or large fallen logs present? How dense is the forest? Do the trees have ample room to grow or are they crowded together? Forests display a vast array of species and structural arrangements (Figure 5.1.)
A common task of the forest technician is to provide data to answer these questions. A survey called a “stand exam” is just that – an examination of the composition and structure of the forest. Once an assessment of the current conditions is completed, then questions about “What’s happening out there?” or “What will the stand look like in the future?” can be addressed more readily.
Stand Structure refers to the overall “look” of the forest stand (Figure 5.1). It is the “horizontal and vertical distribution of components of a stand, including the height, diameter, crown layers and stems of trees, shrubs, herbaceous understory, snags and down woody debris” (Helms 1998).
As one might imagine, the structure of a forest changes over time, as trees grow, as fungi rot the wood, as insects or fire move through, as light conditions change, and so on. Therefore, a stand exam is always a measure of the forest at a point in time; a snapshot, not a hard and fast truism. To successfully manage for wildlife habitat, wood quality, desired growth rates and a myriad of other forest management objectives, foresters often a) assess what is present, b) describe what is desired in the future, then c) develop guidelines for managing toward that future forest structure. We can’t wave a magic wand and proclaim, “Increase photosynthesis,” or “Speed up nutrient cycling,” so our current tools for influencing forest function center on influencing a forest’s species composition and structural elements.
Let’s look at that stand structure definition again. “…horizontal and vertical distribution of components of a stand…..crown layers…” What terms can we use to adequately but briefly describe “distribution?” Horizontal distribution can be expressed in measures of density – trees per acre or basal area per acre (Figure 5.2). The crown is the foliar portion of the plant, and “crown layers” refers to distinct classes or stratification of the canopy. Since trees dominate the canopy of most forests, several forestry terms describe the vertical distribution, or layering of the tree crowns.
An evenaged forest is has one or two distinct age or size classes of trees; thus one or two layers of tree crowns(Figure 5.3A).
An unevenaged forest has three or more distinct age or size classes, thus three or more layers of trees (Figure 5.3B).
Although it is common to have a canopy of trees overhead, shrubs at midstory, and herbs on the forest floor, we would not refer to this as unevenaged unless there are several layers of trees . A multistoried stand is one with multiple layers of trees. As we learned before, tree size does not always indicate tree age; therefore, some foresters try to avoid the terms “evenaged,” “unevenaged” and “age classes,” and instead refer to forest “size classes” or “cohorts” to describe the distinct tree layers of a forest. So when you read “evenaged” think “one dominant layer in the overstory.” When you read “unevenaged” think “three or more tree layers.”


5.2: Perpendicular Bisector
Assume \(M\) is the midpoint of the segment \([AB]\); that is, \(M \in (AB)\) and \(AM = MB\).
The line \(\ell\) that passes thru \(M\) and perpendicular to \((AB)\), is called the perpendicular bisector to the segment \([AB]\).
Given distinct points \(A\) and \(B\), all points equidistant from \(A\) and \(B\) and no others lie on the perpendicular bisector to \([AB]\).
- Proof
-
Let \(M\) be the midpoint of \([AB]\).
Assume \(PA = PB\) and \(P \ne M\). According to SSS ( Theorem 4.4.1 ), \(\triangle AMP \cong \triangle BMP\). Hence
\(\measuredangle AMP = \pm \measuredangle BMP.\)
Since \(A \ne B\), we have "-" in the above formula. Further,
\[\begin{array} {rcl} {\pi} & = & {\measuredangle AMB \equiv} \\ {} & \equiv & {\measuredangle AMP + \measuredangle PMB \equiv} \\ {} & \equiv & {2 \cdot \measuredangle AMP.} \end{array}\]
That is, \(\measuredangle AMP = \pm \dfrac{\pi}{2}\). Therefore, \(P\) lies on the perpendicular bisector.
To prove the converse, suppose \(P\) is any point on the perpendicular bisector to \([AB]\) and \(P \ne M\). Then \(\measuredangle AMP = \pm \dfrac{\pi}{2}\), \(\measuredangle BMP = \pm \dfrac{\pi}{2}\) and \(AM = BM\). By SAS, \(\triangle AMP \cong \triangle BMP\); in particular, \(AP = BP\).
Let \(\ell\) be the perpendicular bisector to the segment \([AB]\) and \(X\) be an arbitrary point on the plane.
Show that \(AX < BX\) if and only if \(X\) and \(A\) lie on the same side from \(\ell\).
- Hint
-
Assume \(X\) and \(A\) lie on the same side of \(\ell\).
Note that \(A\) and \(B\) lie on opposite side of \(\ell\). Therefore, by Corollary 3.4.1 , \([AX]\) does not intersect \(\ell\) and \([BX]\) intersects \(\ell\); suppose that \(Y\) denotes the intersection point.
Note that \(BX = AY + YX \ge AX\). Since \(X \not\in \ell\), by Theorem \(\PageIndex{1}\) we have \(BX \ne BA\). Therefore \(BX > AX\).
This way we proved the "if" part. To prove the "only if" part, you need to switch \(A\) and \(B\) and repeat the above argument.
Let \(ABC\) be a nondegenerate triangle. Show that
\(AC > BC \Leftrightarrow |\measuredangle ABC| > |\measuredangle CAB|.\)
- Hint
-
Apply Exercise \(\PageIndex{1}\), Theorem 4.2.1 and Exercise 3.1.2 .


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1.2.2: The Schrödinger equation, particle in a box, and atomic wavefunctions
1.2: Atomic Structure
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CHEM 361: Inorganic Chemistry (Watson)
1: Intro, Review, and Effective Nuclear Charge
1.2: Atomic Structure
1.2.2: The Schrödinger equation, particle in a box, and atomic wavefunctions
<IP_ADDRESS>: Aufbau Principle
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