3.12: III Glossary
Assimilation: The process of adopting or conforming to the practices, habits, and norms of a cultural group. As a result, the person gradually takes on a new cultural identity and may lose their original identity in the process.
Bias: To carry an attitude, opinion, or inclination (positive or negative) towards a group or members of a group. Bias can be a conscious attitude (explicit), or a person may not be aware of their bias (implicit).
Cultural awareness: A deliberate, cognitive process in which health care providers become appreciative and sensitive to the values, beliefs, lifeways, practices, and problem-solving strategies of a patient’s culture. Cultural awareness goes beyond a simple awareness of the existence of other cultures and involves an interest, curiosity, and appreciation of other cultures.
Cultural competency: The process of applying evidence-based nursing in agreement with the preferred cultural values, beliefs, worldview, and practices of patients to produce improved patient outcomes.
Cultural diversity: Cultural differences in people.
Cultural encounters: A process where the nurse directly engages in face-to-face cultural interactions and other types of encounters with clients from culturally diverse backgrounds in order to modify existing beliefs about a cultural group and to prevent possible stereotyping.
Cultural humility: A humble and respectful attitude toward individuals of other cultures that pushes one to challenge their own cultural biases, realize they cannot know everything about other cultures, and approach learning about other cultures as a lifelong goal and process. [1]
Cultural negotiation: A process where the patient and nurse seek a mutually acceptable way to deal with competing interests of nursing care, prescribed medical care, and the patient’s cultural needs. Cultural negotiation is reciprocal and collaborative. When the patient’s cultural needs do not significantly or adversely affect their treatment plan, the cultural needs can be accommodated.
Culturally responsive care: Nursing actions that integrate a person’s cultural beliefs into their care.
Culturally safe environment: A safe space for patients to interact with health professionals, without judgment or discrimination, where the patient is free to express their cultural beliefs, values, and identity.
Culture: A set of beliefs, attitudes, and practices shared by a group of people or community that is accepted, followed, and passed down to other members of the group.
Discrimination: Unfair and different treatment of another person or group, denying them opportunities and rights to participate fully in society.
Ethnocentrism: The belief that one’s culture (or race, ethnicity, or country) is better and preferable than another’s.
Gender expression: A person’s outward demonstration of gender in relation to societal norms, such as in style of dress, hairstyle, or other mannerisms.
Gender identity: A person’s inner sensibility that they are a man, a woman, or perhaps neither.
Health disparities: Differences in health outcomes resulting from entrenched economic, sociopolitical, or environmental disadvantages.
Health care disparities: Differences in access to health care and insurance coverage.
Holism: Treatment of the whole person, including physical, mental, spiritual, and social needs.
Intersectionality: The many ways in which a person expresses their cultural identity are not separated, but are closely intertwined.
Justice: A principle and moral obligation to act on the basis of equality and equity; a standard linked to fairness for all in society. [2]

Prejudice: To “prejudge”; a preconceived idea, often unfavorable, about a person or group of people.
Race: A socially constructed idea; there are no truly genetically or biologically distinct races. Humans are biologically similar to each other, not different.
Racism: The presumption that races are distinct from one another and there is a hierarchy to race, implying that races are unequal. In racism, expression of one’s cultural beliefs is viewed as a heritable trait.
Sexual orientation: A person’s physical and emotional interest or desire for others. Sexual orientation is on a continuum and is manifested in one’s self-identity and behaviors.
Social determinants of health: Nonmedical factors that influence health outcomes, including conditions in which people are born, grow, work, live, and age, and the wider sets of forces and systems shaping the conditions of daily life. [3]
Social justice: Equal rights, equal treatment, and equitable opportunities for all. [4]
Stereotyping: Assuming that a person has the attributes, traits, beliefs, and values of a group because they are a member of that group.
Subculture: A smaller group of people within a larger culture, often based on a person’s occupation, hobbies, interests, or place of origin.
Transcultural nursing: Incorporating cultural beliefs and practices of people to help them maintain and regain health or to face death in a meaningful way.
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵
- American Nurses Association. (2021). Nursing: Scope and standards of practice (4th ed.). American Nurses Association. ↵


18 .
Briefly describe the three concepts that Peck added to Erikson’s stage of integrity versus despair.
19 .
How do the disengagement theory and the activity theory differ in their explanations of social aging?
20 .
Describe the two types of social support, instrumental and emotional, and provide an example of each.
21 .
How do leisure activities typically change after people retire?
22 .
How does health relate to retirement?
23 .
How has the change from traditional pensions to 401(k) plans affected the retirement outlook for most workers?
24 .
Describe some of the negative consequences of providing caregiving for a loved one.
25 .
What characteristics predict which nursing homes will score low on quality of care?
26 .
Is successful emotional aging a common outcome for older adults? Support your answer.
27 .
How might participating in religious services affect well-being in late life?


4.7: Water Heating
Conventional Water Heating Systems
The most common residential water heater has a 40 or 50 gallon storage tank capacity. They are available in electric, natural gas, and propane (LPG) models. The height and diameter of a water heater varies with the gallon capacity. When selecting water heaters, it is important to know the measurements of the space the appliance will occupy, and to check the water heater manufacturer specifications and codes for space requirements for ventilation, clearances, and other installation requirements. The shorter versions water heaters are known as a “low-boy”. that can fit in closet spaces or under stairs, or as a “squat” water heater that can fit under countertops. Squat water heaters are generally offered in electric configurations ranging from 5 to 40 gallons, and gas is not used due to ventilation and carbon monoxide concerns.
All water heaters must be installed per local codes and manufacturer instructions:
- An isolation valve must be installed on the cold water piping near the inlet.
- A typical residential water heater has 3/4” male or female threaded water supply connections.
- Most codes dictate that any water heater located above a finished area must be installed in a safety pan.
- Most relief valve connections of a residential water heater are 3/4” female and require a plumber to install a 3/4” male adapter.
Electric Water Heaters
An electric water heater only requires a plumber to connect the hot and cold water piping from the rough-in stub-outs to the designated inlet and outlet connection. They do not require venting or gas piping and are less expensive to purchase. Most codes allow the safety pan for an electric water heater to be plastic due to the lack of heat generated externally from the water heater in comparison to gas fired models.
A standard residential electric water heater is classified as a 240-volt, 4500-watt, non-simultaneous water heater. Models with faster recovery rates may have higher wattage elements. The higher the wattage rating, the faster a water heater can heat water. Although most electric water heaters have two elements, each rated at 4500 watts, a non-simultaneous heating cycle has only one of the elements operating at a time. Each wire providing electricity to a water heater is known as a leg. The two different wires (legs) that connect to the high-limit device are identified as line voltage one and two. One wire eventually provides 120 volts of electricity to one side of a heating element and the second wire eventually completes the circuit by providing an additional 120 volts to the same element.The screws that secure the wire connection to the electrical devices are known as terminations, poles, or posts.
No water heater should ever have the ignited gas or electricity energized before filling the system with water and removing all trapped air from the system.
Electric Water Heater Components and Their Functions
Temperature and Pressure (T & P)Relief Valve- Electric, gas & solar applications, relieves at150 psig and/or 210° F. (Temperature (only) relief valves are used on the rooftop collector panels of solar water systems).
Mythbusters- water heater
Query \(\PageIndex{1}\)
Dip Tube- As heat rises, and the outlet supply of hot water to the fixtures feeds from the hottest point at the top of the water heater, manufacturer install a device known as a dip tube into the cold water connection of a top-fed water heater to route the incoming cold water to the bottom of the heater. If a dip tube were not used, the incoming cold water would mix with the hot water located in the top of the water heater, cooling the exiting water during use.
Drain– Used to flush the water heater’s storage tank during maintenance.
Ball valve- Used to shut off the supply water to the tank.
Mixing Valve- Allows cold water to enter hot water stream to temper over-temp water to prevent scalding. Mixing valves are normally required to be installed on solar water heating systems as temperatures can potentially reach in excess of 160°.
Tank Anode Rod- A residential water heater storage tank is manufactured with carbon steel, with most common residential water heaters incorporating a very thin coat of porcelain enamel called a “glass” lining, designed to fill every internal crevice created by the manufacturing process of the tank, to prevent the carbon steel tank from corroding. However, over time, this lining breaks down and the steel tank begins to corrode. An anode rod is a sacrificial device that dissolves (corrodes) over a period of time. An anode rod is installed by the manufacturer to prevent from rusting and corrosion of the inside of the tank due to glass lining imperfections or minor damage during shipping and handling. Magnesium-based and aluminum rods are more vulnerable to rust and minerals that would normally attack the steel walls of a tank and attract corrosives to attack the anode rod instead of the tank.
New versus Expended Anode Rod by Cory Early is licensed under CC BY 4.0
Top Access Cover– The thermostat and the red button are located in here. The upper heating element is also found here.
Bottom Access Cover– The lower thermostat and heating element are located here.
Upper Thermostat– Limits the temperature that the element will heat the water up to. Both, upper and lower, thermostats are secured in place and held against the surface of the tank with a retainer clip, keeping the thermostat in contact with the external portion of the storage tank. Most safety standards do not allow a plumber to set the temperature above 120 degrees Fahrenheit to prevent scalding. The upper thermostat of a seven-pole design only has three posts. This three-post design combined with the four posts of the high limit switch is how the seven-pole design is recognized. The upper thermostat has a temperature setting feature that is typically either identified alphabetically from A through D, or identified as warm, hot, and very hot.
Replacing Upper Thermostat
Query \(\PageIndex{2}\)
Lower Thermostat- The lower thermostat only has two posts and numbered 1 and 2. It also has a temperature setting feature that is the same design as the upper thermostat. Like the upper thermostat, lower thermostats can be adjustable and have similarly marked adjustment points.
Hi-Limit Device (Red Button Over Temperature Thermostat)– The high-limit device halts the electrical current to the thermostats and elements if an unsafe water temperature is present. Most high-limit devices have a reset button designed to pop out if the temperature reaches 190°. These devices can be reset once the temperature of the tank cools below the “pop-off” temperature.
Heating Element- Elements convert electrical energy to heat energy and transfer heat to the water in the tank. They are rated in watts and selected for tanks according to tank size and recovery rate. Residential electric water heating elements are usually a screw-in type, but bolt-in types are used for certain water heater models. The elements can vary in length, with 12” being the typical length provided by a manufacturer in most water heaters. The electrical voltage and wattage ratings of the element are indicated by a manufacturer on the element so its identification of replacement parts specifications is available on the exterior.
Lined and Dielectric Pipe Nipples- Many water heaters are manufactured with pipe nipples that are lined with corrosive-resistant material such as PEX. These pipe nipples are manufactured to resist corrosion due to electrolysis from joining dissimilar metals to the iron tank (copper to iron = rust). Brass pipe fittings can also be used between iron pipe fittings and copper to resist electrolysis. A plumber must take precautions when connecting copper tube to a lined nipple as heating the lining of the pipe nipple directly with a torch or by connecting a fitting that has been soldered and not allowed to cool can melt the internal lining of the nipple. Brass to copper unions provide the same benefit in non-concealed applications.
Expansion Tank- Every water heating system must be protected against dangerous occurrences that exist when water is heated. As a heating cycle occurs, water expands and can cause the pressure relief valve to begin dripping. Expansion tanks are installed, mainly in solar water systems, to absorb the expansion of a system. An expansion tank used for a potable hot water system has an internal rubber membrane known as a bladder. Most codes require that an expansion tank be installed near a water heater to protect the piping system from high pressure caused by the heating cycle.
Gas Water Heaters
Natural gas and propane (LPG) are the two types of gas used for water heating. A water heater designed for natural gas cannot be used with propane unless the particular water heater can be converted. The internal gas regulating orifice is different for each type of gas. The venting requirements of a gas water heater are dictated by code. The exhaust fumes from a gas water heater contain carbon monoxide. Carbon monoxide fumes are odorless and can kill occupants of a home or building. Conventional gas water heaters are vented atmospherically, that is, to the exterior of the building. Atmospheric water heater venting must terminate in specific locations through, and heights above a roof. Gas water heaters must have adequate space around the water heater and replacement air to create proper draft conditions that allow the fumes to be exhausted to the exterior of the building. If adequate air is not provided in the room where an atmospheric vented water heater is located, the exhaust fumes could enter the occupied space.
Selecting a gas water heater based on its capabilities requires manufacturer’s specifications for the specific appliance and knowledge of the load demand. The recovery rate is the most important aspect of determining if a certain gas water heater is capable of being installed for a specific home or use. Gas water heaters are rated by the gallons of hot water they can produce. One British Thermal Unit (BTU) is the amount of heat required to raise one pound of water by one degree Fahrenheit. One gallon of water weighs 8.33 pounds, and 8.33 btus are required to raise the temperature of one gallon of water by one degree Fahrenheit. Temperature rise is the difference between the incoming cold water and the desired temperature expected from a water heater. Temperature rise determines the capabilities of a certain Btu rating of a gas water heater. A length of time is used to clarify the recovery capabilities of a gas water heater, and gallons per hour (gph) is the most common method used in rating a hot water heater.
- Codes vary pertaining to the gas supply connections and venting regulations.
- A gas water heater uses a metal safety pan.
- The gas supply configuration is fairly consistent with most residential water heaters: black iron pipe or approved polyurethane tubing, brass fittings, gas-cock, flex supply line, and regulator
Gas Regulators
A gas regulator is an automated device that controls the gas flow to a burner assembly. The gas supply pipe is connected to the gas regulator which regulates the flow of gas to the burner A thermocouple must sense a pilot flame in order to allow the gas to flow through a gas regulator. The design of a gas regulator is based on safety, and most codes do not allow anyone not certified to repair a gas regulator to disassemble one for repair. Another feature of the regulator is to control the gas flow to a pilot flame. The gas regulator for a residential gas water heater typically has a 1/2” female threaded connection.
How to Relight a Gas Water Heater Pilot Light
Query \(\PageIndex{3}\)
Tankless Water Heaters
A tankless heater design can be suitable for many residential applications. Water flow is regulated to ensure that the desired temperature leaves the heater. They are becoming more desirable and are considered environmentally friendly. Tankless models are also known as an instantaneous water heaters.
The two basic types of tankless water heaters are interior or exterior, defining whether the installation location is inside or outside of the building structure. Tankless water heaters are available in electric models for interior applications. Tankless water heaters can be installed inside or outside a building, but units installed outside must be gas. The units are not interchangeable. Interior units require venting of the flue gas and exterior units may require freeze protection in certain climates.
Solar Water Heaters
Although a solar thermal water heating system can save a residence up to a 40% savings on utility bills, the initial investment to install a solar water heating system may deter many homeowners. Federal, and sometimes state, tax credits are available for installing alternate energy systems and utility savings over time are a good return on investment (ROI).
Solar water systems use solar collector panels, usually mounted on the roof to heat the system water. When the temperature of the water in the collector reaches a specific higher temperature than the water in the tank, a pump circulates the cooler water from the tank which pushes the hotter water in the roof panel back into the tank until the temperature of the panel and the temperature in the tank are equal.
The location and angle of the solar panel (orientation and tilt for optimal sun exposure) is crucial to obtaining optimal efficiency of the entire system. Most solar systems are directly connected to gas or a single electric heating element to provide adequate hot water during non-solar heating periods.
Diagnosing Common Water Heater Problems
Water heaters should be flushed annually to remove contaminates and scale from the tank. Anode rod should be inspected every 2 years and replaced approximately every 5 years, or sooner if conditions show significant corrosion of the rod. Troubleshooting gas and electric water heaters must be performed by qualified individuals. Basic electrical knowledge must be known to safely troubleshoot an electric water heater, and a plumber must have an electrical voltage/amperage meter to diagnose and service an electric water heater. Always remember that a water heater must be always be filled with water before igniting the gas supply or electrically energizing the system and only a certified technician can repair gas regulators.
Electric Water Heater Diagnostics
| SYMPTOM | POSSIBLE CAUSE | POSSIBLE SOLUTION |
|---|---|---|
| No hot water |
No electricity from
source |
Check breaker |
| No hot water |
Electrical problems with
thermostats |
Test and replace appropriate thermostat/s |
| No hot water |
Failed heating
element(s) |
Test and replace appropriate element/s |
| Little hot water | Dip tube failure | Inspect dip tube and replace if necessary |
| Little hot water | Lower thermostat or element failure | Replace lower thermostat and/or element |
| Little hot water | Thermostat/s failure | Test and replace appropriate thermostat/s |
| Water too hot | Thermostat/s failure | Test and replace appropriate thermostat/s |
| Rest button tripped | Water too hot | Test and replace appropriate thermostat/s |
|
Rotten egg smell of
water |
Expended anode rod | Inspect and replace |
|
Popping noise when
heating |
Scale build-up on
element(s) |
Replace element(s) |
Water Heater Not Heating? Thermostat Testing
Query \(\PageIndex{4}\)
Heating Element Diagnostic Procedure
- Switch off the power to the water heater at the main electrical panel.
- Locate the electrical access panels, one near the base of the tank and one nearer the top (solar water systems only have one back-up element and thermostat located in the upper access). Beginning with the top element, remove the panel’s screws with a screwdriver. Take off the panel and remove insulation behind it to uncover the heating element.
- Disconnect either of the two electrical wires screwed into the element terminals. It is not necessary to disconnect both wires to test the element.
- Set the multimeter or Ohmeter to read Ohms, and set the scale to RX1. Touch one probe to each of the two terminal screws. If the needle moves at all, or if there is any reading besides “infinity” on a digital readout, the element is good. If it doesn’t move, or displays “infinity,” no electricity is flowing through the element and it should be replaced.
- Complete replacement or repairs as necessary. Reconnect the wire to the element terminal. Replace the insulation and reinstall the access panel.
- Repeat the process on the bottom element. Restore power to the water heater from the main electrical panel.
How to Replace an Electric Water Heater Heating Element
Query \(\PageIndex{5}\)
Gas Water Heater Diagnostics
| SYMPTOM | POSSIBLE CAUSE | POSSIBLE SOLUTION |
|---|---|---|
| No gas to pilot | No gas from meter or gas cock | Turned off at meter or gas cock |
| No gas to pilot | No gas from regulator | Debris in regulator. Clean or replace defective regulator |
| No gas to pilot | Faulty regulator | Replace regulator |
| No gas to pilot | Crimped pilot tube | Repair or replace tube |
| No gas to pilot | Leak in pilot tube | Replace tube |
| No pilot flame | No gas flow from regulator | Debris in regulator. Clean or replace defective regulator |
| No pilot flame | Defective thermocouple | Replace thermocouple |
| No pilot flame | Air in gas piping | Purge air from piping |
| No gas to burner | Defective regulator | Replace regulator |
| No gas to burner | Crimped burner tube | Repair or replace burner tube |
| No gas to burner | Blockage in burner tube | Remove tube and clean |
| No gas to burner | Remove and clean | Run water to cool tank |
| No gas to burner | Defective high-limit device | Replace device or regulator |
| Temperature and Pressure (T&P) relief valve leaking | Water is too hot | Run water to cool tank |
| Temperature and Pressure (T&P) relief valve leaking | Defective relief valve | Replace relief valve |
| Temperature and Pressure (T&P) relief valve leaking | Excessive pressure | Install expansion tank |
| Temperature and Pressure (T&P) relief valve leaking | Excessive pressure | Check pressure regulator valve to system supply |
| Low water temperature | Thermostat set too low | Adjust temperature setting on thermostat |
| Low water temperature | Dip tube failure | Inspect and replace if necessary |
| Low water temperature | Defective thermostat | Test and replace appropriate thermostat/s |
| Slow recovery time | Sediment in tank | Drain and flush tank |
| Slow recovery time | Dirty burner assembly | Clean burner assembly |
| Slow recovery time | Poor flame | Adjust burner air supply |
| Slow recovery time | Poor flame | Supply more combustion air |
| Not enough hot water | Insufficient size heater | Calculate demand load and replace with an appropriate sized water heating system |
| Not enough hot water | See low water temperature and slow recovery symptoms | See low water temperature and slow recovery symptoms |
| Popping / banging noises | Calcium or sediment build up | Drain and flush tank |
| Banging noise | Check valve slamming when a faucet or valve is being opened or closed | Install a shock absorbing device |
| Fume odor | Poor draft on flue system | Examine flue pipe installation and ensure termination location provides sufficient draft |
| Gas odor | Leak on piping system | Soap test all piping joints and supply lines for leaks and repair as required |
| Soot build up | Poor draft on flue system | See fume odor, install a draft hood fan |
| Soot build up | Insufficient combustion air | Install air supply ducts or vents |
| Soot build up | Poor burner flame | Clean and adjust burner |
| Flame back flash | Negative air pressure | Isolate heater air |


30.E: Atomic Physics (Exercises)
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Conceptual Questions
30.1: Discovery of the Atom
1. Name three different types of evidence for the existence of atoms.
2. Explain why patterns observed in the periodic table of the elements are evidence for the existence of atoms, and why Brownian motion is a more direct type of evidence for their existence.
3. If atoms exist, why can’t we see them with visible light?
30.2: Discovery of the Parts of the Atom: Electrons and Nuclei
4. What two pieces of evidence allowed the first calculation of me, the mass of the electron? Justify your response.
(a) The ratios \(\displaystyle q_e/m_e\) and \(\displaystyle q_p/m_p\).
(b) The values of \(\displaystyle q_e\) and \(\displaystyle E_B\).
(c) The ratio \(\displaystyle q_e/m_e\) and \(\displaystyle q_e\).
5. How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.
30.3: Bohr’s Theory of the Hydrogen Atom
6. How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.
7. Explain how Bohr’s rule for the quantization of electron orbital angular momentum differs from the actual rule.
8. What is a hydrogen-like atom, and how are the energies and radii of its electron orbits related to those in hydrogen?
30.4: X Rays- Atomic Origins and Applications
9. Explain why characteristic x rays are the most energetic in the EM emission spectrum of a given element.
10. Why does the energy of characteristic x rays become increasingly greater for heavier atoms?
11. Observers at a safe distance from an atmospheric test of a nuclear bomb feel its heat but receive none of its copious x rays. Why is air opaque to x rays but transparent to infrared?
12. Lasers are used to burn and read CDs. Explain why a laser that emits blue light would be capable of burning and reading more information than one that emits infrared.
13. Crystal lattices can be examined with x rays but not UV. Why?
14. CT scanners do not detect details smaller than about 0.5 mm. Is this limitation due to the wavelength of x rays? Explain.
30.5: Applications of Atomic Excitations and De-Excitations
15. How do the allowed orbits for electrons in atoms differ from the allowed orbits for planets around the sun? Explain how the correspondence principle applies here.
16. Atomic and molecular spectra are discrete. What does discrete mean, and how are discrete spectra related to the quantization of energy and electron orbits in atoms and molecules?
17. Hydrogen gas can only absorb EM radiation that has an energy corresponding to a transition in the atom, just as it can only emit these discrete energies. When a spectrum is taken of the solar corona, in which a broad range of EM wavelengths are passed through very hot hydrogen gas, the absorption spectrum shows all the features of the emission spectrum. But when such EM radiation passes through room-temperature hydrogen gas, only the Lyman series is absorbed. Explain the difference.
18. Lasers are used to burn and read CDs. Explain why a laser that emits blue light would be capable of burning and reading more information than one that emits infrared.
19. The coating on the inside of fluorescent light tubes absorbs ultraviolet light and subsequently emits visible light. An inventor claims that he is able to do the reverse process. Is the inventor’s claim possible?
20. What is the difference between fluorescence and phosphorescence?
21. How can you tell that a hologram is a true three-dimensional image and that those in 3-D movies are not?
30.6: The Wave Nature of Matter Causes Quantization
22. How is the de Broglie wavelength of electrons related to the quantization of their orbits in atoms and molecules?
30.7: Patterns in Spectra Reveal More Quantization
23. What is the Zeeman effect, and what type of quantization was discovered because of this effect?
30.8: Quantum Numbers and Rules
24. Define the quantum numbers \(\displaystyle n, l,m_l, s\), and \(\displaystyle m_s\).
25. For a given value of \(\displaystyle n\), what are the allowed values of \(\displaystyle l\)?
26. For a given value of \(\displaystyle l\), what are the allowed values of \(\displaystyle m_l\)? What are the allowed values of \(\displaystyle m_l\) for a given value of \(\displaystyle n\)? Give an example in each case.
27. List all the possible values of \(\displaystyle s\) and \(\displaystyle m_s\) for an electron. Are there particles for which these values are different? The same?
30.9: The Pauli Exclusion Principle
28. Identify the shell, subshell, and number of electrons for the following:
(a) \(\displaystyle 2p^3\).
(b) \(\displaystyle 4d^9\).
(c) \(\displaystyle 3s^1\).
(d) \(\displaystyle 5g^{16}\).
29. Which of the following are not allowed? State which rule is violated for any that are not allowed.
(a) \(\displaystyle 1p^3\)
(b) \(\displaystyle 2p^8\)
(c) \(\displaystyle 3g^{11}\)
(d) \(\displaystyle 4f^2\)
Problems & Exercises
30.1: Discovery of the Atom
30. Using the given charge-to-mass ratios for electrons and protons, and knowing the magnitudes of their charges are equal, what is the ratio of the proton’s mass to the electron’s? (Note that since the charge-to-mass ratios are given to only three-digit accuracy, your answer may differ from the accepted ratio in the fourth digit.)
Solution
\(\displaystyle 1.84×10^3\)
31. (a) Calculate the mass of a proton using the charge-to-mass ratio given for it in this chapter and its known charge.
(b) How does your result compare with the proton mass given in this chapter?
32. If someone wanted to build a scale model of the atom with a nucleus 1.00 m in diameter, how far away would the nearest electron need to be?
Solution
50 km
30.2: Discovery of the Parts of the Atom: Electrons and Nuclei
33. Rutherford found the size of the nucleus to be about 10−15m. This implied a huge density. What would this density be for gold?
Solution
\(\displaystyle 6×10^{20}kg/m^3\)
34. In Millikan’s oil-drop experiment, one looks at a small oil drop held motionless between two plates. Take the voltage between the plates to be 2033 V, and the plate separation to be 2.00 cm. The oil drop (of density \(\displaystyle 0.81 g/cm^3\)) has a diameter of \(\displaystyle 4.0×10^{−6}m\). Find the charge on the drop, in terms of electron units.
35. (a) An aspiring physicist wants to build a scale model of a hydrogen atom for her science fair project. If the atom is 1.00 m in diameter, how big should she try to make the nucleus?
(b) How easy will this be to do?
Solution
(a) \(\displaystyle 10.0 μm\)
(b) It isn’t hard to make one of approximately this size. It would be harder to make it exactly 10.0 μm.
30.3: Bohr’s Theory of the Hydrogen Atom
36. By calculating its wavelength, show that the first line in the Lyman series is UV radiation.
Solution
\(\displaystyle \frac{1}{λ}=R(\frac{1}{n^2_f}−\frac{1}{n^2_i})⇒λ=\frac{1}{R}[\frac{(n_i⋅n_f)^2}{n^2_i−n^2_f}];n_i=2,n_f=1,\) so that
\(\displaystyle λ=(\frac{m}{1.097×10^7})[\frac{(2×1)^2}{2^2−1^2}]=1.22×10^{−7}m=122 nm\) , which is UV radiation.
37. Find the wavelength of the third line in the Lyman series, and identify the type of EM radiation.
38. Look up the values of the quantities in \(\displaystyle a_B=\frac{h^2}{4π^2m_ekq^2_e}\), and verify that the Bohr radius \(\displaystyle a_B\) is \(\displaystyle 0.529×10^{−10}m\).
Solution
\(\displaystyle a_B=\frac{h^2}{4π^2mekZq^2_e}=\frac{(6.626×10^{−34}J⋅s)^2}{4π^2(9.109×10^{−31}kg)(8.988×10^9N⋅m^2/C^2)(1)(1.602×10^{−19}C)^2}=0.529×10^{−10}m\)
39. Verify that the ground state energy \(\displaystyle E_0\) is 13.6 eV by using \(\displaystyle E0=\frac{2π^2q^4_em_ek^2}{h^2}\).
40. If a hydrogen atom has its electron in the \(\displaystyle n=4\) state, how much energy in eV is needed to ionize it?
Solution
0.850 eV
41. A hydrogen atom in an excited state can be ionized with less energy than when it is in its ground state. What is n for a hydrogen atom if 0.850 eV of energy can ionize it?
42 . Find the radius of a hydrogen atom in the \(\displaystyle n=2\) state according to Bohr’s theory.
Solution
\(\displaystyle 2.12×10^{–10}m\)
43. Show that \(\displaystyle (13.6 eV)/hc=1.097×10^7m=R\) (Rydberg’s constant), as discussed in the text.
44. What is the smallest-wavelength line in the Balmer series? Is it in the visible part of the spectrum?
Solution
365 nm
It is in the ultraviolet.
45. Show that the entire Paschen series is in the infrared part of the spectrum. To do this, you only need to calculate the shortest wavelength in the series.
46. Do the Balmer and Lyman series overlap? To answer this, calculate the shortest-wavelength Balmer line and the longest-wavelength Lyman line.
Solution
No overlap
365 nm
122 nm
47. (a) Which line in the Balmer series is the first one in the UV part of the spectrum?
(b) How many Balmer series lines are in the visible part of the spectrum?
(c) How many are in the UV?
48. A wavelength of \(\displaystyle 4.653 μm\) is observed in a hydrogen spectrum for a transition that ends in the \(\displaystyle n_f=5\) level. What was \(\displaystyle n_i\) for the initial level of the electron?
Solution
7
49. A singly ionized helium ion has only one electron and is denoted \(\displaystyle He^+\). What is the ion’s radius in the ground state compared to the Bohr radius of hydrogen atom?
50. A beryllium ion with a single electron (denoted \(\displaystyle Be^{3+}\)) is in an excited state with radius the same as that of the ground state of hydrogen.
(a) What is \(\displaystyle n\) for the \(\displaystyle Be^{3+}\) ion?
(b) How much energy in eV is needed to ionize the ion from this excited state?
Solution
(a) 2
(b) 54.4 eV
51. Atoms can be ionized by thermal collisions, such as at the high temperatures found in the solar corona. One such ion is \(\displaystyle C^{+5}\), a carbon atom with only a single electron.
(a) By what factor are the energies of its hydrogen-like levels greater than those of hydrogen?
(b) What is the wavelength of the first line in this ion’s Paschen series?
(c) What type of EM radiation is this?
52. Verify Equations \(\displaystyle r_n=\frac{n^2}{Z}a_B\) and \(\displaystyle a_B=\frac{h^2}{4π^2m_ekq^2_e}=0.529×10^{−10}m\) using the approach stated in the text. That is, equate the Coulomb and centripetal forces and then insert an expression for velocity from the condition for angular momentum quantization.
Solution
\(\displaystyle \frac{kZq^2_e}{r^2_n}=\frac{m_eV^2}{r_n}\), so that \(\displaystyle r_n=\frac{kZq^2_e}{m_eV^2}=\frac{kZq^2_e}{m_e}\frac{1}{V^2}\). From the equation \(\displaystyle m_evr_n=n\frac{h}{2π}\), we can substitute for the velocity, giving: \(\displaystyle r_n=\frac{kZq^2_e}{m_e}⋅\frac{4π^2m^2_er^2_n}{n^2h^2}\) so that \(\displaystyle r_n=\frac{n^2}{Z}\frac{h^2}{4π^2m_ekq^2_e}=\frac{n^2}{Z}a_B\), where \(\displaystyle a_B=\frac{h^2}{4π^2m_ekq^2_e}\).
53. The wavelength of the four Balmer series lines for hydrogen are found to be 410.3, 434.2, 486.3, and 656.5 nm. What average percentage difference is found between these wavelength numbers and those predicted by \(\displaystyle \frac{1}{λ}=R(\frac{1}{n^2_f}−\frac{1}{n^2+i})\)? It is amazing how well a simple formula (disconnected originally from theory) could duplicate this phenomenon.
30.4: X Rays- Atomic Origins and Applications
54. (a) What is the shortest-wavelength x-ray radiation that can be generated in an x-ray tube with an applied voltage of 50.0 kV?
(b) Calculate the photon energy in eV.
(c) Explain the relationship of the photon energy to the applied voltage.
Solution
(a) \(\displaystyle 0.248×10^{−10}m\)
(b) 50.0 keV
(c) The photon energy is simply the applied voltage times the electron charge, so the value of the voltage in volts is the same as the value of the energy in electron volts.
55. A color television tube also generates some x rays when its electron beam strikes the screen. What is the shortest wavelength of these x rays, if a 30.0-kV potential is used to accelerate the electrons? (Note that TVs have shielding to prevent these x rays from exposing viewers.)
56. An x ray tube has an applied voltage of 100 kV.
(a) What is the most energetic x-ray photon it can produce? Express your answer in electron volts and joules.
(b) Find the wavelength of such an X–ray.
Solution
(a) \(\displaystyle 100×10^3eV, 1.60×10^{−14}J\)
(b) \(\displaystyle 0.124×10^{−10}m\)
57. The maximum characteristic x-ray photon energy comes from the capture of a free electron into a \(\displaystyle K\) shell vacancy. What is this photon energy in keV for tungsten, assuming the free electron has no initial kinetic energy?
58. What are the approximate energies of the \(\displaystyle K_α\) and \(\displaystyle K_β\) x rays for copper?
Solution
(a) 8.00 keV
(b) 9.48 keV
30.5: Applications of Atomic Excitations and De-Excitations
59. Figure shows the energy-level diagram for neon.
(a) Verify that the energy of the photon emitted when neon goes from its metastable state to the one immediately below is equal to 1.96 eV.
(b) Show that the wavelength of this radiation is 633 nm.
(c) What wavelength is emitted when the neon makes a direct transition to its ground state?
Solution
(a) 1.96 eV
(b) \(\displaystyle (1240 eV⋅nm)/(1.96 eV)=633 nm\)
(c) 60.0 nm
60. A helium-neon laser is pumped by electric discharge. What wavelength electromagnetic radiation would be needed to pump it? See Figure for energy-level information.
61. Ruby lasers have chromium atoms doped in an aluminum oxide crystal. The energy level diagram for chromium in a ruby is shown in Figure. What wavelength is emitted by a ruby laser?
Chromium atoms in an aluminum oxide crystal have these energy levels, one of which is metastable. This is the basis of a ruby laser. Visible light can pump the atom into an excited state above the metastable state to achieve a population inversion.
Solution
693 nm
62. (a) What energy photons can pump chromium atoms in a ruby laser from the ground state to its second and third excited states?
(b) What are the wavelengths of these photons? Verify that they are in the visible part of the spectrum.
63. Some of the most powerful lasers are based on the energy levels of neodymium in solids, such as glass, as shown in Figure.
(a) What average wavelength light can pump the neodymium into the levels above its metastable state?
(b) Verify that the 1.17 eV transition produces \(\displaystyle 1.06 μm\) radiation.
Neodymium atoms in glass have these energy levels, one of which is metastable. The group of levels above the metastable state is convenient for achieving a population inversion, since photons of many different energies can be absorbed by atoms in the ground state.
Solution
(a) 590 nm
(b) \(\displaystyle (1240 eV⋅nm)/(1.17 eV)=1.06 μm\)
30.8: Quantum Numbers and Rules
64. If an atom has an electron in the \(\displaystyle n=5\) state with \(\displaystyle m_l=3\), what are the possible values of \(\displaystyle l\)?
Solution
\(\displaystyle l=4, 3\) are possible since \(\displaystyle l<n\) and \(\displaystyle ∣m_l∣≤l\).
65. An atom has an electron with \(\displaystyle m_l=2\). What is the smallest value of \(\displaystyle n\) for this electron?
66. What are the possible values of \(\displaystyle m_l\) for an electron in the \(\displaystyle n=4\) state?
Solution
\(\displaystyle n=4⇒l=3, 2, 1, 0⇒m_l=±3,±2,±1, 0\) are possible.
67. What, if any, constraints does a value of \(\displaystyle m_l=1\) place on the other quantum numbers for an electron in an atom?
68. (a) Calculate the magnitude of the angular momentum for an \(\displaystyle l=1\) electron.
(b) Compare your answer to the value Bohr proposed for the \(\displaystyle n=1\) state.
Solution
(a) \(\displaystyle 1.49×10^{−34}J⋅s\)
(b) \(\displaystyle 1.06×10^{−34}J⋅s\)
69. (a) What is the magnitude of the angular momentum for an \(\displaystyle l=1\) electron?
(b) Calculate the magnitude of the electron’s spin angular momentum.
(c) What is the ratio of these angular momenta?
70. Repeat Exercise for \(\displaystyle l=3\).
Solution
(a) \(\displaystyle 3.66×10^{−34}J⋅s\)
(b) \(\displaystyle s=9.13×10^{−35}J⋅s\)
(c) \(\displaystyle \frac{L}{S}=\frac{\sqrt{12}}{\sqrt{3/4}}=4\)
71. (a) How many angles can \(\displaystyle L\) make with the z-axis for an \(\displaystyle l=2\) electron?
(b) Calculate the value of the smallest angle.
72. What angles can the spin \(\displaystyle S\)of an electron make with the z-axis?
Solution
\(\displaystyle θ=54.7º, 125.3º\)
30.9: The Pauli Exclusion Principle
73. (a) How many electrons can be in the \(\displaystyle n=4\) shell?
(b) What are its subshells, and how many electrons can be in each?
Solution
(a) 32.
(b) \(\displaystyle 2\) in \(\displaystyle s\), \(\displaystyle 6\) in \(\displaystyle p\),\(\displaystyle 10\) in \(\displaystyle d\), and 14 in \(\displaystyle f\), for a total of 32.
74. (a) What is the minimum value of 1 for a subshell that has 11 electrons in it?
(b) If this subshell is in the \(\displaystyle n=5\) shell, what is the spectroscopic notation for this atom?
75. (a) If one subshell of an atom has 9 electrons in it, what is the minimum value of \(\displaystyle l\)?
(b) What is the spectroscopic notation for this atom, if this subshell is part of the \(\displaystyle n=3\) shell?
Solution
(a) 2
(b) \(\displaystyle 3d^9\)
76. (a) List all possible sets of quantum numbers \(\displaystyle (n,l,m_l,m_s)\) for the \(\displaystyle n=3\) shell, and determine the number of electrons that can be in the shell and each of its subshells.
(b) Show that the number of electrons in the shell equals \(\displaystyle 2n^2\) and that the number in each subshell is \(\displaystyle 2(2l+1)\).
77. Which of the following spectroscopic notations are not allowed? State which rule is violated for each that is not allowed.
(a) \(\displaystyle 5s^1\)
(b) \(\displaystyle 1d^1\)
(c) \(\displaystyle 4s^3\)
(d) \(\displaystyle 3p^7\)
(e) \(\displaystyle 5g^{15}\).
Solution
(b) \(\displaystyle n≥l\) is violated,
(c) cannot have 3 electrons in \(\displaystyle s\) subshell since \(\displaystyle 3>(2l+1)=2\)
(d) cannot have 7 electrons in \(\displaystyle p\) subshell since \(\displaystyle 7>(2l+1)=2(2+1)=6\)
78. Which of the following spectroscopic notations are allowed (that is, which violate none of the rules regarding values of quantum numbers)?
(a) \(\displaystyle 1s^1\)
(b) \(\displaystyle 1d^3\)
(c) \(\displaystyle 4s^2\)
(d) \(\displaystyle 3p^7\)
(e) \(\displaystyle 6h^{20}\)
79. (a) Using the Pauli exclusion principle and the rules relating the allowed values of the quantum numbers \(\displaystyle (n,l,m_l,m_s)\), prove that the maximum number of electrons in a subshell is \(\displaystyle 2n^2\).
(b) In a similar manner, prove that the maximum number of electrons in a shell is \(\displaystyle 2n^2\).
Solution
(a) The number of different values of \(\displaystyle m_l\) is \(\displaystyle ±l,±(l−1),...,0\) for each \(\displaystyle l>0\) and one for \(\displaystyle l=0⇒(2l+1)\). Also an overall factor of 2 since each \(\displaystyle m_l\) can have \(\displaystyle m_s\) equal to either \(\displaystyle +1/2\) or \(\displaystyle −1/2⇒2(2l+1)\).
(b) for each value of \(\displaystyle l\), you get \(\displaystyle 2(2l+1)=0, 1, 2, ...,(n–1)⇒2{[(2)(0)+1]+[(2)(1)+1]+....+[(2)(n−1)+1]}=2[1+3+...+(2n−3)+(2n−1)]\) to see that the expression in the box is \(\displaystyle =n^2\), imagine taking \(\displaystyle (n−1)\) from the last term and adding it to first term \(\displaystyle =2[1+(n–1)+3+...+(2n−3)+(2n−1)–(n−1)]=2[n+3+....+(2n−3)+n]\). Now take \(\displaystyle (n−3)\) from penultimate term and add to the second term \(\displaystyle 2[n+n+...+n+n]=2n^2\).
80. Integrated Concepts
Estimate the density of a nucleus by calculating the density of a proton, taking it to be a sphere 1.2 fm in diameter. Compare your result with the value estimated in this chapter.
81. Integrated Concepts
The electric and magnetic forces on an electron in the CRT in [link] are supposed to be in opposite directions. Verify this by determining the direction of each force for the situation shown. Explain how you obtain the directions (that is, identify the rules used).
Solution
The electric force on the electron is up (toward the positively charged plate). The magnetic force is down (by the RHR).
82. (a) What is the distance between the slits of a diffraction grating that produces a first-order maximum for the first Balmer line at an angle of \(\displaystyle 20.0º\)?
(b) At what angle will the fourth line of the Balmer series appear in first order?
(c) At what angle will the second-order maximum be for the first line?
83. Integrated Concepts
A galaxy moving away from the earth has a speed of \(\displaystyle 0.0100c\). What wavelength do we observe for an \(\displaystyle n_i=7\) to \(\displaystyle n_f=2\) transition for hydrogen in that galaxy?
Solution
401 nm
84. Integrated Concepts
Calculate the velocity of a star moving relative to the earth if you observe a wavelength of 91.0 nm for ionized hydrogen capturing an electron directly into the lowest orbital (that is, a \(\displaystyle n_i=∞\) to \(\displaystyle n_f=1\), or a Lyman series transition).
85. Integrated Concepts
In a Millikan oil-drop experiment using a setup like that in [link], a 500-V potential difference is applied to plates separated by 2.50 cm.
(a) What is the mass of an oil drop having two extra electrons that is suspended motionless by the field between the plates?
(b) What is the diameter of the drop, assuming it is a sphere with the density of olive oil?
Solution
(a) \(\displaystyle 6.54×10^{−16}kg\)
(b) \(\displaystyle 5.54×10^{−7}m\)
86. Integrated Concepts
What double-slit separation would produce a first-order maximum at \(\displaystyle 3.00º\) for 25.0-keV x rays? The small answer indicates that the wave character of x rays is best determined by having them interact with very small objects such as atoms and molecules.
87. Integrated Concepts
In a laboratory experiment designed to duplicate Thomson’s determination of \(\displaystyle q_e/m_e\), a beam of electrons having a velocity of \(\displaystyle 6.00×10^7m/s\) enters a \(\displaystyle 5.00×10^{−3}T\) magnetic field. The beam moves perpendicular to the field in a path having a 6.80-cm radius of curvature. Determine \(\displaystyle q_e/m_e\) from these observations, and compare the result with the known value.
Solution
\(\displaystyle 1.76×10^{11}C/kg\), which agrees with the known value of \(\displaystyle 1.759×10^{11}C/kg\) to within the precision of the measurement
88. Integrated Concepts
Find the value of \(\displaystyle l\), the orbital angular momentum quantum number, for the moon around the earth. The extremely large value obtained implies that it is impossible to tell the difference between adjacent quantized orbits for macroscopic objects.
89. Integrated Concepts
Particles called muons exist in cosmic rays and can be created in particle accelerators. Muons are very similar to electrons, having the same charge and spin, but they have a mass 207 times greater. When muons are captured by an atom, they orbit just like an electron but with a smaller radius, since the mass in \(\displaystyle a_B=\frac{h^2}{4π^2m_ekq^2_e}=0.529×10^{−10}m\) is \(\displaystyle 207 m_e\).
(a) Calculate the radius of the \(\displaystyle n=1\) orbit for a muon in a uranium ion \(\displaystyle (Z=92)\).
(b) Compare this with the 7.5-fm radius of a uranium nucleus. Note that since the muon orbits inside the electron, it falls into a hydrogen-like orbit. Since your answer is less than the radius of the nucleus, you can see that the photons emitted as the muon falls into its lowest orbit can give information about the nucleus.
Solution
(a) 2.78 fm
(b) 0.37 of the nuclear radius.
90. Integrated Concepts
Calculate the minimum amount of energy in joules needed to create a population inversion in a helium-neon laser containing \(\displaystyle 1.00×10^{−4}\) moles of neon.
91. Integrated Concepts
A carbon dioxide laser used in surgery emits infrared radiation with a wavelength of \(\displaystyle 10.6 μm\). In 1.00 ms, this laser raised the temperature of \(\displaystyle 1.00 cm^3\) of flesh to \(\displaystyle 100ºC\) and evaporated it.
(a) How many photons were required? You may assume flesh has the same heat of vaporization as water.
(b) What was the minimum power output during the flash?
Solution
(a) \(\displaystyle 1.34×10^{23}\)
(b) 2.52 MW
92. Integrated Concepts
Suppose an MRI scanner uses 100-MHz radio waves.
(a) Calculate the photon energy.
(b) How does this compare to typical molecular binding energies?
93. Integrated Concepts
(a) An excimer laser used for vision correction emits 193-nm UV. Calculate the photon energy in eV.
(b) These photons are used to evaporate corneal tissue, which is very similar to water in its properties. Calculate the amount of energy needed per molecule of water to make the phase change from liquid to gas. That is, divide the heat of vaporization in kJ/kg by the number of water molecules in a kilogram.
(c) Convert this to eV and compare to the photon energy. Discuss the implications.
Solution
(a) 6.42 eV
(b) \(\displaystyle 7.27×10^{−20}J/molecule\)
(c) 0.454 eV, 14.1 times less than a single UV photon. Therefore, each photon will evaporate approximately 14 molecules of tissue. This gives the surgeon a rather precise method of removing corneal tissue from the surface of the eye.
94. Integrated Concepts
A neighboring galaxy rotates on its axis so that stars on one side move toward us as fast as 200 km/s, while those on the other side move away as fast as 200 km/s. This causes the EM radiation we receive to be Doppler shifted by velocities over the entire range of ±200 km/s. What range of wavelengths will we observe for the 656.0-nm line in the Balmer series of hydrogen emitted by stars in this galaxy. (This is called line broadening.)
95. Integrated Concepts
A pulsar is a rapidly spinning remnant of a supernova. It rotates on its axis, sweeping hydrogen along with it so that hydrogen on one side moves toward us as fast as 50.0 km/s, while that on the other side moves away as fast as 50.0 km/s. This means that the EM radiation we receive will be Doppler shifted over a range of \(\displaystyle ±50.0 km/s\). What range of wavelengths will we observe for the 91.20-nm line in the Lyman series of hydrogen? (Such line broadening is observed and actually provides part of the evidence for rapid rotation.)
Solution
91.18 nm to 91.22 nm
96. Integrated Concepts
Prove that the velocity of charged particles moving along a straight path through perpendicular electric and magnetic fields is \(\displaystyle v=E/B\). Thus crossed electric and magnetic fields can be used as a velocity selector independent of the charge and mass of the particle involved.
97. Unreasonable Results
(a) What voltage must be applied to an X-ray tube to obtain 0.0100-fm-wavelength X-rays for use in exploring the details of nuclei?
(b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
Solution
(a) \(\displaystyle 1.24×10^{11}V\)
(b) The voltage is extremely large compared with any practical value.
(c) The assumption of such a short wavelength by this method is unreasonable.
98. Unreasonable Results
A student in a physics laboratory observes a hydrogen spectrum with a diffraction grating for the purpose of measuring the wavelengths of the emitted radiation. In the spectrum, she observes a yellow line and finds its wavelength to be 589 nm.
(a) Assuming this is part of the Balmer series, determine \(\displaystyle n_i\), the principal quantum number of the initial state.
(b) What is unreasonable about this result?
(c) Which assumptions are unreasonable or inconsistent?
99. Construct Your Own Problem
The solar corona is so hot that most atoms in it are ionized. Consider a hydrogen-like atom in the corona that has only a single electron. Construct a problem in which you calculate selected spectral energies and wavelengths of the Lyman, Balmer, or other series of this atom that could be used to identify its presence in a very hot gas. You will need to choose the atomic number of the atom, identify the element, and choose which spectral lines to consider.
100. Construct Your Own Problem
Consider the Doppler-shifted hydrogen spectrum received from a rapidly receding galaxy. Construct a problem in which you calculate the energies of selected spectral lines in the Balmer series and examine whether they can be described with a formula like that in the equation \(\displaystyle \frac{1}{λ}=R(\frac{1}{n^2_f}−\frac{1}{n^2_i})\), but with a different constant \(\displaystyle R\).
Contributors and Attributions
-
Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0) .


5.1.2D: Conflict
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Learning Objectives
- Discuss how various groups in society compete for resources, status and power within society, known as conflict theory
Social conflict is the struggle for agency or power within a society. It occurs when two or more people oppose one another in social interactions, reciprocally exerting social power in an effort to attain scarce or incompatible goals, and prevent the opponent from attaining them.
Conflict theory emphasizes interests deployed in conflict, rather than the norms and values. This perspective argues that the pursuit of interests is what motivates conflict. Resources are scarce and individuals naturally fight to gain control of them. Thus, the theory sees conflict as a normal part of social life, rather than an abnormal occurrence. The three tenets of conflict theory are as follows:
- Society is composed of different groups that compete for resources.
- While societies may portray a sense of cooperation, a continual power struggle exists between social groups as they pursue their own interests.
- Social groups will use resources to their own advantage in the pursuit of their goals, frequently leading powerful groups to take advantage of less powerful groups.
Conflict theory relies upon the notion of a zero sum game, meaning that if group A acquires any given resource, group B will be unable to acquire it. Thus, any gain for group A is automatically a loss for group B. Conflict theory further argues that group A will continue to search for resources in order to keep group B from getting them, leading to the exploitation of the powerless. The idea that those who have control will maintain control is called the Matthew Effect.
According to the principles of conflict theory, all cooperation is only for the purpose of acquiring individual or group resources. This motivation for behavior restructures day-to-day interactions among people in a given society.
Key Points
- Conflict theory argues that conflict is a normal and necessary part of social interaction. In other words, conflict is seen as part of the social landscape rather than an anomaly.
- According to the theory, conflict is motivated by pursuit of personal interests. All individuals and groups are interested in gaining control over scarce resources, and this leads to conflict.
- Once one party gets control of resources, that party is unlikely to release them. The Matthew Effect is the idea that those in control will remain in control.
Key Terms
- Matthew Effect : The idea that those who have control will maintain control.
- Zero Sum Game : The idea that if group A acquires any given resource, group B will be unable to acquire it.
- Social Conflict : The struggle for agency or power within a society.


7.1: Random Access Memory (RAM) and Read Only Memory (ROM)
Memory Basics
Memory is the most essential element of a computing system because without it computer can’t perform simple tasks. Computer memory is of two basic type – Primary memory(RAM and ROM) and Secondary memory(hard drive,CD,etc.). Random Access Memory (RAM) is primary-volatile memory and Read Only Memory (ROM) is primary-non-volatile memory.
- Random Access Memory (RAM)
- It is also called as read write memory or the main memory or the primary memory .
- The programs and data that the CPU requires during execution of a program are stored in this memory.
- It is a volatile memory as the data loses when the power is turned off.
- RAM is further classified into two types- SRAM (Static Random Access Memory) and DRAM (Dynamic Random Access Memory) .
- Read Only Memory (ROM)
- Stores crucial information essential to operate the system, like the program essential to boot the computer.
- It is not volatile.
- Always retains its data.
- Used in embedded systems or where the programming needs no change.
- Used in calculators and peripheral devices.
- ROM is further classified into 4 types- ROM , PROM , EPROM , and EEPROM .
Types of Read Only Memory (ROM) –
- PROM (Programmable read-only memory) – It can be programmed by user. Once programmed, the data and instructions in it cannot be changed.
- EPROM (Erasable Programmable read only memory) – It can be reprogrammed. To erase data from it, expose it to ultra violet light. To reprogram it, erase all the previous data.
- EEPROM (Electrically erasable programmable read only memory) – The data can be erased by applying electric field, no need of ultra violet light. We can erase only portions of the chip.
Adapted from:
"Random Access Memory (RAM) and Read Only Memory (ROM)"
by
Deepanshi_Mittal
,
Geeks for Geeks
is licensed under
CC BY-SA 4.0


12.A: Review Questions
Complete as directed.
Fill in the Blanks:
- Where manufacturer s specifications are not available, the crane lift limitations assigned to the equipment shall be based on the determinations of a ________ ________ competent in this field and such determinations will be appropriately documented and recorded.
- The employer shall designate a ________ who shall inspect all machinery and equipment prior to each use, and during use, to make sure it is in safe operating condition.
- Accessible areas within the swing radius of the rear of the rotating superstructure of the crane either permanently or temporarily mounted shall be ________ in such a manner as to prevent an employee from being struck or crushed by the crane.
- An accessible ________ of 5BC rating, or higher, shall be available at all operations or cabs of equipment.
- Following assembly and erection of hoists, and before being put in service, an inspection and test of all functions and safety devices shall be made under the supervision of a ________ . A similar inspection and test is required following major alteration of an existing installation. All hoists shall be inspected and tested at not more than ________ month intervals.
- A ________ shall be worn and a lanyard attached to the boom or basket when working from an aerial lift.
Multiple Choice:
- For energized transmission and distribution lines rated 50kV or below, minimum clearance between the lines and any part of the crane or load shall be ________ feet. a. 3 b. 5 c. 10 d. 15
- When crane or derrick personnel platforms are used, they shall be capable of supporting, without failure, their own weight and at least ________ times the maximum intended load. a. 2 b. 3 c. 4 d. 5
- Hoist towers outside the structure shall be enclosed for the full height on the side or sides used for entrance and exit to the structure. At the lowest landing, the enclosure on the sides not used for exit or entrance to the structure shall be enclosed to a height of at least ________ feet. a. 4 b. 8 c. 10 d. 12
General:
- Describe the minimum personal protective equipment required by OSHA for employees receiving the load lifted by a helicopter.


7.7E: Exercises
Solve Rational Inequalities
In the following exercises, solve each rational inequality and write the solution in interval notation.
1. \(\dfrac{x-3}{x+4} \geq 0\)
- Answer
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\((-\infty,-4) \cup[3, \infty)\)
2. \(\dfrac{x+6}{x-5} \geq 0\)
3. \(\dfrac{x+1}{x-3} \leq 0\)
- Answer
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\([-1,3)\)
4. \(\dfrac{x-4}{x+2} \leq 0\)
5. \(\dfrac{x-7}{x-1}>0\)
- Answer
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\((-\infty, 1) \cup(7, \infty)\)
6. \(\dfrac{x+8}{x+3}>0\)
7. \(\dfrac{x-6}{x+5}<0\)
- Answer
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\((-5,6)\)
8. \(\dfrac{x+5}{x-2}<0\)
9. \(\dfrac{3 x}{x-5}<1\)
- Answer
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\(\left(-\dfrac{5}{2}, 5\right)\)
10. \(\dfrac{5 x}{x-2}<1\)
11. \(\dfrac{6 x}{x-6}>2\)
- Answer
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\((-\infty,-3) \cup(6, \infty)\)
12. \(\dfrac{3 x}{x-4}>2\)
13. \(\dfrac{2 x+3}{x-6} \leq 1\)
- Answer
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\([-9,6)\)
14. \(\dfrac{4 x-1}{x-4} \leq 1\)
15. \(\dfrac{3 x-2}{x-4} \geq 2\)
- Answer
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\((-\infty,-6] \cup(4, \infty)\)
16. \(\dfrac{4 x-3}{x-3} \geq 2\)
17. \(\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}\)
- Answer
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\(a=10\)
18. \(\dfrac{1}{x^{2}-4 x-12}>0\)
19. \(\dfrac{3}{x^{2}-5 x+4}<0\)
- Answer
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\((1,4)\)
20. \(\dfrac{4}{x^{2}+7 x+12}<0\)
21. \(\dfrac{2}{2 x^{2}+x-15} \geq 0\)
- Answer
-
\((-\infty,-3) \cup\left(\dfrac{5}{2}, \infty\right)\)
22. \(\dfrac{6}{3 x^{2}-2 x-5} \geq 0\)
23. \(\dfrac{-2}{6 x^{2}-13 x+6} \leq 0\)
- Answer
-
\(\left(-\infty, \dfrac{2}{3}\right) \cup\left(\dfrac{3}{2}, \infty\right)\)
24. \(\dfrac{-1}{10 x^{2}+11 x-6} \leq 0\)
17. \(\dfrac{1}{a}+\dfrac{2}{5}=\dfrac{1}{2}\)
- Answer
-
\(a=10\)
18. \(\dfrac{1}{x^{2}-4 x-12}>0\)
19. \(\dfrac{3}{x^{2}-5 x+4}<0\)
- Answer
-
\((1,4)\)
20. \(\dfrac{4}{x^{2}+7 x+12}<0\)
25. \(\dfrac{1}{2}+\dfrac{12}{x^{2}}>\dfrac{5}{x}\)
- Answer
-
\((-\infty, 0) \cup(0,4) \cup(6, \infty)\)
26. \(\dfrac{1}{3}+\dfrac{1}{x^{2}}>\dfrac{4}{3 x}\)
27. \(\dfrac{1}{2}-\dfrac{4}{x^{2}} \leq \dfrac{1}{x}\)
- Answer
-
\([-2,0) \cup(0,4]\)
28. \(\dfrac{1}{2}-\dfrac{3}{2 x^{2}} \geq \dfrac{1}{x}\)
29. \(\dfrac{1}{x^{2}-16}<0\)
- Answer
-
\((-4,4)\)
30. \(\dfrac{4}{x^{2}-25}>0\)
31. \(\dfrac{4}{x-2} \geq \dfrac{3}{x+1}\)
- Answer
-
\([-10,-1) \cup(2, \infty)\)
32. \(\dfrac{5}{x-1} \leq \dfrac{4}{x+2}\)
Solve an Inequality with Rational Functions
In the following exercises, solve each rational function inequality and write the solution in interval notation.
33. Given the function \(R(x)=\dfrac{x-5}{x-2}\), find the values of \(x\) that make the function less than or equal to 0.
- Answer
-
\((2,5]\)
34. Given the function \(R(x)=\dfrac{x+1}{x+3}\), find the values of \(x\) that make the function less than or equal to 0.
35. Given the function \(R(x)=\dfrac{x-6}{x+2}\), find the values of \(x\) that make the function less than or equal to 0.
- Answer
-
\((-\infty,-2) \cup[6, \infty)\)
36. Given the function \(R(x)=\dfrac{x+1}{x-4}\), find the values of \(x\) that make the function less than or equal to 0.
Writing Exercises
37. Write the steps you would use to explain solving rational inequalities to your little brother.
- Answer
-
Answers will vary
38. Create a rational inequality whose solution is \((-\infty,-2] \cup[4, \infty)\).


3.5: Interval Notation and Infinity
What is infinity? Infinity is not a real number. Infinity is larger than any number that can be imagined. It is an idea of having no bounds. A line is an example of having no bounds. For instance the number line has arrows at the end to represent this idea of having no bounds. The symbol used to represent infinity is \(\infty\). On the left side of the number line is \(−\infty\) and on the right side of the number line is \(\infty\) to describe the boundless behavior of the number line.
| Interval Notation | Number Line |
| a) \((12,\infty)\) | |
| b) \([-5,\infty)\) | |
| c) \((−\infty,4)\) | |
| d) \((−\infty,0]\) | |
| e) \((−\infty,\infty)\) |
Note : Since \(\infty\) is not a real number, it is required to use parentheses (,). \(\infty\) can’t be included in the interval.
Draw a number line that corresponds to the following intervals.
- \((−\infty,5)\)
- \([-5,\infty)\)
- \((−\infty,\infty)\)
- \((-3,\infty)\)
- \((−\infty,-2]\)


10.4: Polygons, Perimeter, and Circumference
- Identify polygons by their sides.
- Identify polygons by their characteristics.
- Calculate the perimeter of a polygon.
- Calculate the sum of the measures of a polygon’s interior angles.
- Calculate the sum of the measures of a polygon’s exterior angles.
- Calculate the circumference of a circle.
- Solve application problems involving perimeter and circumference.
In our homes, on the road, everywhere we go, polygonal shapes are so common that we cannot count the many uses. Traffic signs, furniture, lighting, clocks, books, computers, phones, and so on, the list is endless. Many applications of polygonal shapes are for practical use, because the shapes chosen are the best for the purpose.
Modern geometric patterns in fabric design have become more popular with time, and they are used for the beauty they lend to the material, the window coverings, the dresses, or the upholstery. This art is not done for any practical reason, but only for the interest these shapes can create, for the pure aesthetics of design.
When designing fabrics, one has to consider the perimeter of the shapes, the triangles, the hexagons, and all polygons used in the pattern, including the circumference of any circular shapes. Additionally, it is the relationship of one object to another and experimenting with different shapes, changing perimeters, or changing angle measurements that we find the best overall design for the intended use of the fabric. In this section, we will explore these properties of polygons, the perimeter, the calculation of interior and exterior angles of polygons, and the circumference of a circle.
Identifying Polygons
A polygon is a closed, two-dimensional shape classified by the number of straight-line sides. See Figure 10.82 for some examples. We show only up to eight-sided polygons, but there are many, many more.
If all the sides of a polygon have equal lengths and all the angles are equal, they are called regular polygons . However, any shape with sides that are line segments can classify as a polygon. For example, the first two shapes, shown in Figure
, are both pentagons because they each have five sides and five vertices. The third shape Figure is a hexagon because it has six sides and six vertices. We should note here that the hexagon in Figure is a concave hexagon, as opposed to the first two shapes, which are convex pentagons. Technically, what makes a polygon concave is having an interior angle that measures greater than . They are hollowed out, or cave in, so to speak. Convex refers to the opposite effect where the shape is rounded out or pushed out.While there are variations of all polygons, quadrilaterals contain an additional set of figures classified by angles and whether there are one or more pairs of parallel sides. See Figure
.Identify each polygon.
- Answer
-
- This shape has six sides. Therefore, it is a hexagon.
- This shape has four sides, so it is a quadrilateral.
- This shape has eight sides making it an octagon.
- This is an equilateral triangle, as all three sides are equal.
- This is a rhombus; all four sides are equal.
- This is a regular octagon, eight sides of equal length and equal angles.
Identify the shape.
What polygons make up Figure 10.85?
- Answer
-
Shapes 1 and 5 are hexagons; shapes 2, 3, 4, 6, 7, 9, 10, 12, 13, 14, 15, and 16 are triangles; shapes 8 and 17 are parallelograms; and shape 11 is a trapezoid.
What polygons make up the figure shown?
Perimeter
Perimeter refers to the outside measurements of some area or region given in linear units. For example, to find out how much fencing you would need to enclose your backyard, you will need the perimeter. The general definition of perimeter is the sum of the lengths of the sides of an enclosed region. For some geometric shapes, such as rectangles and circles, we have formulas. For other shapes, it is a matter of just adding up the side lengths.
A rectangle is defined as part of the group known as quadrilaterals, or shapes with four sides. A rectangle has two sets of parallel sides with four angles. To find the perimeter of a rectangle, we use the following formula:
The formula for the perimeter \(P\) of a rectangle is \(P=2L+2W\), twice the length \(L\) plus twice the width \(W\).
For example, to find the length of a rectangle that has a perimeter of 24 inches and a width of 4 inches, we use the formula. Thus,
\[\begin{aligned}
24 & =2 l+2(4) \\
& =2 l+8 \\
24-8 & =2 l \\
16 & =2 l \\
8 & =l
\end{aligned} \nonumber \]
The length is 8 units.
The perimeter of a regular polygon with \(n\) sides is given as
\[P=n \cdot s. \nonumber \]
For example, the perimeter of an equilateral triangle, a triangle with three equal sides, and a side length of 7 cm is \(P=3(7)=21~\text{cm}\).
Find the perimeter of a regular pentagon with a side length of 7 cm (Figure
).- Answer
-
A regular pentagon has five equal sides. Therefore, the perimeter is equal to \(P=5(7)=35~\text{cm}\).
Find the perimeter of a square table that measures 30 inches across from one side to its opposite side.
Find the perimeter of a regular octagon with a side length of 14 cm (Figure
).- Answer
-
A regular octagon has eight sides of equal length. Therefore, the perimeter of a regular octagon with a side length of 14 cm is \(P=8(14)=112~\text{cm}\).
Find the perimeter of a regular heptagon with a side length of 3.2 as shown in the figure.
Sum of Interior and Exterior Angles
To find the sum of the measurements of interior angles of a regular polygon, we have the following formula.
The sum of the interior angles of a polygon with sides is given by
\(S=(n-2) 180^{\circ}\).
For example, if we want to find the sum of the interior angles in a parallelogram, we have
\[\begin{aligned}
S & =(4-2) 180^{\circ} \\
& =2(180)=360^{\circ}
\end{aligned} \nonumber \]
Similarly, to find the sum of the interior angles inside a regular heptagon, we have
\[\begin{aligned}
S & =(7-2) 180^{\circ} \\
& =5(180) \\
& =900^{\circ} .
\end{aligned} \nonumber \]
To find the measure of each interior angle of a regular polygon with \(n\) sides, we have the following formula.
The measure of each interior angle of a regular polygon with \(n\) sides is given by
\[a=\frac{(n-2) 180^{\circ}}{n} \nonumber \]
For example, find the measure of an interior angle of a regular heptagon, as shown in Figure
. We have
\[a=\frac{(7-2) 180^{\circ}}{7}=128.57^{\circ} \nonumber \]
Find the measure of an interior angle in a regular octagon using the formula, and then find the sum of all the interior angles using the sum formula.
- Answer
-
An octagon has eight sides, so \(n=8\).
Step 1 : Using the formula \(a=\frac{(n-2) 180^{\circ}}{8}\) :
\[\begin{aligned}
a & =\frac{(8-2) 180^{\circ}}{8} \\
& =\frac{(6) 180^{\circ}}{8} \\
& =135^{\circ} .
\end{aligned} \nonumber \]So, the measure of each interior angle in a regular octagon is \(135^{\circ}\).
Step 2 : The sum of the angles inside an octagon, so using the formula:
\[\begin{aligned}
S & =(n-2) 180^{\circ} \\
& =(8-2) 180^{\circ} \\
& =6(180) \\
& =1,080^{\circ} .
\end{aligned} \nonumber \]Step 3 : We can test this, as we already know the measure of each angle is \(135^{\circ}\). Thus, \(8\left(135^{\circ}\right)=1,080^{\circ}\).
Find the measure of each interior angle of a regular pentagon and then find the sum of the interior angles.
Use algebra to calculate the measure of each interior angle of the five-sided polygon (Figure
).- Answer
-
Step 1 : Let us find out what the total of the sum of the interior angles should be. Use the formula for the sum of the angles in a polygon with \(n\) sides: \(S=(n-2) 180^{\circ}\). So, \(S=(5-2) 180^{\circ}=540^{\circ}\).
Step 2 : We add up all the angles and solve for \(x\) :
\[\begin{aligned}
5(x+7)+120+(6 x+25)+5(2 x+5)+5(3 x-5) & =540 \\
5 x+6 x+10 x+15 x+180 & =540 \\
36 x & =360 \\
x & =10
\end{aligned} \nonumber \]Step 3 : We can then find the measure of each interior angle:
\[\begin{array}{l}
m \measuredangle A=5(10+7)=85^{\circ} \\
m \measuredangle B=120^{\circ} \\
m \measuredangle C=6(10)+25=85^{\circ} \\
m \measuredangle D=5\left(2^* 10+5\right)=125^{\circ} \\
m \measuredangle E=5\left(3^* 10-5\right)=125^{\circ}
\end{array} \nonumber \]
Find the sum of the measures of the interior angles and then find the measure of each interior angle in the figure shown.
An exterior angle of a regular polygon is an angle formed by extending a side length beyond the closed figure. The measure of an exterior angle of a regular polygon with sides is found using the following formula:
To find the measure of an exterior angle of a regular polygon with sides we use the formula
In Figure
, we have a regular hexagon \(A B C D E F\). By extending the lines of each side, an angle is formed on the exterior of the hexagon at each vertex. The measure of each exterior angle is found using the formula, \(b=\frac{360^{\circ}}{6}=60^{\circ}\).Now, an important point is that the sum of the exterior angles of a regular polygon with \(n\) sides equals \(360^{\circ}\). This implies that when we multiply the measure of one exterior angle by the number of sides of the regular polygon, we should get \(360^{\circ}\). For the example in Figure
, we multiply the measure of each exterior angle, \(60^{\circ}\), by the number of sides, six. Thus, the sum of the exterior angles is \(6\left(60^{\circ}\right)=360^{\circ}\).Find the sum of the measure of the exterior angles of the pentagon (Figure
).- Answer
-
Each individual angle measures \(\frac{360}{5}=72^{\circ}\). Then, the sum of the exterior angles is \(5\left(72^{\circ}\right)=360^{\circ}\).
Find the sum of the measures of the exterior angles in the figure shown.
Circles and Circumference
The perimeter of a circle is called the circumference . To find the circumference, we use the formula
\[C=\pi d \nonumber \]
where \(d\) is the diameter, the distance across the center, or
\[C=2\pi r \nonumber \]
where \(r\) is the radius.
The circumference of a circle is found using the formula \(C=πd\), where \(d\) is the diameter of the circle, or \(C=2\pi r\), where \(𝑟\) is the radius.
The radius is \(1 / 2\) of the diameter of a circle. The symbol \(\pi=3.141592654 \ldots\) is the ratio of the circumference to the diameter. Because this ratio is constant, our formula is accurate for any size circle. See Figure
.Let the radius be equal to 3.5 inches. Then, the circumference is
\[\begin{aligned}
C & =2 \pi(3.5) \\
& =21.99 \mathrm{in} .
\end{aligned} \nonumber \]
Find the circumference of a circle with diameter 10 cm.
- Answer
-
If the diameter is 10 cm, the circumference is \(C=10 \pi =31.42~\text{cm}\).
Find the circumference of a circle with a radius of 2.25 cm.
Find the radius of a circle with a circumference of 12 in.
- Answer
-
If the circumference is 12 in, then the radius is
\[\begin{aligned}
& 12=2 \pi r \\
& \frac{12}{2 \pi}=r=1.91 \mathrm{in} .
\end{aligned} \nonumber \]
Find the radius of a circle with a circumference of \(15.71\,\text{cm.}\)
You decide to make a trim for the window in Figure
. How many feet of trim do you need to buy?- Answer
-
The trim will cover the 6 feet along the bottom and the two 12-ft sides plus the half circle on top. The circumference of a semicircle is ½ the circumference of a circle. The diameter of the semicircle is 6 ft. Then, the circumference of the semicircle would be
1 2 π d = 1 2 π ( 6 ) = 3 π ft = 9.4 ft . 1 2 π d = 1 2 π ( 6 ) = 3 π ft = 9.4 ft . Therefore, the total perimeter of the window is You need to buy 39.4 ft of trim.
To make a trim for the window in the figure shown, you need the perimeter. How many feet of trim do you need to buy?
The overwhelming consensus is that Archimedes (287–212 BCE) was the greatest mathematician of classical antiquity, if not of all time. A Greek scientist, inventor, philosopher, astronomer, physicist, and mathematician, Archimedes flourished in Syracuse, Sicily. He is credited with the invention of various types of pulley systems and screw pumps based on the center of gravity. He advanced numerous mathematical concepts, including theorems for finding surface area and volume. Archimedes anticipated modern calculus and developed the idea of the “infinitely small” and the method of exhaustion. The method of exhaustion is a technique for finding the area of a shape inscribed within a sequence of polygons. The areas of the polygons converge to the area of the inscribed shape. This technique evolved to the concept of limits, which we use today.
One of the more interesting achievements of Archimedes is the way he estimated the number pi, the ratio of the circumference of a circle to its diameter. He was the first to find a valid approximation. He started with a circle having a diameter of 1 inch. His method involved drawing a polygon inscribed inside this circle and a polygon circumscribed around this circle. He knew that the perimeter of the inscribed polygon was smaller than the circumference of the circle, and the perimeter of the circumscribed polygon was larger than the circumference of the circle. This is shown in the drawing of an eight-sided polygon. He increased the number of sides of the polygon each time as he got closer to the real value of pi. The following table is an example of how he did this.
| Sides | Inscribed Perimeter | Circumscribed Perimeter |
|---|---|---|
| 4 | 2.8284 | 4.00 |
| 8 | 3.0615 | 3.3137 |
| 16 | 3.1214 | 3.1826 |
| 32 | 3.1365 | 3.1517 |
| 64 | 3.1403 | 3.1441 |
Archimedes settled on an approximation of after an iteration of 96 sides. Because pi is an irrational number, it cannot be written exactly. However, the capability of the supercomputer can compute pi to billions of decimal digits. As of 2002, the most precise approximation of pi includes 1.2 trillion decimal digits.
The Platonic solids (Figure 10.100) have been known since antiquity. A polyhedron is a three-dimensional object constructed with congruent regular polygonal faces. Named for the philosopher, Plato believed that each one of the solids is associated with one of the four elements: Fire is associated with the tetrahedron or pyramid, earth with the cube, air with the octahedron, and water with the icosahedron. Of the fifth Platonic solid, the dodecahedron, Plato said, “… God used it for arranging the constellations on the whole heaven.”
Plato believed that the combination of these five polyhedra formed all matter in the universe. Later, Euclid proved that exactly five regular polyhedra exist and devoted the last book of the Elements to this theory. These ideas were resuscitated by Johannes Kepler about 2,000 years later. Kepler used the solids to explain the geometry of the universe. The beauty and symmetry of the Platonic solids have inspired architects and artists from antiquity to the present.
In the following exercises, identify the regular polygons.
Find the perimeter of the regular pentagon with side length of 6 cm.
Find the sum of the interior angles of a regular hexagon.
Find the measure of each interior angle of a regular hexagon.
Find the measurements of each angle in the given figure.
Find the circumference of the circle with radius equal to 3 cm.


5.13: Galant Schemata - Opens and Closes
Schemata (pl. of schema) are “stock musical phrases” (Gjerdingen 2007, p. 6) that act as melodic, harmonic, and rhythmic/metric skeletons for passages of music in the Galant style. We can apply the term schema in three specific ways. First, a schema is a prototype —an idealized version of a common pattern. Second, a schema can be an exemplar —a single pattern that resembles the prototype. Third, a schema can be a theory —an explanation of a commonly occurring musical event. All of these ideas go into how we understand schemata. We understand an individual pattern (exemplar) as a version of an ideal general pattern (prototype), and that relationship helps us understand how that pattern is functioning within a particular passage of music (theory).
Schemata are often give names, sometimes based on descriptions from earlier theorists (“Monte, Fonte, Ponte,” were described by Joseph Riepel) or at other times, named after theorists themselves (“Meyer” is named after Leonard Meyer).
Schemata have both internal defining characteristics and normative placements within a series of musical events. Internal characteristics describe a schema’s (1) melodic features—shown with scale degrees; (2) harmonic features—shown with figured bass, and (3) metric features—indicating whether a “stage” in the schema occurs in a weak or strong metric position. A schema’s normative placement describes it temporal location. For example, an “opening gambit” such as the “Meyer” is associated with the beginning of theme, perhaps constituting the whole of the presentation phrase or of a basic idea. A closing “riposte” like the “Prinner” is used as closing gesture.
##Opening Gambits##
###“The Meyer”###
A Meyer is archetypal”opening” schema in the galant style that prolongs the tonic contrapuntally. It begins and closes on a tonic chord, with non-tonic sonorities occuring in the middle two stages. Because the Meyer closes with mi/me in the top voice, it is not so strong as to imply the finality we associate with a cadence. So it works well at the beginning of a theme. This melodic skeleton is the most important feature of the Meyer, and occurs invariably even when the bass is somewhat altered.
When each sonority receives one “measure” of music, it is commonly found in the presentation or antecedent part of an opening theme. If those stages occur at the rate of two per measure, the Meyer may form a basic idea that would be followed by a closing gesture, such as the “Prinner” described below.
A fairly exhaustive list of opening schemata can be found on the Galant schemata – summary page. Many of them follow the same general pattern as the Meyer. For example, “The Pastorella,” “The Jupiter,” and “The Aprile,” all prolong tonic by moving away for the two middle stages before a tonic return. Like the Meyer, they are ideal prototypes for the first phrase of an opening theme. As with the Meyer, the bass and harmonic structure are less fixed than is the melody. The two central stages may articulate dominant harmony in all three schemas, and the second stage is also commonly accompanied by predominant harmony.
##Closing gestures##
###“The Prinner”###
The Prinner is a typical response to an opening schema. It often occurs in a sentence (or a hybrid theme type) as the continuation phrase. Or if the harmonic rhythm is quicker, it may be used as the basis for a constrasting idea in an antecedent or consequent phrase.
The Prinner has four stages corresponding to four bass notes: fa –mi/me – re – do. The skeleton of the Prinner’s melody typically accompanies the bass in parallel tenths: la/le – sol – fa – mi/me. Harmonically, the fa and do bass notes tend to take 5/3 chords while the two middle bass notes, mi/me and re, take 6/3 chords.
Some Prinner exemplars insert a sol bass note before the last chord, resulting in an authentic cadence. Prinners that operate as continuation phrases often contain this move in order to end the sentence or hybrid theme satisfactorily.
To complete a modulation to the dominant, the Prinner schema can be transposed up a fifth. When this occurs, the first stage of the Prinner (the 5/3 chord on fa) is the tonic of the home key and the subdominant of the dominant key, making a particularly smooth transition. Modulating Prinners are used in sentence or hybrid themes either to modulate to the dominant key or to effect a strong half cadence. They also commonly appear at the beginning of the Transition (TR) zone in a sonata movement, effecting the same move to the dominant.
For more continuation/cadential schemata, see the Galant schemata – summary page.
References
Gjerdingen, Robert O. Music in the Galant Style . Oxford University Press, 2007.


3.2: Definitions of Homomorphisms and Isomorphisms
Definition: Homomorphism and Isomorphism
Let \(\langle S,*\rangle\) and \(\langle S',*'\rangle\) be binary structures. A function \(\phi\) from \(S\) to \(S'\) is a homomorphism if
\begin{equation*} \phi(a* b)=\phi(a)*'\phi(b) \end{equation*}
for all \(a,b\in S\text{.}\) An isomorphism is a homomorphism that is also a bijection.
Intuitively, you can think of a homomorphism \(\phi\) as a “structure-preserving” map: if you multiply and then apply \(\phi\text{,}\) you get the same result as when you first apply \(\phi\) and then multiply. Isomorphisms, then, are both structure-preserving and cardinality-preserving.
Note
We may omit the \(*\) and \(*'\text{,}\) as per our group conventions, but we include them here to emphasize that the operations in the structures may be distinct from one another. When we omit them and write \(\phi(st)=\phi(s)\phi(t)\text{,}\) then it is the writers' and readers' responsibility to keep in mind that \(s\) and \(t\) are being operated together using the operation in \(S\text{,}\) while \(\phi(s)\) and \(\phi(t)\) are being operated together using the operation in \(S'\text{.}\)
Remark
There may be more than one homomorphism [isomorphism] from one binary structure to another (see Example \(3.2.1\)).
Example \(\PageIndex{1}\)
For each of the following, decide whether or not the given function \(\phi\) from one binary structure to another is a homomorphism, and, if so, if it is an isomorphism. Prove or disprove your answers! For Parts 6 and 7, \(C^0\) is the set of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\text{;}\) \(C^1\) is the set of all differentiable functions from \(\mathbb{R}\) to \(\mathbb{R}\) whose derivatives are continuous; and each \(+\) indicates pointwise addition on \(C^0\) and \(C^1\text{.}\)
-
\(\phi:\langle \mathbb{Z},+\rangle \to \langle \mathbb{Z},+\rangle\) defined by \(\phi(x)=x\text{;}\)
-
\(\phi:\langle \mathbb{Z},+\rangle \to \langle \mathbb{Z},+\rangle\) defined by \(\phi(x)=-x\text{;}\)
-
\(\phi:\langle \mathbb{Z},+\rangle \to \langle \mathbb{Z},+\rangle\) defined by \(\phi(x)=2x\text{;}\)
-
\(\phi:\langle \mathbb{R},+\rangle \to \langle \mathbb{R}^+,\cdot\,\rangle\) defined by \(\phi(x)=e^x\text{;}\)
-
\(\phi:\langle \mathbb{R},+\rangle \to \langle \mathbb{R}^*,\cdot\,\rangle\) defined by \(\phi(x)=e^x\text{;}\)
-
\(\phi:\langle C^1,+\rangle \to \langle C^0,+\rangle\) defined by \(\phi(f)=f'\) (the derivative of \(f\));
-
\(\phi:\langle C^0,+\rangle \to \langle \mathbb{R},+\rangle\) defined by \(\phi(f)=\displaystyle{\int_0^1 f(x)\, dx}\text{.}\)
Example \(\PageIndex{2}\)
Let \(\langle G,\;\cdot\rangle\) be a group and let \(a\in G\text{.}\) Then the function \(c_a\) from \(G\) to \(G\) defined by \(c_a(x)=axa^{-1}\) (for all \(x\in G\)) is a homomorphism. Indeed, let \(x,y\in G\text{.}\) Then
\(\begin{array} &c_a(xy)& =a(xy)a^{-1}\\ & =(ax)e(ya^{-1})\\ & =(ax)(a^{-1}a)(ya^{-1})\\ & =(axa^{-1})(aya^{-1})\\ & =c_a(x)c_a(y). \end{array}\)
The homomorphism \(c_a\) is called conjugation by \(a\).
Definition: Endomorphism and Automorphism
Homomorphisms from a group \(G\) to itself are called endomorphisms , and isomorphisms from a group to itself are called automorphisms .
It can be shown that conjugation by any element \(a\) of a group \(G\) is a bijection from \(G\) to itself (can you prove this?), so such conjugation is an automorphism of \(G\text{.}\) (Beware: Some texts use “conjugation by \(a\)” to refer to the function \(x\mapsto a^{-1}xa\text{.}\)) Both versions of conjugation by \(a\) in group \(G\) are automorphisms of \(G\text{.}\))
We end with a theorem stating basic facts about homomorphisms from one group to another. ( Note. This doesn't apply to arbitrary binary structures, which may or may not even have identity elements.)
Theorem \(\PageIndex{1}\)
Let \(\langle G,\cdot\rangle\) and \(\langle G',\cdot'\rangle\) be groups with identity elements \(e\) and \(e'\text{,}\) respectively, and let \(\phi\) be a homomorphism from \(G\) to \(G'\text{.}\) Then:
- \(\phi(e)=e'\text{;}\) and
- For every \(a\in G\text{,}\) \(\phi(a)^{-1}=\phi(a^{-1})\text{.}\)
- Proof
-
For Part 1, note that
\(\begin{array}&ϕ(e)⋅′e′&=ϕ(e) &(\text{by definition of }e′)\\ &=ϕ(e⋅e) &(\text{by definition of } e) \\&=ϕ(e)⋅′ϕ(e) &(\text{since \(ϕ\) is a homomorphism}). \end{array} \)
Thus, by left cancellation, \(e′=ϕ(e)\). The proof of Part 2 is left as an exercise for the reader.


6.2: The Standard Normal Distribution
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The standard normal distribution is a normal distribution of standardized values called z-scores . A z -score is measured in units of the standard deviation.
The mean for the standard normal distribution is 0, and the standard deviation is 1. What this does is dramatically simplify the mathematical calculation of probabilities. Take a moment and substitute 0 and 1 in the appropriate places in the above formula and you can see that the equation collapses into one that can be much more easily solved using integral calculus. The transformation \(z=\frac{x-\mu}{\sigma}\) produces the distribution \(Z \sim N(0, 1)\). The value \(x\) in the given equation comes from a known normal distribution with known mean \(\mu\) and known standard deviation \(\sigma\). The z-score tells how many standard deviations a particular \(x\) is away from the mean.
Z-Scores
If \(X\) is a normally distributed random variable and \(X \sim N(\mu, \sigma)\), then the z-score for a particular value \(x\) is:
\[z=\frac{x-\mu}{\sigma}\nonumber\]
The z -score tells you how many standard deviations the value \(\bf{x}\) is above (to the right of) or below (to the left of) the mean, \(\bf{\mu}\) . Values of \(x\) that are larger than the mean have positive z-scores, and values of \(x\) that are smaller than the mean have negative z-scores. If \(x\) equals the mean, then \(x\) has a z-score of zero.
Example \(\PageIndex{1}\)
Suppose \(X \sim N(5, 6)\). This says that \(X\) is a normally distributed random variable with mean \(\mu = 5\) and standard deviation \(\sigma = 6\). Suppose \(x = 17\). Then:
\[z=\frac{x-\mu}{\sigma}=\frac{17-5}{6}=2\nonumber\]
This means that \(x = 17\) is two standard deviations \((2\sigma)\) above or to the right of the mean \(\mu = 5\).
Now suppose \(x = 1\). Then: \(z=\frac{x-\mu}{\sigma}=\frac{1-5}{6}=-0.67\) (rounded to two decimal places)
This means that \(\bf{x = 1}\) is 0.67 standard deviations \(\bf{(–0.67\sigma)}\) below or to the left of the mean \(\bf{\mu = 5}\) .
The Empirical Rule
If \(X\) is a random variable and has a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), then the Empirical Rule states the following:
- About 68% of the \(x\) values lie between \(–1\sigma\) and \(+1\sigma\) of the mean \(\mu\) (within one standard deviation of the mean).
- About 95% of the \(x\) values lie between \(–2\sigma\) and \(+2\sigma\) of the mean \(\mu\) (within two standard deviations of the mean).
- About 99.7% of the \(x\) values lie between \(–3\sigma\) and \(+3\sigma\) of the mean \(\mu\) (within three standard deviations of the mean). Notice that almost all the x values lie within three standard deviations of the mean.
- The z-scores for \(+1\sigma\) and \(–1\sigma\) are \(+1\) and \(–1\), respectively.
- The z-scores for \(+2\sigma\) and \(–2\sigma\) are \(+2\) and \(–2\), respectively.
- The z-scores for \(+3\sigma\) and \(–3\sigma\) are \(+3\) and \(–3\) respectively.
Example \(\PageIndex{2}\)
Suppose \(X\) has a normal distribution with mean 50 and standard deviation 6.
- About 68% of the \(X\) values lie within one standard deviation of the mean. Therefore, about 68% of the \(X\) values lie between \(–1\sigma = (–1)(6) = –6\) and \(1\sigma = (1)(6) = 6\) of the mean 50. The values \(50 – 6 = 44\) and \(50 + 6 = 56\) are within one standard deviation from the mean 50. The z-scores are –1 and +1 for 44 and 56, respectively.
- About 95% of the \(X\) values lie within two standard deviations of the mean. Therefore, about 95% of the \(X\) values lie between \(–2\sigma = (–2)(6) = –12\) and \(2\sigma = (2)(6) = 12\). The values \(50 – 12 = 38\) and \(50 + 12 = 62\) are within two standard deviations from the mean 50. The z-scores are –2 and +2 for 38 and 62, respectively.
- About 99.7% of the \(X\) values lie within three standard deviations of the mean. Therefore, about 99.7% of the \(X\) values lie between \(–3\sigma = (–3)(6) = –18\) and \(3\sigma = (3)(6) = 18\) of the mean 50. The values \(50 – 18 = 32\) and \(50 + 18 = 68\) are within three standard deviations from the mean 50. The z-scores are –3 and +3 for 32 and 68, respectively.