 Consequently, the basic operation and care of microscopes is an important skill in biology.
Two basic types of microscopes are used in our introductory biology labs: compound light microscopes and stereo microscopes (aka dissecting microscopes).
Compound light microscopes pass visible light through two sets of magnifying lenses (the ocular and the objective lenses) to magnify specimens mounted on a glass slide and placed on the flat surface ( stage ) of the microscope. The specimens must be sliced very thin in order for light coming from the light source to pass up through them so that they can be viewed. Specimens are sometimes stained with dyes to add contrast and to make structures more easily identifiable. Microscopic examination of cells and tissues allows students to observe the principle of complementarity - how cell and tissue structure determines function.
Exercise 1: Parts of the Microscope
In this exercise, you will identify and learn the functions of various microscope parts. Proper practice for handling and use of compound light microscopes is as follows:
Materials:
- Compound light microscope
Procedure:
- Obtain a compound light microscope as directed by your instructor.
- Carry the microscope upright with one hand supporting the base and the other hand grasping the arm. Care should be taken not to bump the microscope on the microscope cabinet, chairs, tables, or other obstacles. Gently place your microscope on your laboratory bench and remove its protective plastic cover (if present).
- Do not push or slide the microscope across the table. This causes vibrations that can loosen screws or misalign microscope parts.
- Clean the lenses with lens paper (if needed). The lenses on the microscope scratch easily. If you need to clean them, use ONLY lens paper (KIM Wipes are not lens paper).
- Before you use the microscope, locate the parts of the microscope and label Figure \(\PageIndex{1}\) . Additionally, Table 1 provides the function of each part.
|
Microscope Part |
Function |
|---|---|
|
Head |
Upper part of microscope that extends from arm and contains the ocular lenses and revolving nosepiece with objective lenses. |
|
Arm |
Supports the body tube and lenses. Use the arm and base to carry the microscope. |
|
Base |
Supports the entire microscope. |
|
Ocular (eyepiece) |
The lens in the upper part of the microscope. Monocular microscopes have one ocular, while binocular microscopes have two oculars. Ocular magnification is 10x . |
|
Body tube (turret) |
Holds the ocular at one end and the nosepiece at the other. Conducts light rays. |
|
Revolving nosepiece |
Located at the lower end of the body tube. A revolving device that holds the objective lenses ( aka objectives). |
|
Objective lenses |
Located on the revolving nosepiece. Each lens has a different magnifying power. The smallest objective ( scanning objective ) is the smallest magnification at 4x, followed by the low power objective at 10x, the high power objective at 40x, and the highest magnification (immersion oil objective) at 100x. Only one objective may be used at a time. The selected lens is rotated into position by turning the nosepiece. |
|
Stage |
The horizontal platform upon which the slide rests. |
|
Condenser |
Lens beneath the stage that concentrates and focuses the light before it passes through the specimen to be viewed. |
|
Iris diaphragm lever |
Small lever beneath the condenser which controls the amount of light passing through the condenser. |
|
Light source |
Provides a beam of visible light to be passed through the specimen. |
|
Mechanical stage |
Moveable part of the stage controlled by stage adjustment dials located below the stage. Allows the observer to move the stage forwards, backwards, left, and right. |
|
Coarse adjustment knob |
Located on either side of the arm. Moves the stage to bring the object into focus. This knob should only be used when the scanning (4x) objective is positioned above the stage! |
|
Fine adjustment knob |
Located within the coarse adjustment knob. Allows fine focusing of the specimen. |
- The head of the microscope supports the first set of magnifying lenses called the ocular lenses . These are located at the top of the microscope. These lenses are where we place our eyes and are also referred to as eyepieces. The ocular lenses magnify a specimen ten times (10x). If your microscope is binocular, the interpupillary distance (the distance between the eyepieces) can be adjusted to accommodate this distance.
- The second set of four lenses is located on a revolving nosepiece . Collectively known as objective lenses , these lenses can magnify a specimen four times (4x), ten times (10x), forty times (40x), or 100 times (100x), respectively. Magnification values are etched on the sides of the metal casings that protect the lenses.
- Closely related to the topic of magnification is resolution. Resolution (or resolving power ) is the ability of a microscope to distinguish two adjacent structures as distinct, or separate. The higher the resolution, the better the clarity and detail of the image.
- The bending of light through objective and ocular lenses is called refraction. This is what can give us our magnification power. At even higher magnifications, excessive refraction can cause distortion of the image but the oil immersion objective lens (100x) can remedy this problem by eliminating the gap between the specimen and the objective lens with a drop of oil. We will NOT be using the oil immersion lens in this class unless otherwise noted by your instructor.
- The light source in these microscopes can be controlled at a variety of points on the microscope, which you will see as part of this lab. Adjusting the light source correctly can be as important to see your image as adjusting focus.
- Calculating Magnification: When using a compound light microscope, both the ocular lens and objective lens help magnify the image. Therefore, to calculate the total magnification of an image, the contribution of each lens must be taken into consideration. Total magnification can be calculated using a simple formula:
\[\text{Total magnification} = \text{Ocular lens power} \times \text{Objective lens power}\]
Calculate total magnification values for your microscope and record the values in Table 2 below.
- Always begin viewing a slide using the scanning (4x) objective . Never begin an observation with the higher-powered (10x, 40x, 100x) objectives. Doing so could result in broken slides or scratched lenses.
- Never use the coarse focus adjustment at high magnification. Once a specimen is brought into focus using the lowest power, you can rotate to a higher power objective lens to increase the magnification. You will probably need to focus only slightly, as most light microscopes are parfocal , meaning that the image remains nearly in focus as you change lenses (from lowest to highest power).
- Replace the microscope properly. When you are finished using the microscope, turn off the light, remove the last slide from the stage, and wipe any material from the stage. Lower the stage and move the lowest power (4x) objective into position. Bundle the electrical cord securely (not around the arm of the microscope), replace the plastic cover, and put the microscope back in the storage cabinet.
Exercise 2: Focusing & Image Inversion
The optics of a light microscope’s lenses change the orientation of the image the user sees. A specimen that is right-side up and facing right on the microscope slide will appear upside-down and facing left when viewed through a microscope, and vice versa. Similarly, if the slide is moved left while looking through the microscope, it will appear to move right. If moved down, it will seem to move up. This occurs because microscopes use two sets of lenses to magnify the image. Because of the manner by which light travels through the lenses, this system of two lenses produces an inverted image.
You can get a slide nearly into focus without ever looking into the ocular lens by following these simple steps:
- Turn the light on and adjust the objective lens to the lowest power scanning lens.
- With the light on, place your slide on the stage and center the specimen in the light using the stage adjustment knobs.
- Raise the stage the whole way up, then bring it down with half a rotation of the coarse adjustment knob.
- Compound light microscope
- Prepared slides: letter “e”
Procedure:
- Obtain a slide of the letter “e.”
- Look at the letter “e” and draw what you see in the circle provided.
Draw the letter “e” as seen on the slide while holding it in your hand.
- Place the slide on the stage . Secure it with the stage clips . Make sure that the scanning objective (4x) is clicked into place. Use the stage adjustment knobs to center the letter “e” over the light source and directly in the center of the field of view (the lighted circular area you see as you look through the ocular lenses).
- While looking into the ocular lenses , use the coarse focus adjustment to bring the letter “e” into focus. Use the fine focus adjustment to “fine tune” the image. Try adjusting the light with the iris diaphragm lever. Decreasing the aperture size decreases the amount of light on the specimen and increases contrast.
- Observe the position of the letter “e” as it appears in the field of view (the circular area that can be seen when looking through the ocular). Draw the letter as it appears through the microscope in the circle provided.
Letter “e” as seen through microscope
- Compare your drawing from step (b) to that of step (e). How do the two drawings differ? __________________________________________________________
- While observing the letter “e” through the ocular lenses, describe the movement that you observe as you move the slide:
- to the left - _____________________________________
- to the right - ____________________________________
- forward - ______________________________________
- backward - _____________________________________
- When you view the letter “e” using the scanning objective , what is the total magnification ? _________________________________________
- Center the letter “e” in the field of view. Look at your microscope from the side (not through the oculars) and rotate the revolving nosepiece so the low power (10x ) objective clicks into place.
- Focus the letter using the fine focus adjustment. (Do not use the coarse focus adjustment.) Most microscopes are parfocal . This means that when you focus the specimen under the lowest magnification, that the specimen is “almost” in focus at any higher magnification, such that you will only need to use the fine focus adjustment to focus the image.
- Under low power , how many times has the letter “e” been magnified? __________________________________________________________
- Compared to scanning magnification , does the letter appear larger? __________________________________________________________
- Compared to scanning magnification, can you see more detail in the letter or the paper it is printed on? ___________________________________
Exercise 3: Diameter of the Field of View
Since the scanning objective is 4x and the low power objective is 10x, images will be magnified more with low power than with scanning power. Because objects will appear larger, the low power “field of view” will be smaller than the scanning power field . Therefore the relationship between the diameter of the field of view and the magnification is inversely proportional. Not only does measuring the field of view at different magnifications demonstrate this property, but once you know the diameter of the field of view in millimeters (mm) at various magnifications, you will be able to estimate the size of the cells or other structures being viewed.
- Compound light microscope
- Clear metric ruler
Procedure:
- Lower the stage, and bring the scanning objective (4x) back to the center position. Remove the letter “e” slide if you haven’t already removed it.
- Position a clear plastic ruler across the stage, and focus on its edge, so that the edge of the ruler, with millimeter markings, is visible in the field of view and equally bisects the field of view.
- The distance between two lines on the ruler is 1mm. Draw what you see in the circle provided. Measure the diameter of the field of view (in mm). ________ mm.
- Leaving the ruler in place, rotate the low power (10x) objective into position.
- Use the fine focus if necessary to bring the ruler into focus.
- Estimate the number of millimeters that you see in the field of view:
_______mm
- Leaving the ruler in place, rotate the high power (40x) objective into position.
- Use the fine focus if necessary to bring the ruler into focus.
- Estimate the number of millimeters that you see in the field of view:
_______mm
- What happens to the field of view when you increase magnification? Explain. _________________________________________________
- Do you see MORE or LESS of an object if you increase magnification? Explain. _________________________________________________
- Move the scanning (4x) objective lens back into place, lower the stage with the coarse adjustment knob, and remove the ruler.
- Place the letter “e” slide back onto the stage and secure it with the stage clip. Using the scanning objective lens, estimate the diameter of the letter “e”. If it occupies ½ of the field of view, then it is ½ times the measurement you recorded in step (c) above. Record the diameter of the letter “e”. _____________________ mm
Exercise 4: Depth of field
Depth of field is the area (top to bottom) of an object that comes into focus while slowly moving the fine adjustment knob up and down. Because the depth of focus is very short in the compound microscope, you must focus up and down to clearly view all of the planes of a specimen.
- Compound light microscope
- Slide of crossed colored threads
Procedure:
- Obtain a slide with colored threads mounted together. Place it on the stage and focus using the scanning (4x) objective . The center of your field should be the point where the three fibers cross each other.
- Focus up and down using your coarse focus adjustment . Under scanning magnification, are all three fibers in focus at the same time? ______________________________________________________
- Can you easily tell which fiber is on top and which is on the bottom? ______________________________________________________
- Rotate the nosepiece so that the low power (10x) objective clicks into place.
- Focus up and down using your fine focus adjustment . Under low magnification, are all three fibers in focus at the same time? ______________________________________________________
- Can you easily tell which fiber is on top and which is on the bottom? ______________________________________________________
- At which magnification is there a greater depth of field ? ______________________________________________________
- Use the iris diaphragm to change the amount of light passing across the fibers. Note how changing the position of the diaphragm helps to increase (or decrease) your ability to view objects.
- Does the 4x or the 10x objective have a shorter depth of field? _____________________________________________________
- Using the fine focus adjustment, slowly rotate the knob until all three threads are just out of focus. Slowly refocus using the fine focus adjustment.
- Which thread comes into focus first? _______________________
- Is this thread lying under or over the next thread? _____________
- Slowly rotate the high power (40x) objective into position and focus up and down using the fine focus adjustment only. Does the 10x or the 40x objective have a shorter depth of field? ___________________________
Exercise 5: Wet Mount of Human Epithelial Cell (Cheek Cell)
According to the cell theory, the cell is the fundamental biological unit, the smallest and simplest biological structure which possesses all of the traits of something that is living. All living organisms are composed of one or more cells, and every activity taking place in any living organism is ultimately related to the metabolic processes that are taking place in cells. Therefore, to understand the processes of life, it is necessary to understand the structure and function of the cell. Cells that line the interior of the mouth and cheeks are situated very close together, similar to tiles on a floor. These thin cells form a thick layer that protects the underlying tissue from abrasion and foreign pathogens (e.g. viruses and bacteria). The cells comprising the most superficial layer are continually sloughed off and replaced by underlying cells. Gently scraping the lining of the cheek removes the superficial cells. In this activity, you will prepare a wet mount slide of cheek cells and observe them under the compound light microscope.
- Compound light microscope
- Clean microscope slides and coverslips
- Toothpicks
- Methylene blue dye
- Water with dropper
Procedure:
- Obtain a clean microscope slide and coverslip. Add one drop of water to the slide.
- Obtain a flat toothpick and obtain a sample of your cheek epithelial cells from the inside lining of the oral cavity. The goal is to gently scrape loose cells free, not to draw blood from your cheek.
- Apply the cells to the slide, rotating the toothpick between your thumb and forefinger to dislodge the cells into the water. Dispose of the toothpick in the biohazard trash.
- Obtain a small bottle of methylene blue dye. Make sure that the dropper does not actually touch the slide ( do not contaminate the dropper bottle with your cheek cells ). Let a VERY small drop of methylene blue dye fall onto the slide. (*Methylene blue will stain just about anything it touches a deep blue, so be careful not to get it on your skin or clothing.)
- What is the purpose of adding stain to the biological specimen? __________________________________________________
- Place a clear cover slip on top of the specimen. You have just prepared a
wet mount slide !
- Bring the epithelial cells into focus using the scanning (4x) objective and the coarse focus adjustment. Zoom in on a few cells by switching to the low power (10x) objective lens.
- Confirm with your instructor that you have viewed a cheek epithelial cell.
- Switch the objective lens to high power (40x) . Sketch one cell in the circle below.
Total Magnification_______
- What is the genetic material found in the nucleus of these eukaryotic cells? _________________________________________________
- What is the approximate size of one cell? ___________________ mm
- How do you know this? ____________________________________
Exercise 6: Observation of Plant Cells (Elodea Cells)
Leaves of Elodea, a common aquatic water plant, are great for observing the major characteristics of a typical plant cell. In this activity you will prepare a wet mount and examine one of the leaves from the Elodea under the compound light microscope.
- Compound light microscope
- Clean microscope slides and coverslips
- Elodea cells
- Water with dropper
- Optional: Onion epithelial cells or prepared slide of onion root tip
- Methylene blue dye (only if using onion epithelial cells as dye is not needed to see the cellular structures in the Elodea leaf)
Procedure for Elodea Cells:
- Obtain a clean microscope slide and coverslip.
- Using forceps, remove an Elodea leaf (smallest, youngest leaves are best) from the sprig of Elodea provided and place it on the slide. Make sure it is pressed down and flat.
- Add a drop of water to cover the top of the leaf.
- Place a clear cover slip on top of the specimen. You have just prepared a
wet mount slide !
- Bring the plant cells into focus using the scanning (4x) objective and the coarse focus adjustment. Zoom in on a few cells by switching to the low power (10x) objective lens.
- Switch the objective lens to high power (40x) . Sketch several Elodea cells in the circle below.
Total Magnification_______
- Identify the following structures:
- Cell wall - the rigid framework consisting of the carbohydrate cellulose that surrounds the cell and lies outside the cell membrane. Its function is to give the cell a definite shape and provide support. Cell walls are not found in animal cells.
- Protoplasm - the organized contents of the cell, exclusive of the cell wall.
- Central vacuole - the membrane bound sac, filled with water and dissolved substances, that lies within the cytoplasm. Its functions include the storage of metabolic wastes as well as providing turgor pressure which gives the cells support.
- Cytoplasm - the protoplasm of the cell exclusive of the nucleus.
- Chloroplasts - the green spherical organelles containing the pigment chlorophyll that is involved in photosynthesis. As the microscope light heats up the water on the slide, you may observe the cytoplasm and the chloroplasts moving around the central vacuole in a process called cytoplasmic streaming .
- Nucleus - the organelle within the cytoplasm that contains the genetic information in the form of DNA. Since dye has not been used to visualize cellular structures, the nucleus may be transparent and difficult to observe.
- What three structures are observed in the Elodea cells that are unique to plants and not observed in animal cells ? ___________________________________________________________
Procedure for Onion Epithelial Cells: (Optional)
- Remove the thin, transparent epidermis (skin) from an onion leaf to prepare a wet mount slide. Alternatively, you may view a prepared slide of onion root tip. Do not discard commercially prepared slides.
- Place on a clean slide and add a drop of methylene blue. Do not contaminate the dropper (do not touch the onion skin with the dropper). Cover with a clear coverslip.
- Observe with the scanning objective lens using the coarse focus adjustment first, then the fine adjustment knob.
- Observe using the low power objective lens. Make sure you see the rectangular shaped onion cells. Confirm with the instructor, if necessary. Sketch the onion cells in the circle below. On your drawing label the cell wall (lacking in animal cells), cell membrane, cytoplasm, central vacuole, and nucleus. Be sure to indicate the total magnification used in your drawing.
Sketch onion cells
Total Mag. ____
- Estimate the size of one cell: ____________________________ mm
Exercise 7: Stereo Microscope
Stereo microscopes (also called dissecting microscopes; Figure 2 ) also contain 2 sets of lenses (ocular and objective lenses). The ocular lenses on a stereo microscope, like a compound light microscope, magnify by a factor of 10x. The objective lenses, however, have relatively low magnification. There is a great deal of variation in stereo microscopes and the manner in which they achieve higher magnification. Stereo microscopes include an additional magnification system that makes the final image appear to be upright.
They are used for viewing and manipulating relatively large specimens which can be viewed in three dimensions. They have a binocular feature that creates a stereoscopic effect. They can be used to study entire small organisms since their depth of field is much greater than the compound light microscope. The light source can be directed down onto a specimen (reflected light) as well as up through the specimen (transmitted light), which permits the viewing of objects too thick to allow for the transmission of light. It is also possible to view multiple samples in a petri dish placed on the microscope’s stage with light that is projected from below.
- Stereo microscopes
- Variety of objects to be viewed (e.g. feather, shell, penny, preserved insects, crystals)
Procedure:
- Your instructor will describe how to use the stereomicroscope.
- Use the stereomicroscope to view the variety of materials provided for you.
Make a drawing of 2 of the items you viewed with the dissecting microscope.
Exercise 8: Electron Microscope
Electron microscopes, first developed in the 1940s, use a beam of electrons instead of light to magnify an object. They can magnify objects as small as 2 nanometers (0.00000004 inches) over 100,000 times.
Transmission electron microscopes are used to study internal cell structure and are analogous to compound light microscopes in that regard. Specimens are cut into thin sections, usually “stained” with heavy metal atoms (atoms with large atomic numbers) that attach to cell structures. An electron beam is then focused through the specimen. Figure 3 is an example of the type of image an electron microscope can produce.
Scanning electron microscopes , analogous to stereo microscopes, allow a specimen’s surfaces to be observed in detail. The object is chemically frozen and then coated with a thin film of metal. An electron beam excites surface electrons on the specimen and produces a three-dimensional image. Figure 4 is an example image that is produced by an SEM.
Electron microscopes are expensive and require special training. Even though the preparation techniques used to prepare specimens for electron microscopy kill most living cells, some cells such as Tardigrades can withstand these harsh treatments.
Questions For Review:
- Why, specifically, did we look at the letter “e” slide, threads slide, and the ruler?
- What cellular parts of your cheek cell could you observe?
- What does it mean if a microscope is parfocal?
- What is the relationship between the magnification and the diameter of the field of view? Explain.
- What two parts of the microscope do you use to carry it?
Practical Challenge Questions:
- Imagine you are looking at a slide on high magnification and you lose focus on it. What should you do?
- You are using a microscope you haven't used before and you notice the microscope has a 10x ocular as well as a 73x objective lens. What will the total magnification be if this objective lens is positioned over the stage?
- You are given several specimens to observe with a microscope but you are unsure what type of scope to use. What are some reasons that you may choose different microscopes?
- Why is the resolving power of the transmission electron microscope so much greater than that of the compound light microscope?
References
Belwood, Jacqueline; Rogers, Brandy; and Christian, Jason, Foundations of Biology Lab Manual (Georgia Highlands College). “Lab 3: Microscopy,” (2019). Biological Sciences Open Textbooks . 18. CC-BY https://oer.galileo.usg.edu/biology-textbooks/18


26.2: Test #2
Note: This test may need to be updated and/or revised to accurately reflect current statistics, research findings, and health trends.
1. The CDC (Centers for Disease Control and Prevention) estimates that approximately how many new cases of STIs occur each year in the United States?
a. 2 million
b. 12 million
c. 20 million
d. 40 million
2. It is estimated that there are more than how many total STIs among men and women in the U.S.?
a. 75 million
b. 110 million
c. 150 million
d. 210 million
3. Which of the following STIs cannot be cured?
a. chlamydia
b. syphilis
c. herpes
d. gonorrhea
4. Which statement below is false regarding STI testing?
a. There are separate tests for different STIs.
b. You must request that an STI test be performed.
c. If you do not have any symptoms, it is not necessary to get tested.
d. STI tests are relatively easy to get.
5. Antibiotic treatment can be used for which STI?
a. herpes
b. HPV
c. chlamydia
d. HIV
6. If a condom is used during intercourse, there is no risk for getting an STI.
a. True
b. False
7. About one out of _____ adults in the U.S. has genital herpes, but as many as 90% of those infected don’t know it.
a. 2
b. 4
c. 6
d. 8
8. If you are trying to protect yourself from getting genital herpes, oral sex is a safe alternative to vaginal sex.
a. True
b. False
9. Which four bodily fluids can transmit HIV to others?
a.  A diagnosis of AIDS ensues when an HIV-infected person’s CD4 cell count drops below…?
a. 700 cells/uL
b. 500 cells/uL
c. 200 cells/uL
d. 1000 cells/uL
11. The most effective way to protect yourself against STIs EACH time you have a new sexual partner (and before you first engage in any sexual activity with that person) is…?
a. to determine how many previous sexual partners he/she has been with
b. to ask if he/she has any STIs
c. to go get tested for STIs together and see each other’s results
d. to rely on the effectiveness of using a condom when you decide to have sex
12. Hepatitis can severely damage which organ?
a. kidney
b. colon
c. liver
d. spleen
13. Most sexually active men and women will get HPV at some point in their lives.
a. True
b. False
14. The HPV vaccines for females and males are most effective when all doses are…?
a. given within a 1 month time period
b. given before a person first has sexual contact
c. given at the time the infection first occurs
d. given directly after sexual contact has occurred
15. Explain why “I” statements are generally a more effective approach to communication than “you” statements, and provide at least one example that demonstrates the difference between the two:
16. Directions: For each scenario below, circle the choice that most closely resembles how you would likely respond to the situation, and then answer the question that follows:
a. Your roommate/significant other/family member rarely, if ever, cleans up anything in your dorm room/apartment/house. You…
i. don’t say anything and just clean up after them
ii. talk to them about a plan/schedule for taking turns cleaning shared spaces
iii. start placing clutter and dirty dishes in the person’s space
iv. call them a slob, yell at them, and/or threaten to kick them out
b. Your classmate is not pulling his/her weight on your assigned group project. You…
i. pick up the slack for that person without addressing the issue
ii. discuss with that person/the whole group how to best proceed/divide tasks
iii. rip the person apart/call them lazy and useless
c. Your friend says something that hurts your feelings. You…
i. say nothing and hope it doesn’t happen again
ii. tell your friend that what he/she said hurt your feelings
iii. get even with your friend by saying something hurtful too
d. Your friend borrows your watch/jewelry/piece of clothing without returning it. You…
i. let them keep it even though it is one of your favorites
ii. ask your friend to return the item
iii. get even by taking something that belongs to them
iv. accuse your friend of trying to steal from you
Question : In addition to the previous scenarios, take a moment to reflect on how you typically respond to various situations in your life. Do you have a difficult time speaking up? Or, do you tend to lash out at others very quickly (e.g. name-calling, raised voice, etc.)? If you chose “ii” for most of the scenarios, then you are probably on the “right” track to communicating effectively. Briefly (and honestly!) discuss how satisfied or dissatisfied you are with your current level of communication with others, as well as areas that could potentially use improvement. Keep in mind, communicating effectively involves respecting others as well as yourself.
Your response should be at least 75 words in length :
17. What are the 3 key elements of Sternberg’s Triangular Theory of Love (write them below)?
a.
b.
c.
18. HPV can cause cancer in both men and women.
a. True
b. False
19. People who have an STI are more likely to become infected with HIV.
a. True
b. False
- Test #2. Authored by : Trina DiGregorio, M.S.. Provided by : Monroe Community College. License : CC BY: Attribution


18.15: Fractals
1.
3.
5.
9. Four copies of the Koch curve are needed to create a curve scaled by 3.
\(D=\frac{\log (4)}{\log (3)} \approx 1.262\)
11. Eight copies of the shape are needed to make a copy scaled by 3. \(D=\frac{\log (8)}{\log (3)} \approx 1.893\)
13.
15. a) \(5-i\) b) \(5-4 i\)
17. a) \(6+12 i\) b) \(10-2 i\) c) \(14+2 i\)
19. \((2+3 i)(1-i)=5+i\). It appears that multiplying by \(1-i\) both scaled the number away from the origin, and rotated it clockwise about 45°.
21.
\(z_{1}=i z_{0}+1=i(2)+1=1+2 i\)
\(z_{2}=i z_{1}+1=i(1+2 i)+1=i-2+1=-1+i\)
\(z_{3}=i z_{2}+1=i(-1+i)+1=-i-1+1=-i\)
23.
\(z_{0}=0\)
\(z_{1}=z_{0}^{2}-0.25=0-0.25=-0.25\)
\(z_{2}=z_{1}^{2}-0.25=(-0.25)^{2}-0.25=-0.1875\)
\(z_{3}=z_{2}^{2}-0.25=(-0.1875)^{2}-0.25=-0.21484\)
\(z_{4}=z_{3}^{2}-0.25=(-0.21484)^{2}-0.25=-0.20384\)
25. attracted, to approximately \(-0.37766+0.14242 i\)
27. periodic 2-cycle 29. Escaping 31. periodic 3-cycle
33. a) Yes, periodic 3-cycle b) Yes, periodic 3-cycle c) No


7.5: Culture
In this section
- You will learn about prayer as a part of the daily routine in the Arab world.
Daily and weekly prayer in the Arab world
- Prayer is the second pillar of Islam.
- Muslims perform five main prayers : Fajr (dawn), Zuhr (Noon), Asr (late afternoon), Maghrib (sunset) and Isha (at dusk).
- Some people pray all five prayers in a the mosque المسجد = الجامع.
- On Friday, the noon prayer is called Friday prayer صلاة الجمعة which must be performed in congregation preceded by a sermon. This is the reason why Friday is not a business day in the Arab Muslim world.
- For Muslims, it is a part of the daily routine to attend Friday prayer in congregation. It is more common for men but women can also go and they usually have separate sections in the mosque.
- There are special prayers that occur during Ramdan nights (the month of fasting) and the two Eid celebrations in the Muslim world.
- Muslims follow a moon calendar for their religious observance of fasting and pilgrimage.
- Christians observe their prayers at the church الكنيسة usually on Sundays.
- Watch this video on how Friday prayer was resumed under COVID-19 conditions in Egypt.
Key Takeaways
- You can search YouTube for Friday prayer congregations.
- You can conduct a search on prayer times and how they change over the year in different countries.
- Discuss with your classmates the five pillars of Islam and provide your reflection.


11.6: Adding H5P (formative) Activities
LibreTexts content can be enhanced with interactive H5P technology ( https://h5p.org/ ). On this page we will discuss how this technology operates formatively (i.e., without a gradebook) within the LibreTexts platform. LibreText Adapt can be used to make H5P problems into summative activities. The H5P activities are stored on a dedicated Wordpress server, not on the Library servers. Embedding an activity into a LibreTexts page is a three-step process:
Step 1: Create or locate the activity
Browse the LibreStudio server ( https://studio.libretexts.org ) to see if there may already be an H5P activity that you can use. If your credentials do not allow logging into the server, consult the LibreTexts development team for access<EMAIL_ADDRESS>).
You can also upload the desired activity into the H5P LibreStudio server. There are many public domain H5P activities on the H5P project server and elsewhere that you may find useful and want to import into LibreStudio to use in a LibreText.
And in the likely event you cannot find a problem that does exactly what you have in mind, you can create your own problem in LibreStudio! See the H5P project site ( https://h5p.org/ ) for full instructions on this process, and look at example problems to get ideas.
Step 2: Copy the ID Number of the desired activity
Once you have created an H5P activity, look for the H5P ID number. Click on the activity and the following information should be displayed on the right. Note the H5P ID at the bottom of this list.
Step 3: Insert ID Number into H5P Template on page
The H5P template lets authors embed the H5P activity into a page (page > elements > template).
Once the template is added to the page, all you have to do is change the ID number of the activity here to the one you located in Step 2 (in the place of 5 in below template).
Save the page. An example activity is below:


4.1.6: Levels of Culture
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Anthropologists recognize three levels of culture: international , national , and subculture . Keep in mind that while anthropologists have classified these three general patterns, it is acknowledged that there is variation within any given culture. Even at the individual level, there may be differences from the dominant culture. While most people don’t think about their culture at the most general levels, these levels do impact us even if we’re not aware of it. One of the criticisms of the culture concept is that it generalizes and stereotypes groups of people. Indeed as you read about the levels of culture you may agree with this criticism. However, these generalizations can be used to develop a starting point in understanding a culture.
INTERNATIONAL CULTURE
International culture refers to cultural traits that extend beyond national boundaries. These cultural traits and patterns spread through diffusion, migration, colonization, and globalization (Kottak and Kozaitis 2012). Once again, an obvious example would be Canada, the United States, Australia, and Great Britain. These countries share common traits because they have a common language and ancestral cultural heritage. Other examples are evident in the adoption and use of technology and social media across continents. For example, computers and mobile devices allow people to live and operate across national boundaries enabling them to create and sustain an international culture around a common interest or purpose (i.e., Olympics, United Nations, World Cup, etc.).
Definition: international culture
Cultural traits that extend beyond national boundaries.
NATIONAL CULTURE
In contrast, national culture is the beliefs, behavior patterns, values, cultural traits, and institutions shared within a country. National culture is most easily recognizable in the form of symbols such as flags, logos, and colors as well as sound including national anthems and musical styles. Although it is a mistake to automatically assume that everyone in a large multicultural country like the United States shares a common culture, generally most people could agree on some core values on some level. Core values are "the key, basic, or central values that integrate culture and help distinguish it from others" (Kottak 2012, p. 21). Gary Althen (2003), in the essay “American Values and Assumptions”, identifies some American core values such as individualism, freedom, equality, competitiveness, privacy, achievement, and work. Japanese culture, on the other hand, stresses different core values such as belonging, harmony, group orientation (interdependence over independence), politeness, modesty, gentleness, patience, and formality (Evason 2016). Again, it is important to remember, when speaking of national culture that all members within that society may not agree with all of the core values.
Definition: national culture
The beliefs, behavior patterns, values, cultural traits, and institutions shared within a country.
Definition: core values
The key, basic, or central values that integrate culture and help distinguish it from others.
Within the national culture, there is also popular culture. The term popular culture refers to the pattern of cultural experiences and attitudes that exist in mainstream society. Popular culture events might include a parade, a baseball game, or the season finale of a television show. Rock and pop music—“pop” is short for “popular”—are part of popular culture. Popular culture is often expressed and spread via commercial media such as radio, television, movies, the music industry, publishers, and corporate-run websites.
Definition: popular culture
The pattern of cultural experiences and attitudes that exist in mainstream society.
SUBCULTURES
Subcultures , another level of culture, are smaller groups of people who share cultural traits and patterns within the same country. Subcultures have shared experiences and common cultural distinctions, but they are a part of the larger society or cultural system. Subcultures have their own set of symbols, meanings, and behavioral norms, which develop by interacting with one another. Subcultures develop their own self-culture, or idioculture, that has significant meaning to members of the group and creates social boundaries for membership and social acceptance (Griswold 2013).
Definition: subculture
Smaller groups of people who share cultural traits and patterns within the same country.
Thousands of subcultures exist within the United States. Ethnic, racial groups, and geographic regions share the language or dialect, food, and customs of their heritage. Other subcultures are united by shared experiences. Biker culture revolves around a dedication to motorcycles. Some subcultures are formed by members who possess traits or preferences that differ from the majority of a society’s population. The body modification community embraces aesthetical additions to the human body, such as tattoos, piercings, and certain forms of plastic surgery. In the United States, adolescents often form subcultures to develop a shared youth identity. Alcoholics Anonymous offers support to those suffering from alcoholism. But even as members of a subculture band together, they still identify with and participate in the larger society.
BIBLIOGRAPHY
Althen, G. American ways: A guide for foreigners in the United States. Yarmouth, ME: Intercultural Press, 2003.
Evason, Nina. Japanese Culture, 2016. Retrieved from https://culturalatlas.sbs.com.au/jap...ure-references
Kottak, Conrad P. Mirror for Humanity: A Concise Introduction to Cultural Anthropology . New York, N.Y: McGraw-Hill, 2012.
Kottak, Conrad P. and Kathryn A. Kozaitis. On Being Different: Diversity and Multiculturalism in the North American Mainstream . 4th ed. New York: McGraw-Hill Companies, Inc., 2012.
Derived From
"What is Culture?" by OpenStax under CC-BY 4.0. Access for free at https://openstax.org/books/introduction-sociology-2e/pages/1-introduction-to-sociology .
"Chapter 8: Roots of American National Culture" in Speaking of Culture by Nolan Weil under CC BY-NC 4.0 .
"Levels of Culture" in Beyond Race - Cultural Influences on Human Social Life by Vera Kennedy under CC BY-NC-SA 4.0 .


5.7: Linear Recurrence Relations
A couple interesting illustrations
Before defining terminology we will experiment with the following problems. Except where directed you should write code to perform the calculations so we concentrate on the results rather than the calculations.
Checkpoint \(\PageIndex{1}\)
The following is called a continued radical.
- Evaluate using technology \(a_1=\sqrt{2}\) to produce an approximation.
- Take that result and calculate \(a_2=\sqrt{2+a_1}.\)
- Take that result and calculate \(a_3=\sqrt{2+a_2}.\)
- Repeat this seven (7) more times.
- Is there a pattern to the numbers? What do you think will happen if you continue this process?
Checkpoint \(\PageIndex{2}\)
The following is called a continued fraction.
- Let \(a_1=1\text{.}\)
- By hand calculate \(a_2= \displaystyle \frac{1}{1+a_1}.\)
- By hand calculate \(a_3= \displaystyle \frac{1}{1+a_2}.\)
- Repeat this seven (7) more times.
- Is there a pattern to the numbers? What do you think will happen if you continue this process?
- Ask the instructor to provide a number to compare.
Sometimes directly calculating the value of a function can be difficult. In the examples above each value of the function is based on the previous. The first example was \(f(n)=\sqrt{2+f(n-1)}.\) The second example was \(g(n)=\frac{1}{1+g(n-1)}.\) This is known as recursive definition.
Recurrence Relations
A famous example comes from Fibonacci's Liber Abaci.
The first month there was one pair of rabbits. The second month there was still one pair of rabbits. Starting the third month and continuing every month thereafter, the number of pairs of rabbits is the sum of the previous two months. Thus the third month there are \(1+1=2\) pairs of rabbits. The fourth month there are \(1+2=3\) pairs of rabbits. This produces the following sequence.
\[ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \ldots \nonumber \]
The Fibonacci sequence in standard notation is
\[ a_1=1, \; a_2=1, \; a_n=a_{n-1}+a_{n-2}. \nonumber \]
It is possible to count using recursive functions as well. Consider the number of words from the alphabet \(\{\)a,b,c,d,e\(\}\) with no two, consecutive a's. For words of length one there are no restrictions, so there are 5. For words of length two all combinations except ‘aa’ are acceptable. Thus there are \(25-1=24\) words of length two. For words of length three all combinations except ‘aaa’, ‘aaX,’ and ‘Xaa’ are allowable. This removes \(1+4+4=9\) words leaving \(5^3-9=116\) acceptable words.
For longer words of length \(n\text{,}\) note that removing the last letter produces an acceptable word of length \(n-1.\) Appending b,c,d,e to the end of these length \(n-1\) words produces another acceptable word. This constructs \(4a_{n-1}\) acceptable words of length \(n.\)
The only words to which an ‘a’ can be appended are those that ended in b,c,d, or e. These could have been appended to any word of length \(n-2.\) Thus appending ‘a’ constructs an additional \(4a_{n-2}.\)
Thus the total number of acceptable words of length \(n\) is
\[ a_n=4a_{n-1}+4a_{n-2}. \nonumber \]
While it is possible to produce a function that provides the \(n\)th term, this is generally not easy. For linear recurrence relations the technique demonstrated here will always work.
Find a closed form for the recurrence relation
Practice
Checkpoint \(\PageIndex{5}\)
Guido is superstitious and will not take food from two, adjacent bins at a buffet. If the buffet is a line having \(n\) bins each containing distinct food, how many different combinations of food can he choose? Note he can eat as many items as he wants.
Checkpoint \(\PageIndex{6}\)
Guido walks up stairs taking one or two steps at a time. Find a recursive formula for the number of ways he could end up at step \(n.\) Note he starts at step 0 (not on the stairs).


6.7A: Microbial Growth Cycle
Increases in cell size are tightly linked in unicellular organisms and under optimal conditions bacteria can grow and divide rapidly.
LEARNING OBJECTIVES
Duplicate the requirements of microbial growth cycles
Key Takeaways
Microbial Growth Cycle
All microbial metabolisms can be arranged according to three principles: 1) How the organism obtains carbon for synthesizing cell mass. 2) How the organism obtains reducing equivalents used either in energy conservation or in biosynthetic reactions. 3) How the organism obtains energy for living and growing (for more detail on this topic see atom on Growth Terminology). Unlike in multicellular organisms, increases in cell size (cell growth and reproduction by cell division) are tightly linked in unicellular organisms. Bacteria grow to a fixed size and then reproduce through binary fission which is a form of asexual reproduction. Under optimal conditions, bacteria can grow and divide extremely rapidly. These optimal conditions are discussed below.
Oxygen Requirements
Different kinds of bacteria need different amounts of oxygen to survive, which determines which bacteria can infect which parts of the body. They are not able infect the skin because oxygen is present, and they can only grow in the presence of oxygen. Conversely, obligate anaerobes are killed by oxygen and carry out fermentation. Tetanus is an obligate anaerobe so it will infect areas where oxygen in limited. Aerotolerant anaerobes breath anaerobically (without oxygen), but they are able to survive when oxygen is present.
Nutrient Requirements
For microbial growth to process, microorganisms require certain nutrients including carbon, nitrogen, phosphorus, sulfur, and metal ions.
Temperature Requirements
Various types of bacteria thrive at different temperatures. Microorganisms that grow best at moderate temperatures are called mesophiles. Those surviving at high temperatures are thermophiles and microorganisms surviving at very low temperatures are called psychrophiles.
Key Points
- Bacteria grow to a fixed size and then reproduce through binary fission which a form of asexual reproduction. Under optimal conditions, bacteria can grow and divide extremely rapidly.
- Different kinds of bacteria need different amounts of oxygen to survive.
- For microbial growth to process, microorganisms require certain nutrients including carbon, nitrogen, phosphorus, sulfur, and metal ions.
- Various types of bacteria thrive at different temperatures.
Key Terms
- binary fission : The process whereby a cell divides asexually to produce two daughter cells.
- Anaerobe : An anaerobic organism; one that does not require oxygen to sustain its metabolic processes.
- Aerobe : Any organism (but especially a bacterium) that can tolerate the presence of oxygen or that needs oxygen to survive.


10.6: Angles after inversion
In the inversive plane, the inverse of an arc is an arc.
- Proof
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Consider four distinct points \(A, B, C\), and \(D\); let \(A', B', C'\), and \(D'\) be their inverses. We need to show that \(D\) lies on the arc \(ABC\) if and only if \(D'\) lies on the arc \(A'B'C'\). According to Proposition 9.5.1 , the latter is equivalent to the following:
\(\measuredangle ADC = \measuredangle ABC \Leftrightarrow \measuredangle A'D'C' = \measuredangle A'B'C'.\)
The latter follows from Theorem 10.2.1b .
The following theorem states that the angle between arcs changes only its sign after the inversion.
Let \(AB_1C_1\), \(AB_2C_2\) be two arcs in the inversive plane, and the arcs \(A'B_1'C_1'\), \(A'B_2'C_2'\) be their inverses. Let \([AX_1)\) and \([AX_2)\) be the half-lines tangent to \(AB_1C_1\) and \(AB_2C_2\) at \(A\), and \([A'Y_1)\) and \([A'Y_2)\) be the half-lines tangent to \(A'B_1'C_1'\) and \(A'B_2'C_2'\) at \(A'\). Then
\(\measuredangle X_1AX_2 \equiv -\measuredangle Y_1A'Y_2\).
- Proof
-
Applying to Proposition 9.6.1 ,
\(\begin{array} {rcl} {\measuredangle X_1AX_2} & \equiv & {\measuredangle X_1AC_1 + \measuredangle C_1AC_2 + \measuredangle C_2AX_2 \equiv} \\ {} & \equiv & {(\pi - \measuredangle C_1B_1A) + \measuredangle C_1AC_2 + (\pi - \measuredangle AB_2C_2) \equiv} \\ {} & \equiv & {-(\measuredangle C_1B_1A + \measuredangle AB_2C_2 + \measuredangle C_2AC_1) \equiv} \\ {} & \equiv & {-(\measuredangle C_1B_1A + \measuredangle AB_2C_1) -(\measuredangle C_1B_2C_2 + \measuredangle C_2AC_1).} \end{array}\)
The same way we get that
\(\measuredangle Y_1A'Y_2 \equiv -(\measuredangle C_1'B_1'A' + \measuredangle A'B_2'C_1') - (\measuredangle C_1'B_2'C_2' + \measuredangle C_2'A'C_1').\)
By Theorem 10.2.1b ,
\(\begin{array} {rcl} {\measuredangle C_1B_1A + \measuredangle AB_2C_1} & \equiv & {-(\measuredangle C_1'B_1'A' + \measuredangle A'B_2'C_1'),} \\ {\measuredangle C_1B_2C_2 + \measuredangle C_2AC_1} & \equiv & {-(\measuredangle C_1'B_2'C_2' + \measuredangle C_2'A'C_1')} \end{array}\).
and hence the result.
The angle between arcs can be defined as the angle between its tangent half-lines at the common endpoint. Therefore under inversion, the angles between arcs are preserved up to sign.
From Exercise 5.7.4 , it follows that the angle between arcs with common endpoint A is the limit of \(\measuredangle P_1AP_2\) where \(P_1\) and \(P_2\) are points approaching \(A\) along the corresponding arcs. This observation can be used to define the angle between a pair of curves emerging from one point. It turns out that under inversion, angles between curves are also preserved up to sign.
Let \(P\) be the inverse of point \(Q\) in a circle \(\Gamma\). Assume that \(P'\), \(Q'\), and \(\Gamma'\) are the inverses of \(P, Q\), and \(\Gamma\) in another circle \(\Omega\). Then \(P'\) is the inverse of \(Q'\) in \(\Gamma'\).
- Proof
-
If \(P = Q\), then \(P'=Q' \in \Gamma'\). Therefore, \(P'\) is the inverse of \(Q'\) in \(\Gamma'\).
It remains to consider the case \(P \ne Q\). Let \(\Delta_2\) and \(\Delta_2\) be two distinct circles that intersect at \(P\) and \(Q\). According to Corollary 10.5.2 , \(\Delta_1 \perp \Gamma\) and \(\Delta_2 \perp \Gamma\).
Let \(\Delta_1'\) and \(\Delta_2'\) denote the inverses of \(\Delta_1\) and \(\Delta_2\) in \(Omega\). Clearly, \(\Delta_1'\) meets \(\Delta_2'\) at \(P'\) and \(Q'\).
By Theorem \(\PageIndex{1}\), \(\Delta_1' \perp \Gamma'\) and \(\Delta_2' \perp \Gamma'\). By Corollary 10.5.1 , \(P'\) is the inverse of \(Q'\) in \(\Gamma'\).


2.1.8: Samuel Taylor Coleridge (1772-1834)
Selected Poems British Romanticism Coleridge is the first critic of the Romantic literary movement in England. His father was a school-master and a vicar, and Coleridge grew up in a household full of books, which he read voraciously. His father's fascination with astronomy created in the young man a vision of the vastness of the world, a vision that would later inform his own work. His character and literary style were formed when he was an adolescent student at Christ's Hospital school in London; there, he immersed himself in the classics and English poets such as Shakespeare and Milton. From the English poets, he drew the significance of sound and imagery. Coleridge saw poetry as a means of enjoyment and science as a means to scientific truth; according to Coleridge, however, the best poetry uses metaphor and imagery to express truth. He is responsible for bringing the ideas of Immanuel Kant and human understanding to the literary circles of England. He also introduces an innovative supernatural context in much of his poetry, which requires that his audience release their grip on reason. He collaborated with William Wordsworth in Lyrical Ballads , a foundational book of verse for the Romantic period. In his Biographia Literaria (1817), Coleridge introduces the concept of a "willing suspension of disbelief": In this idea originated the plan of the Lyrical Ballads ; in which it was agreed, that my endeavours should be directed to persons and characters supernatural, or at least romantic, yet so as to transfer from our inward nature a human interest and a semblance of truth sufficient to procure for these shadows of imagination that willing suspension of disbelief for the moment, which constitutes poetic faith. The suspension of disbelief or poetic faith is necessary for the enjoyment of two of Coleridge's famous poems, "The Rime of the Ancient Mariner" (1798) and "Kubla Khan" (1817). The Rime of the Ancient Mariner Coleridge's poem is written in tight rhyme and meter, yet his subject is otherworldly. In the poem, he introduces the psychological concept of mesmerizing with the Mariner's "glittering eye." The concept of the eye that can mesmerize, or hold the mind of its object captive, is one he borrowed from the German physician Frank Mesmer, who proposed that all living creatures have "animal magnetism." Coleridge's verse is indeed haunting and mesmerizing throughout the poem, holding both the Wedding Guest and the poem's audience captive to the tragic story of the sole survivor of a fateful voyage. The poem, full of metaphor and symbolism, introduces the Romantic era to the disturbing sensations that disturb the human imagination. Kubla Kahn; or, A Vision in a Dream: A Fragment Coleridge's "Kubla Kahn" appeared in his imagination in the waking moments of an opium-induced sleep; the poem represents the poet's fascination with the human imagination and the vastness of nature, the unconscious and fantasy. This poem, though not Coleridge's most prized work, has stimulated scholarly conversation for its potent and vivid imagery and language, and foreshadows the work of poets like William Blake and the American poet Edgar Allan Poe. Consider while reading:
- Discuss the significance of the Wedding Guest being kept from a celebration.
- What is the significance of the albatross around the neck of the Mariner?
- How does Coleridge depict the voyage as the life of every human being?
- Choose and discuss an image from "Kubla Khan."
Written by Karen Dodson


3.13: Roxanna Panufnik
In this chapter you will
- learn about the composer’s diverse religious background
- consider how Jewish and Christian texts outside the Bible continue the exploration of biblical stories and themes in a manner similar to what composers do
If you have read this book through in order, you have already heard examples of the music of Roxanna Panufnik. One is linked to in chapter 17 on the Magnificat and Nunc Dimittis, the other in chapter 9 on the story of Isaac. Even a significant amount of her instrumental work draws on and explores biblical stories. In her setting of Psalm 136, Love Endureth , she draws on Jewish chant. As the daughter of a Jewish mother and a Catholic father (her father was also a well-known composer, Andrzej Panufnik), it is no surprise to find she explores and addresses religious texts and themes in and through her music and has clearly articulated thoughts on subjects such as religious diversity and commonality. In an interview with the Jewish Chronicle , she said of her violin concerto Abraham , “I was five months pregnant with my first child and I suddenly began to panic about what kind of a world I was bringing a baby into. There’s so much common ground between the faiths. I knew I couldn’t change the world by trying to express my feelings in music, but it started me on a quest to build musical bridges between faiths. Our various faiths may take different paths, but they all go in the same direction, towards the same one God.” [1] She says more about this in her interview with Classic FM . Listen closely to the ways that she engages not only with biblical text but with some of the diverse forms of musical expression you have read about in this book.
In the interview with the Jewish Chronicle already mentioned, Panufnik also shares that she began exploring her Jewish roots while a student at the Royal Academy of Music. She says there, “My first Jewish work was a setting of the Shemah, and it was performed at my father’s funeral.” [2] The Shema is Deuteronomy 6:4 , one of the most important texts that is felt to capture and sum up what is central to Judaism: “Hear O Israel, the Lord our God, the Lord is one.” It features prominently in Jewish prayer .
Panufnik has also explored biblical texts in ways that transcend any one religious tradition, as you will have heard in the interview. She has also explored beyond the canon of Scripture into extracanonical texts, such as the Infancy Gospel of Thomas . That text is a collection of stories that imagines what Jesus might have been like as a child in between the stories found in the New Testament Gospels. It provided the basis for Jessica Duchen’s libretto for Panufnik’s oratorio “Let Me In.” As a contemporary composer with this kind of background and musical output, she makes for a natural place to draw our exploration of music and the Bible to a close.
- “Why Her Music Is a Religious Experience, Three Times Over,” Jewish Chronicle , April 19, 2012, https://www.thejc.com/culture/features/why-her-music-is-a-religious-experience-three-times-over-1.32950 . ↵
- See also her interview with Stephen Darlington in Roxanna Panufnik, “Beyond a Mass for Westminster,” in Composing Music for Worship , ed. Stephen Darlington and Alan Kreider (Norwich: Canterbury Press, 2003), 76–85. ↵


5.4: Bone Formation and Development
By the end of this section, you will be able to:
- Explain the function of cartilage
- List the steps of intramembranous ossification
- List the steps of endochondral ossification
- Explain the growth activity at the epiphyseal plate
- Compare and contrast the processes of modeling and remodeling
In the early stages of embryonic development, the embryo’s skeleton consists of fibrous membranes and hyaline cartilage. By the sixth or seventh week of embryonic life, the actual process of bone development, ossification (osteogenesis), begins. There are two osteogenic pathways—intramembranous ossification and endochondral ossification—but bone is the same regardless of the pathway that produces it.
Cartilage Templates
Bone is a replacement tissue; that is, it uses a model tissue on which to lay down its mineral matrix. For skeletal development, the most common template is cartilage. During fetal development, a framework is laid down that determines where bones will form. This framework is a flexible, semi-solid matrix produced by chondroblasts and consists of hyaluronic acid, chondroitin sulfate, collagen fibers, and water. As the matrix surrounds and isolates chondroblasts, they are called chondrocytes, which are surrounded by a lacuna. Unlike most connective tissues, cartilage is avascular, meaning that it has no blood vessels supplying nutrients and removing metabolic wastes. These functions are carried on by diffusion through the matrix. This is why damaged cartilage does not repair itself as readily as most tissues do.
Throughout fetal development and into childhood growth and development, bone forms on the cartilaginous matrix. By the time a fetus is born, most of the cartilage has been replaced with bone. Some additional cartilage will be replaced throughout childhood, and some cartilage remains in the adult skeleton.
Intramembranous Ossification
During intramembranous ossification , compact and spongy bone develops directly from sheets of mesenchymal (undifferentiated) connective tissue. The flat bones of the face, most of the cranial bones, and the clavicles (collarbones) are formed via intramembranous ossification.
The process begins when mesenchymal cells in the embryonic skeleton gather together and begin to differentiate into specialized cells (Figure \(\PageIndex{1.a}\)). Some of these cells differentiate and form the tissues of capillaries, while others will become osteogenic cells and then osteoblasts. Although they will ultimately be spread out by the formation of bone tissue, early osteoblasts appear in a cluster called an ossification center .
The osteoblasts secrete osteoid , uncalcified matrix, which calcifies (hardens) within a few days as mineral salts are deposited on it, thereby entrapping the osteoblasts within. Once entrapped, the osteoblasts become osteocytes (Figure \(\PageIndex{1.b}\)). As osteoblasts transform into osteocytes, osteogenic cells in the surrounding connective tissue differentiate into new osteoblasts.
Osteoid (unmineralized bone matrix) secreted around the capillaries results in a trabecular matrix. At the surfaces of the forming osseus tissue mesenchymal cells differentiate into fibroblasts which surround the forming structure with fibrous periosteum while osteoblasts, osteoclasts and osteogenic cells form the cellular (osteogenic) layer. (Figure \(\PageIndex{1.c}\)). The periosteum then creates a protective layer of compact bone superficial to the trabecular bone. The trabecular bone crowds nearby blood vessels, which eventually condense into red marrow (Figure \(\PageIndex{1.d}\)).
Intramembranous ossification begins in utero during fetal development and continues on into adolescence. At birth, the skull and clavicles are not fully ossified nor are the sutures of the skull closed. This allows the skull and shoulders to deform during passage through the birth canal. The last bones to ossify via intramembranous ossification are the flat bones of the face, which reach their adult size at the end of the adolescent growth spurt.
Endochondral Ossification
In endochondral ossification , bone develops by replacing hyaline cartilage. Cartilage does not become bone. Instead, cartilage serves as a template to be completely replaced by new bone. Endochondral ossification takes much longer than intramembranous ossification. Bones at the base of the skull and long bones are examples of bones that form via endochondral ossification.
In a long bone, for example, at about 6 to 8 weeks after conception, some of the mesenchymal cells differentiate into chondrocytes (cartilage cells) that form the cartilaginous skeletal precursor of the bones (Figure \(\PageIndex{2.a}\)). Soon after, cellular differentiation produces cells that form the perichondrium , a membrane that covers the cartilage (Figure \(\PageIndex{2.b}\)).
As more matrix is produced, the chondrocytes in the center of the cartilaginous model grow in size. As the matrix calcifies, nutrients can no longer reach the chondrocytes. This results in their death and the disintegration of the surrounding cartilage. Blood vessels invade the resulting spaces, not only enlarging the cavities but also carrying osteogenic cells with them, many of which will become osteoblasts. These enlarging spaces eventually combine to become the medullary cavity.
This invasion of blood vessels initiates the transformation of the perichondrium into the bone-producing periosteum. Here, the osteoblasts form a periosteal collar of compact bone around the cartilage of the diaphysis. By the second or third month of fetal life, bone cell development and ossification ramps up and creates the primary ossification center , a region deep in the periosteal collar where ossification begins (Figure \(\PageIndex{2.c}\)).
While these deep changes are occurring, chondrocytes and cartilage continue to grow at the ends of the bone (the future epiphyses), which increases the bone’s length at the same time bone is replacing cartilage in the diaphyses. By the time the fetal skeleton is fully formed, cartilage only remains at the joint surface as articular cartilage and between the diaphysis and epiphysis as the epiphyseal plate, the latter of which is responsible for the longitudinal growth of bones. After birth, this same sequence of events (matrix mineralization, death of chondrocytes, invasion of blood vessels from the periosteum, and seeding with osteogenic cells that become osteoblasts) occurs in the epiphyseal regions, and each of these centers of activity is referred to as a secondary ossification center (Figure \(\PageIndex{2.e}\)).
How Bones Grow in Length
The epiphyseal plate is the area of growth in a long bone. It is a layer of hyaline cartilage where ossification occurs in immature bones. On the epiphyseal side of the epiphyseal plate, cartilage is formed. On the diaphyseal side, cartilage is ossified, and the diaphysis grows in length. The epiphyseal plate is composed of four zones of cells and activity (Figure \(\PageIndex{3}\)). The reserve zone is the region closest to the epiphyseal end of the plate and contains small chondrocytes within the matrix. These chondrocytes do not participate in bone growth but secure the epiphyseal plate to the osseous tissue of the epiphysis.
The proliferative zone is the next layer toward the diaphysis and contains stacks of slightly larger chondrocytes. It makes new chondrocytes (via mitosis) to replace those that die at the diaphyseal end of the plate. Chondrocytes in the next layer, the zone of maturation and hypertrophy , are older and larger than those in the proliferative zone. The more mature cells are situated closer to the diaphyseal end of the plate. The longitudinal growth of bone is a result of cellular division in the proliferative zone and the maturation of cells in the zone of maturation and hypertrophy.
Most of the chondrocytes in the zone of calcified matrix , the zone closest to the diaphysis, are dead because the matrix around them has calcified. Capillaries and osteoblasts from the diaphysis penetrate this zone, and the osteoblasts secrete bone tissue on the remaining calcified cartilage. Thus, the zone of calcified matrix connects the epiphyseal plate to the diaphysis. A bone grows in length when osseous tissue is added to the diaphysis.
Bones continue to grow in length until early adulthood. The rate of growth is controlled by hormones, which will be discussed later. When the chondrocytes in the epiphyseal plate cease their proliferation and bone replaces the cartilage, longitudinal growth stops. All that remains of the epiphyseal plate is the epiphyseal line (Figure \(\PageIndex{4}\)).
How Bones Grow in Diameter
While bones are increasing in length, they are also increasing in diameter; growth in diameter can continue even after longitudinal growth ceases. This is called appositional growth (Figure \(\PageIndex{5}\)). Osteoclasts resorb old bone that lines the medullary cavity, while osteoblasts produce new bone tissue beneath the periosteum. The erosion of old bone along the medullary cavity and the deposition of new bone beneath the periosteum not only increase the diameter of the diaphysis but also increase the diameter of the medullary cavity. This process is called modeling .
Bone Remodeling
The process in which matrix is resorbed on one surface of a bone and deposited on another is known as bone modeling. Modeling primarily takes place during a bone’s growth. However, in adult life, bone undergoes remodeling , in which resorption of old or damaged bone takes place on the same surface where osteoblasts lay new bone to replace that which is resorbed. Injury, exercise, and other activities lead to remodeling. Those influences are discussed later in the chapter, but even without injury or exercise, about 5 to 10 percent of the skeleton is remodeled annually by destroying old bone and renewing it with fresh bone.
DISEASES OF THE...
Skeletal System
Osteogenesis imperfecta (OI) is a genetic disease in which bones do not form properly and therefore are fragile and break easily. It is also called brittle bone disease . The disease is present from birth and affects a person throughout life.
The genetic mutation that causes OI affects the body’s production of collagen, one of the critical components of bone matrix. The severity of the disease can range from mild to severe. Those with the most severe forms of the disease sustain many more fractures than those with a mild form. Frequent and multiple fractures typically lead to bone deformities and short stature. Bowing of the long bones and curvature of the spine are also common in people afflicted with OI. Curvature of the spine makes breathing difficult because the lungs are compressed.
Because collagen is such an important structural protein in many parts of the body, people with OI may also experience fragile skin, weak muscles, loose joints, easy bruising, frequent nosebleeds, brittle teeth, blue sclera, and hearing loss. There is no known cure for OI. Treatment focuses on helping the person retain as much independence as possible while minimizing fractures and maximizing mobility. Toward that end, safe exercises, like swimming, in which the body is less likely to experience collisions or compressive forces, are recommended. Braces to support legs, ankles, knees, and wrists are used as needed. Canes, walkers, or wheelchairs can also help compensate for weaknesses.
When bones do break, casts, splints, or wraps are used. In some cases, metal rods may be surgically implanted into the long bones of the arms and legs. Research is currently being conducted on using bisphosphonates to treat OI. Smoking and being overweight are especially risky in people with OI, since smoking is known to weaken bones, and extra body weight puts additional stress on the bones.
Concept Review
All bone formation is a replacement process. Embryos develop a cartilaginous skeleton and various membranes. During development, these are replaced by bone during the ossification process. In intramembranous ossification, bone develops directly from sheets of mesenchymal connective tissue. In endochondral ossification, bone develops by replacing hyaline cartilage. Activity in the epiphyseal plate enables bones to grow in length. Modeling allows bones to grow in diameter. Remodeling occurs as bone is resorbed and replaced by new bone. Osteogenesis imperfecta is a genetic disease in which collagen production is altered, resulting in fragile, brittle bones.
Review Questions
Q. Why is cartilage slow to heal?
A. because it eventually develops into bone
B. because it is semi-solid and flexible
C. because it does not have a blood supply
D. because endochondral ossification replaces all cartilage with bone
- Answer
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Answer: C
Q. Why are osteocytes spread out in bone tissue?
A. They develop from mesenchymal cells.
B. They are surrounded by osteoblasts.
C. They travel through the capillaries.
D. Formation of osteoid spreads out the osteoblasts that formed the ossification centers.
- Answer
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Answer: D
Q. In endochondral ossification, what happens to the chondrocytes?
A. They develop into osteocytes.
B.
C. They grow and form the periosteum.
D. They group together to form the primary ossification center.
- Answer
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Answer: B
Q. Which of the following bones is (are) formed by intramembranous ossification?
A. the metatarsals
B. the femur
C. the ribs
D. the flat bones of the cranium
- Answer
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Answer: D
Q. Bones grow in length due to activity in the ________.
A. epiphyseal plate
B. perichondrium
C. periosteum
D. medullary cavity
- Answer
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Answer: A
Q. Bones grow in diameter due to bone formation ________.
A. in the medullary cavity
B. beneath the periosteum
C. in the epiphyseal plate
D. within the metaphysis
- Answer
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Answer: B
Q. Which of the following represents the correct sequence of zones in the epiphyseal plate?
A. proliferation, reserved, maturation, calcification
B. maturation, proliferation, reserved, calcification
C. calcification, maturation, proliferation, reserved
D. calcification, reserved, proliferation, maturation
- Answer
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Answer: C
Critical Thinking Questions
Q. In what ways do intramembranous and endochondral ossification differ?
- Answer
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A. In intramembranous ossification, bone develops directly from sheets of mesenchymal connective tissue, but in endochondral ossification, bone develops by replacing hyaline cartilage. Intramembranous ossification is complete by the end of the adolescent growth spurt, while endochondral ossification lasts into young adulthood. The flat bones of the face, most of the cranial bones, and a good deal of the clavicles (collarbones) are formed via intramembranous ossification, while bones at the base of the skull and the long bones form via endochondral ossification.
Q. Considering how a long bone develops, what are the similarities and differences between a primary and a secondary ossification center?
- Answer
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A. A single primary ossification center is present, during endochondral ossification, deep in the periosteal collar. Like the primary ossification center, secondary ossification centers are present during endochondral ossification, but they form later, and there are two of them, one in each epiphysis.
Glossary
- endochondral ossification
- process in which bone forms by replacing hyaline cartilage
- epiphyseal line
- completely ossified remnant of the epiphyseal plate
- intramembranous ossification
- process by which bone forms directly from mesenchymal tissue
- modeling
- process, during bone growth, by which bone is resorbed on one surface of a bone and deposited on another
- ossification
- (also, osteogenesis) bone formation
- ossification center
- cluster of osteoblasts found in the early stages of intramembranous ossification
- osteoid
- uncalcified bone matrix secreted by osteoblasts
- perichondrium
- membrane that covers cartilage
- primary ossification center
- region, deep in the periosteal collar, where bone development starts during endochondral ossification
- proliferative zone
- region of the epiphyseal plate that makes new chondrocytes to replace those that die at the diaphyseal end of the plate and contributes to longitudinal growth of the epiphyseal plate
- remodeling
- process by which osteoclasts resorb old or damaged bone at the same time as and on the same surface where osteoblasts form new bone to replace that which is resorbed
- reserve zone
- region of the epiphyseal plate that anchors the plate to the osseous tissue of the epiphysis
- secondary ossification center
- region of bone development in the epiphyses
- zone of calcified matrix
- region of the epiphyseal plate closest to the diaphyseal end; functions to connect the epiphyseal plate to the diaphysis
- zone of maturation and hypertrophy
- region of the epiphyseal plate where chondrocytes from the proliferative zone grow and mature and contribute to the longitudinal growth of the epiphyseal plate
Contributors and Attributions
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OpenStax Anatomy & Physiology (CC BY 4.0). Access for free at https://openstax.org/books/anatomy-and-physiology


3.3: Examples of Integumentary Terms Easily Defined By Their Word Components
Here are examples of integumentary medical terms that can be easily defined by breaking them into word components.
Dermatologist
- Break down the medical term into word components: Dermat / o / logist
- Label the word parts: Dermat = WR; o = CV; logist = S
- Define the word components: Dermat = skin; logist = specialist who studies and treats disorders
- Create a final definition of the medical term: A specialist who studies and treats disorders of the skin
Epidermal
- Break down the medical term into word components: Epi / derm / al
- Label the word parts: Epi = P; derm = WR; al = S
- Define the word components: Epi = upon/on; derm = skin; al = pertaining to
- Create a final definition of the medical term: Pertaining to upon/on the skin
Hypodermic
- Break down the medical term into word components: Hypo / derm / ic
- Label the word parts: Hypo = P; derm = WR; ic = S
- Define the word components: Hypo = below; derm = skin; ic = pertaining to
- Create a final definition of the medical term: Pertaining to below the skin
Onychomycosis
- Break down the medical term into word components: Onych / o / myc / osis
- Label the word parts: Oncyh = WR; o = CV; myc = WR; osis = S
- Define the word components: Onych = nail; myc = fungus; osis = abnormal condition
- Create a final definition of the medical term: Abnorm al condition of a fungus in the nails
Interactive Learning Activity:
Practice pronouncing and defining medical terms related to the integumentary system by breaking them into word components.
Query \(\PageIndex{1}\)
You can also print this as a Chapter 3 Student Companion Worksheet and check your answers with this Answer Key .


16.7: Eukaryotic Gene Regulation - Transcriptional Enhancers and Repressors
- Explain how enhancers and repressors regulate gene expression
Enhancers and Transcription
In some eukaryotic genes, there are regions that help increase or enhance transcription. These regions, called enhancers, are not necessarily close to the genes they enhance. They can be located upstream of a gene, within the coding region of the gene, downstream of a gene, or may be thousands of nucleotides away.
Enhancer regions are binding sequences, or sites, for transcription factors. When a DNA-bending protein binds to an enhancer, the shape of the DNA changes. This shape change allows the interaction between the activators bound to the enhancers and the transcription factors bound to the promoter region and the RNA polymerase to occur. Whereas DNA is generally depicted as a straight line in two dimensions, it is actually a three-dimensional object. Therefore, a nucleotide sequence thousands of nucleotides away can fold over and interact with a specific promoter.
Turning Genes Off: Transcriptional Repressors
Like prokaryotic cells, eukaryotic cells also have mechanisms to prevent transcription. Transcriptional repressors can bind to promoter or enhancer regions and block transcription. Like the transcriptional activators, repressors respond to external stimuli to prevent the binding of activating transcription factors.
A corepressor is a protein that decreases gene expression by binding to a transcription factor that contains a DNA-binding domain. The corepressor is unable to bind DNA by itself. The corepressor can repress transcriptional initiation by recruiting histone deacetylase, which catalyzes the removal of acetyl groups from lysine residues. This increases the positive charge on histones, which strengthens the interaction between the histones and DNA, making the DNA less accessible to the process of transcription.
Key Points
- Enhancers can be located upstream of a gene, within the coding region of the gene, downstream of a gene, or thousands of nucleotides away.
- When a DNA -bending protein binds to the enhancer, the shape of the DNA changes, which allows interactions between the activators and transcription factors to occur.
- Repressors respond to external stimuli to prevent the binding of activating transcription factors.
- Corepressors can repress transcriptional initiation by recruiting histone deacetylase.
- Histone deactylation increases the positive charge on histones, which strengthens the interaction between the histones and DNA, making the DNA less accessible to transcription.
Key Terms
- enhancer : a short region of DNA that can increase transcription of genes
- repressor : any protein that binds to DNA and thus regulates the expression of genes by decreasing the rate of transcription
- activator : any chemical or agent which regulates one or more genes by increasing the rate of transcription


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1.1: Functions and Function Notation
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A function is a rule for a relationship between an input, or independent, quantity and an output, or dependent, quantity in which each input value uniquely determines one output value. We say "the output is a function of the input."
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1.2: Domain and Range
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One of our main goals in mathematics is to model the real world with mathematical functions. In doing so, it is important to keep in mind the limitations of those models we create.
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1.3: Rates of Change and Behavior of Graphs
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Since functions represent how an output quantity varies with an input quantity, it is natural to ask about the rate at which the values of the function are changing.
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1.4: Composition of Functions
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When the output of one function is used as the input of another, we call the entire operation a composition of functions.
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1.5: Transformation of Functions
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There are systematic ways to shift, stretch, compress, flip and combine functions to help them become better models for the problems we are trying to solve. We can transform what we already know into what we need, hence the name, “Transformation of functions.” When we have a story problem, formula, graph, or table, we can then transform that function in a variety of ways to form new functions.
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1.6: Inverse Functions
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Thumbnail: (CC BY; Openstax)


8.1: Simple Harmonic Motion
Equation of Motion for an Elastic Force
We have discussed the idea of a restoring force a few times already. If such a force counteracts displacements in both directions of one-dimensional motion, then it can cause the object to move back-and forth across the equilibrium point: An object subject to a restoring force is displaced from its equilibrium point and released. It accelerates toward the equilibrium point thanks to the restoring force. Upon arrival at the equilibrium point, it doesn't stop, because the restoring force is zero there. As it continues past, the restoring force acts to slow down and eventually stop the object, whereupon the object accelerates back toward the equilibrium point and the motion repeats in the opposite direction. This is called oscillatory motion , and it results from all two-way one-dimensional restoring forces.
The most common sort of restoring force we study is the elastic force. Indeed, other restoring forces occurring in nature (such as those between particles exhibiting a Lennard-Jones potential energy, as discussed at the end of Section 3.7 ) are often modeled as masses on springs. The oscillatory motion induced by the elastic restoring force is quite special, as we will see, and is called simple harmonic motion . We seek here the equation that relates the position of the mass as a function of time (with the equilibrium point being the origin), usually referred to as the equation of motion for this force.
Start (naturally) with Newton's second law, where the net force is simply that of a spring (Hooke's law). As we are working in one dimension, we once again have the luxury of treating our vector directions as simply (+) or (–):
\[ F_{net} = -kx = ma \;\;\; \Rightarrow \;\;\; \dfrac{d^2x}{dt^2}+\dfrac{k}{m}x = 0 \]
We seek to determine the function \(x\left(t\right)\) that satisfies this differential equation. This is actually simpler than it might at first appear, if thought about in the following way: First, let's imagine that the ratio \(\frac{k}{m}\) is just the number 1. Can we think of a function that after two derivatives becomes the negative of itself? We don't know a whole lot of special functions, but amazingly, there are actually a couple that do satisfy this: sine and cosine. The derivative of sine is cosine, and then a second derivative brings it back to negative sine.
There are lots of different features we can include with a sine (or cosine) function, so let's write one out in all its glory:
\[ x\left(t\right) = A \sin\left(\omega t + \phi\right) \]
Two derivatives of this function gives:
\[ \dfrac{d^2}{dt^2}\left[A \sin\left(\omega t + \phi\right)\right] = \dfrac{d}{dt}\left[\omega A \cos\left(\omega t + \phi\right)\right] = -\omega^2 A \sin\left(\omega t + \phi\right) \]
Plugging this into the differential equation gives a solution if we have:
\[ \omega = \sqrt{\dfrac{k}{m}} \]
Total Phase
It may seem crazy at this point to introduce the greek letter \(\omega\) as a constant here when we so recently used it as a measure of angular velocity, and there is no rotational motion going on here. But there is a good reason to do this. Consider a bead moving at a constant speed on a circular loop of wire of radius \(A\).
Figure 8.1.1 – Bead Moving on a Circular Loop of Wire
The equation describing its motion is something we are quite familiar with:
\[ \Phi\left(t\right) = \omega t + \phi \]
We have changed notation a bit here from what we used previously for circular or rotational motion: \(\Phi\left(t\right)\) represents the total angle traversed, in place of our previous \(\theta\left(t\right)\), and \(\phi\) represents the starting angle at \(t=0\), in place of our previous \(\theta_o\).
Next let's imagine placing a light source to the left of this loop, and a screen (the plane of which is perpendicular to the plane of the loop) to the right of the loop. The light would project a shadow of the bead onto the screen, and the motion of this shadow can be described mathematically:
Figure 8.1.2 – Motion of a Projection of a Bead moving on a Circular Loop
The motion of the shadow is simply a component of the motion of the bead – if we know the angle the bead makes on the circule, we know the height of the shadow on the screen. In terms of the starting angle and angular velocity of the bead, the motion of the shadow can be written explicitly as:
\[x\left(t\right) = A\sin\left(\omega t+\phi\right) \]
This is precisely the same as the equation of motion of a mass on a spring, Equation 8.1.2 . That is, if we placed a mass on a spring at the screen with the equilibrium position at \(x=0\), pulled the spring to a maximum stretch (or pushed it to a maximum compression) of \(x=A\), and then waited for the shadow to land on the mass before releasing it, the shadow would remain on the mass as it moves, if the angular velocity of the bead happens to equal \(\sqrt{k/m}\).
We can now discard our bead-on-circular-loop model, but we keep the mathematical structure it leaves behind. The argument of the sine function \(\Phi\left(t\right)\) is called the total phase of the harmonic motion of the mass-on-spring. The maximum expansion/compression of the spring \(A\) is called the amplitude of the harmonic motion. The constant \(\omega\) is referred to as the angular frequency (not angular velocity – we've left the bead model behind), and still is expressed in units of radians per second. This constant sometimes gives way to a frequency \(f\) that is measured in cycles per second (or hertz ), with the translation between the two being:
\[\omega = 2\pi f \]
The period of oscillation is the time it takes the system to come all the way back to where it started, and as the time per cycle, it is the inverse of the frequency:
\[T = \dfrac{1}{f} = \dfrac{2 \pi}{\omega} = 2\pi \sqrt{\dfrac{m}{k}} \]
Alert
The period is the time required for the system to complete a full cycle, which is not the same as the time required for the mass to return to a previous position. The mass must return to the same position and it must be moving in the same direction . In other words, the total phase \(\Phi\) must change by \(2\pi\).
And finally, the constant \(\phi\) is called the phase constant , and it carries the information of where the mass is at time \(t=0\).
Kinematics of Harmonic Motion
Once we have a formula for the position of an object following simple harmonic motion, we can use the usual calculus tools to determine the velocity and acceleration at various times as well. The velocity as a function of time is:
We note a couple of features of this result. First, since the cosine function never exceeds 1, we have the maximum speed of the object:
\[v_{max} = A\omega =A\sqrt{\dfrac{k}{m}} \]
And second, this maximum speed is achieved at \(x=0\) (the equilibrium point), which makes sense, since the spring was accelerating it toward that point, and immediately after passing it, the spring starts slowing it down.
The acceleration of the mass as a function of time we get from another derivative:
\[a\left(t\right) = \dfrac{d}{dt} v\left(t\right) = \dfrac{d}{dt} \left[A\omega\cos\left(\omega t+\phi\right)\right] = -A\omega^2\sin\left(\omega t+\phi\right)\]
The fact that this result is only different from the position function by a factor of \(\left(-\dfrac{k}{m}\right)\) brings us back to what started all of this, Equation 8.1.1 .
Mechanical Energy
We already know that the elastic force is conservative, so mechanical energy is conserved during simple harmonic motion. At any given time during the motion, the mass will have kinetic and potential energy, with its total energy remaining constant. It's easy to write an expression for the total energy in this system by choosing a convenient point in the motion – when the mass is stationary. This occurs when it reaches its maximum separation from the equilibrium point, i.e. when the displacement equals the amplitude:
\[E_{tot} = \frac{1}{2}kx_{max}^2 = \frac{1}{2}kA^2 \]
We can double-check this result by looking at the moment in time when there is zero potential energy and all of the mechanical energy is kinetic – at the equilibrium point. Using Equation 8.1.9 , we get:
\[E_{tot} = \frac{1}{2}mv_{max}^2 = \frac{1}{2}m\left(A\sqrt{\dfrac{k}{m}}\right)^2 = \frac{1}{2}kA^2 \]


2.2: Redrawing Complex Circuits
Typically, complex circuits are not arranged in nice, neat, clean schematic diagrams for us to follow. They are often drawn in such a way that makes it difficult to follow which components are in series and which are in parallel with each other. The purpose of this section is to show you a method useful for re-drawing circuit schematics in a neat and orderly fashion. Like the stage-reduction strategy for solving series-parallel combination circuits, it is a method easier demonstrated than described.
Let’s start with the following (convoluted) circuit diagram. Perhaps this diagram was originally drawn this way by a technician or engineer. Perhaps it was sketched as someone traced the wires and connections of a real circuit. In any case, here it is in all its ugliness:
With electric circuits and circuit diagrams, the length and routing of wire connecting components in a circuit matters little. (Actually, in some AC circuits it becomes critical, and very long wire lengths can contribute unwanted resistance to both AC and DC circuits, but in most cases wire length is irrelevant.) What this means for us is that we can lengthen, shrink, and/or bend connecting wires without affecting the operation of our circuit.
The strategy I have found easiest to apply is to start by tracing the current from one terminal of the battery around to the other terminal, following the loop of components closest to the battery and ignoring all other wires and components for the time being. While tracing the path of the loop, mark each resistor with the appropriate polarity for voltage drop.
In this case, I’ll begin my tracing of this circuit at the negative terminal of the battery and finish at the positive terminal, in the same general direction as the electrons would flow. When tracing this direction, I will mark each resistor with the polarity of negative on the entering side and positive on the exiting side, for that is how the actual polarity will be as electrons (negative in charge) enter and exit a resistor:
Any components encountered along this short loop are drawn vertically in order:
Now, proceed to trace any loops of components connected around components that were just traced.  Noting the polarity of voltage drops across R 3 and R 1 , I mark R 4 and R 2 likewise:
Now we have a circuit that is very easily understood and analyzed. In this case, it is identical to the four-resistor series-parallel configuration we examined earlier in the chapter.
Let’s look at another example, even uglier than the one before:
The first loop I’ll trace is from the negative (-) side of the battery, through R 6 , through R 1 , and back to the positive (+) end of the battery:
Re-drawing vertically and keeping track of voltage drop polarities along the way, our equivalent circuit starts out looking like this:
Next, we can proceed to follow the next loop around one of the traced resistors (R 6 ), in this case, the loop formed by R 5 and R 7 . As before, we start at the negative end of R 6 and proceed to the positive end of R 6 , marking voltage drop polarities across R 7 and R 5 as we go:
Now we add the R 5 —R 7 loop to the vertical drawing. Notice how the voltage drop polarities across R 7 and R 5 correspond with that of R 6 , and how this is the same as what we found tracing R 7 and R 5 in the original circuit:
We repeat the process again, identifying and tracing another loop around an already-traced resistor. In this case, the R 3 —R 4 loop around R 5 looks like a good loop to trace next:
Adding the R 3 —R 4 loop to the vertical drawing, marking the correct polarities as well:
With only one remaining resistor left to trace, then next step is obvious: trace the loop formed by R 2 around R 3 :
Adding R 2 to the vertical drawing, and we’re finished! The result is a diagram that’s very easy to understand compared to the original:
This simplified layout greatly eases the task of determining where to start and how to proceed in reducing the circuit down to a single equivalent (total) resistance. Notice how the circuit has been re-drawn, all we have to do is start from the right-hand side and work our way left, reducing simple-series and simple-parallel resistor combinations one group at a time until we’re done.
In this particular case, we would start with the simple parallel combination of R 2 and R 3 , reducing it to a single resistance. Then, we would take that equivalent resistance (R 2 //R 3 ) and the one in series with it (R 4 ), reducing them to another equivalent resistance (R 2 //R 3 —R 4 ). Next, we would proceed to calculate the parallel equivalent of that resistance (R 2 //R 3 —R 4 ) with R 5 , then in series with R 7 , then in parallel with R 6 , then in series with R 1 to give us a grand total resistance for the circuit as a whole.
From there we could calculate total current from total voltage and total resistance (I=E/R), then “expand” the circuit back into its original form one stage at a time, distributing the appropriate values of voltage and current to the resistances as we go.