9.14: Current Issues in Corrections- Aging and Overcrowding
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- Alison S. Burke, David Carter, Brian Fedorek, Tiffany Morey, Lore Rutz-Burri, & Shanell Sanchez
- Southern Oregon University via OpenOregon
One of the side effects of lengthier sentences is that the individuals in prison getting older, in prison. Thus, the amounts of individuals in prisons over 55 years old has dramatically increased. As McKillop and Boucher (2018) relate in the graphic below, based on BJS data, there has been a 280% increase in prisoners, age 55 and older. [1]
And as the title of their article depicts, there is a growing cost within this subpopulation of inmates. McKillop and Boucher (2018) relay that the cost of this group of inmates can be upwards of three times the cost of the normal inmate (30k to 100k per inmate). [2]
Beyond just the cost of these inmates, a more philosophical question has risen, in regards to how to treat prisoners as they enter their last phase in life. Some have articulated for compassionate releases for individuals that are entering hospice care or in need of assisted living conditions. Other articulate that this is unfair to put the burden on the inmate, as they have been incarcerated for long periods of time and have few self-support options available. In a powerful documentary on this matter, Edgar Barens details these issues. Information about this film, The Prison Terminal (2013), can be seen here: https://www.prisonterminal.com/ .
Overcrowding
These issues and others have all contributed to the rising correctional population in the United States. It is estimated that we have over 8 million people in correctional control, and that number does not seem to be subsiding. Yes, there are reductions in certain areas, such as a decline in the prison population in the last few years, but this does not mean that they are not still under control. In one of the more detailed examples of just where individuals are at in corrections, Alexi Jones (2018) of the Prison Policy Initiative provides a graphic, based on State and Federal data to demonstrate this impact.
Not only does this graphic demonstrate the overall volume of correctional control, but it also highlights how states are handling their populations differentially. The second half of Jones’ (2018) [3] report details the volume of individuals with each state. Please take a moment to review the last portion of this report to see how many are under correctional control, found here: https://www.prisonpolicy.org/reports/correctionalcontrol2018.html#statedata
Prison overcrowding is problematic for multiple reasons. First, when there are too many individuals (especially antisocial ones) within a facility, there are more assaults and injuries that occur within the institution. Moreover, there is a safety concern for not only the inmate on inmate violence but also inmate on staff crimes. Second, the more people you have in a facility, the faster that facility wears down. Operating a jail or prison at maximum (or over maximum) capacity causes more items to break or wear out within the facility at a fast rate. Finally, when individuals are unable to access adequate health care because of the excessively long waits, due to overcrowding, it is a violation of their constitutional rights.
As prisoners, we (the public and the State) have a responsibility to house and properly care for the prisoners overseen. This is not to say that offenders are getting premier care, but that they are at least receiving a modicum of care. When this low level of care is deliberately denied due to excessive volumes of individuals, it is a violation of a person’s 8th amendment rights against cruel and unusual punishment. As was found in the case of Estelle v. Gamble, (1976). [4] This similar issue was presented in California over ten years ago. A three-judge panel ruled with the prisoners, citing the need for California to reduce its prison population to a level where the individuals could effectively be managed, and cared for [emphasis on the latter]. Dealing with overcrowding is a constant issue for most prisons and jails. Some have resolved to release more out into the community at a higher volume, on parole or just release. However, this too has its own set of problems, as reentry is now becoming the current issue within corrections.
- McKillop, M., & Boucher, A. (2018). Aging prison populations drive up costs. Older individuals have more chronic illnesses and other ailments that necessitate greater spending. Available at: https://www.pewtrusts.org/en/research-and-analysis/articles/2018/02/20/aging-prison-populations-drive-up-costs ↵
- McKillop, M., & Boucher, A. (2018). Aging prison populations drive up costs. Older individuals have more chronic illnesses and other ailments that necessitate greater spending. Available at: https://www.pewtrusts.org/en/research-and-analysis/articles/2018/02/20/aging-prison-populations-drive-up-costs ↵
- Jones, A. (2018). Correctional control 2018: Incarceration and supervision by state Prison Policy Initiative. Available at: https://www.prisonpolicy.org/reports/correctionalcontrol2018.html#statedata . ↵
- Estelle v. Gamble , 429 U.S. 97 (1976). ↵


13.2.1: Tropical Forests
The Tropical Rain forest
The Tropical Rain forest contains trees standing 30 to 55 meters in height, creating a continuous canopy of foliage. The enclosed canopy shades the forest floor, creating an open forest formation. Few pure stands of trees exist in the rain forest. Instead, individual trees are widely dispersed throughout the forest. Mahogany, teak and other tropical hardwoods are harvested for creating fine furniture. The lush vegetation and great animal diversity should be expected in the tropical rain forest climate.
Peering into the canopy reveals a multistory appearance of broad leaf, evergreen vegetation called "selva". The typical three-tiered zonation includes an emergent layer of solitary, giant trees reaching heights of 55 meters (180 ft) for the sunlight they require. Beneath is an intermediate zone ( canopy ) of continuous foliage 9 to 18 meters (30 to 60 ft) high. The forest floor is relatively open as the shade of the closed canopy inhibits plant growth. It is a damp and dim world as one walks through the aroma of decaying vegetation. Giant woody vines called " lianas " snake their way up the trunks of trees. Epiphytes ("air plants") , like the brilliantly colored bromeliad, grow in the hollows of trees or the upper surfaces of horizontally-growing branches capturing nutrients and moisture from the air.
A cloud forest exists on extremely moist mountain slopes above the elevation of the true rain forest. The cloud forest differs from the rain forest found at lower elevations in that trees are much shorter and the forest floor is virtually impenetrable.
Plants have adapted to this environment in unique ways:
“The canopy itself, the ceiling of the jungle, is a dense continuous layer of greenery some 6 or 7 meters deep. Each leaf is accurately angled to ensure that it will collect the maximum amount of light. Many have a special joint at the base of the stalk that enables them to twist and follow the sun as it swings overhead from east to west each day. All except the topmost layer is screened from the wind, so the around them is warm and humid”.
The Living Planet
D. Attenborough
The rain forest is a treasure trove of different animal and plant species. The numerous species that inhabit the rain forests are not well documented. The intense precipitation of the tropical rain forest climate heavily leaches the soil. Oxisol soil common to the rain forest is relatively infertile due to intense weathering and a lack of available nutrients. Deforestation and habitat destruction is severely crippling the rain forest ecosystem. View the " Secret Life of the Rainforest" provided by the Smithsonian Channel.
Tropical Monsoon/Seasonal Forest and Shrub
The Tropical Monsoon/Seasonal Forest and Shrub contains trees of smaller stature than those found in the rain forest. The Monsoon forest may include deciduous trees, as well as, broadleaf evergreen trees reflecting the seasonal precipitation of the monsoon climate. The trees of the Monsoon Forest have a more open canopy than the rain forest, creating a dense, closed forest at the floor, or what we think of as a "tropical jungle". The thick surface undergrowth makes navigating through the forest difficult. Jungle growth is also found along streams, and in openings created by humans.
Like the tropical rain forest, the monsoon forest supports a diversity of plant and animal species. And like the rain forest, it's biotic system is being stressed by human activities.
Investigate the effects of human activities on the tropical forest biome by "Digging Deeper: Human Activities and the tropical Forests" or continue to the next topic.
Species Endangerment in the Tropical Forests
Habitat encroachment and destruction have been a primary cause of declining animal and plant species, threatening many with extinction. For example, habitat encroachment and sport hunting has led to the rapid decline of the Bengal tiger and are considered a threaten species. Solitary animals, tigers can weigh up to 500 pounds and measure nine feet in length. The main diet of the tiger consists of deer, antelope, and wild pig.
Logging and agriculture in the densely populated countries of southeast Asia have literally squeezed the tiger out of its home. Human encroachment on the tiger's habitat has led to more frequent conflicts between tigers and humans. However, human fatalities from tigers are a result of injured or sick animals to weak to hunt wild animals.
Sport hunting and killing to use body parts for traditional medicines since 1900 has significantly contributed to the decline of the tiger. In the early 1900's it was estimated that 100,000 tigers roamed Asia. In 1900, approximately 40,000 lived in India. By the 1960's this number fell to 4,000. By the early seventies fewer that 2,000 remained in India. Recognizing the potential loss of their national treasure, intensive conservation efforts by the Indian government and the world community has doubled their numbers in recent years. The world-wide population of tigers is estimated to be 5,100 to 7,500 individuals (World Wildlife Fund).
Illegal hunting to support the bush meat trade has taken a devastating toll on animal species. The bushmeat trade ranks among the greatest threats to tropical wildlife according to some environmentalists. Research has shown that increased poaching in Ghana has resulted in significant declines in 41 wild animal species. This research speculates that the bushmeat trade has grown partly in response to over fishing off West Africa by foreign and domestic industrial fleets. With dwindling resources for protein, people turn to the forest for food.
Deforestation in the Tropical Forests
Human activity has drastically altered the natural distribution of forests through history. Deforestation and habitat destruction is severely crippling the rain forest ecosystem. Rain forests are being destroyed at a rate of 78 million acres (31 million hectares) per year; an area larger than Poland. With habitat destruction comes loss of species . The World Resources Institute predicts that deforestation rates of 15 million hectares would reduce species in the closed canopy forests by 35% by 2040. In Brazil, the estimated average rate of destruction between 1979 - 1990 was 5.4 million acres per year. In 2003 a record 10,000 square miles of Brazil was cleared (Lobe, 2004).
Deforestation causes a multitude of effects on the natural environment as shown below. Vegetation is degraded due to a loss of nutrient-rich litter and microorganisms to decompose organic matter. The loss of shade accelerates leaching, soil erosion and drying. The hard surface caused by baking impedes water infiltration causing excess runoff and flooding. In addition to the effect on the local vegetation and hydrology, forest removal impacts atmospheric composition, evapotranspiration, and precipitation.
Soil degradation from forest clearing for agriculture happens quickly. Soil fertility can decrease by 80% in only a few years after forest removal.
Soil fertility loss due to forest clearing ( Drew,1983 )
|
Land use |
Organic Content |
Cation
|
Nitrogen |
Phosphorus |
|---|---|---|---|---|
|
Virgin Forest |
100 |
100 |
100 |
100 |
|
1 Year after clearing, unused |
104 |
82 |
66 |
120 |
|
After 2 years cultivation |
46 |
51 |
36 |
75 |
Government policies in some countries have fostered the exploitation of forest resources. Encouraging the population to develop forest land has opened the forest to continued degradation. In the 1990's the Bolivian Government began a large-scale program to increase the rate of forest removal for commercial agriculture (primarily soy and sugar cane, but also cocoa) on the Amazon Basin side of the Andean highlands. The dramatic effect of this action is seen in Figures BE21a and BE21b. Forest clearings began as small tracts perpendicular to access roads taking on a herringbone pattern when seen from the Space Shuttle in 1995. By 2008, the cleared area has greatly expanded. The impact of clearing on soil erosion and stream sediment load is evident in the brown, silt-laden Rio Parapetí that flows through the region.
The heavy seasonal rain of the monsoon forest poses a great danger, generating rapid runoff, mudflows and flash flooding. A result of unregulated deforestation, rivers become choked with sediment degrading the aquatic environment.
Deforestation has had a significant impact on indigenous cultures too. Six to nine million indigenous people inhabited the Brazilian rain forest in 1500. Today less than 1% of Brazil's 177 million people are full-blooded indigenous Indians (Source: US State Department). The loss of indigenous cultures destroys a wealth of knowledge about the environment in which they lived.


19.5: Cardiac Physiology
By the end of this section, you will be able to:
- Relate heart rate to cardiac output
- Describe the effect of exercise on heart rate
- Identify cardiovascular centers and cardiac reflexes that regulate heart function
- Describe factors affecting heart rate
- Distinguish between positive and negative factors that affect heart contractility
- Summarize factors affecting stroke volume and cardiac output
- Describe the cardiac response to variations in blood flow and pressure
The autorhythmicity inherent in cardiac cells keeps the heart beating at a regular pace; however, the heart is regulated by and responds to outside influences as well. Neural and endocrine controls are vital to the regulation of cardiac function. In addition, the heart is sensitive to several environmental factors, including electrolytes.
Resting Cardiac Output
Cardiac output (CO) is a measurement of the amount of blood pumped by each ventricle in one minute. To calculate this value, multiply stroke volume (SV) , the amount of blood pumped by each ventricle, by heart rate (HR) , in contractions per minute (or beats per minute, bpm). It can be represented mathematically by the following equation:
CO = HR × SV
SV is normally measured using an echocardiogram to record EDV and ESV, and calculating the difference: SV = EDV – ESV. SV can also be measured using a specialized catheter, but this is an invasive procedure and far more dangerous to the patient. A mean SV for a resting 70-kg (150-lb) individual would be approximately 70 mL. There are several important variables, including size of the heart, physical and mental condition of the individual, sex, contractility, duration of contraction, preload or EDV, and afterload or resistance. Normal range for SV would be 55–100 mL. An average resting HR would be approximately 75 bpm but could range from 60–100 in some individuals.
Using these numbers, the mean CO is 5.25 L/min, with a range of 4.0–8.0 L/min. Remember, however, that these numbers refer to CO from each ventricle separately, not the total for the heart. Factors influencing CO are summarized in Figure \(\PageIndex{1}\).
SVs are also used to calculate ejection fraction , which is the portion of the blood that is pumped or ejected from the heart with each contraction. To calculate ejection fraction, SV is divided by EDV. Despite the name, the ejection fraction is normally expressed as a percentage. Ejection fractions range from approximately 55–70 percent, with a mean of 58 percent.
Exercise and Maximum Cardiac Output
In healthy young individuals, HR may increase to 150 bpm during exercise. SV can also increase from 70 to approximately 130 mL due to increased strength of contraction. This would increase CO to approximately 19.5 L/min, 4–5 times the resting rate. Top cardiovascular athletes can achieve even higher levels. At their peak performance, they may increase resting CO by 7–8 times.
Since the heart is a muscle, exercising it increases its efficiency. The difference between maximum and resting CO is known as the cardiac reserve . It measures the residual capacity of the heart to pump blood.
Heart Rates
HRs vary considerably, not only with exercise and fitness levels, but also with age. Newborn resting HRs may be 120 bpm. HR gradually decreases until young adulthood and then gradually increases again with age.
Maximum HRs are normally in the range of 200–220 bpm, although there are some extreme cases in which they may reach higher levels. As one ages, the ability to generate maximum rates decreases. This may be estimated by taking the maximal value of 220 bpm and subtracting the individual’s age. So a 40-year-old individual would be expected to hit a maximum rate of approximately 180, and a 60-year-old person would achieve a HR of 160.
Disorders of the...
Heart: Abnormal Heart Rates
For an adult, normal resting HR will be in the range of 60–100 bpm. Bradycardia is the condition in which resting rate drops below 60 bpm, and tachycardia is the condition in which the resting rate is above 100 bpm. Trained athletes typically have very low HRs. If the patient is not exhibiting other symptoms, such as weakness, fatigue, dizziness, fainting, chest discomfort, palpitations, or respiratory distress, bradycardia is not considered clinically significant. However, if any of these symptoms are present, they may indicate that the heart is not providing sufficient oxygenated blood to the tissues. The term relative bradycardia may be used with a patient who has a HR in the normal range but is still suffering from these symptoms. Most patients remain asymptomatic as long as the HR remains above 50 bpm.
Bradycardia may be caused by either inherent factors or causes external to the heart. While the condition may be inherited, typically it is acquired in older individuals. Inherent causes include abnormalities in either the SA or AV node. If the condition is serious, a pacemaker may be required. Other causes include ischemia to the heart muscle or diseases of the heart vessels or valves. External causes include metabolic disorders, pathologies of the endocrine system often involving the thyroid, electrolyte imbalances, neurological disorders including inappropriate autonomic responses, autoimmune pathologies, over-prescription of beta blocker drugs that reduce HR, recreational drug use, or even prolonged bed rest. Treatment relies upon establishing the underlying cause of the disorder and may necessitate supplemental oxygen.
Tachycardia is not normal in a resting patient but may be detected in pregnant people or individuals experiencing extreme stress. In the latter case, it would likely be triggered by stimulation from the limbic system or disorders of the autonomic nervous system. In some cases, tachycardia may involve only the atria. Some individuals may remain asymptomatic, but when present, symptoms may include dizziness, shortness of breath, lightheadedness, rapid pulse, heart palpations, chest pain, or fainting (syncope). While tachycardia is defined as a HR above 100 bpm, there is considerable variation among people. Further, the normal resting HRs of children are often above 100 bpm, but this is not considered to be tachycardia. Many causes of tachycardia may be benign, but the condition may also be correlated with fever, anemia, hypoxia, hyperthyroidism, hypersecretion of catecholamines, some cardiomyopathies, some disorders of the valves, and acute exposure to radiation. Elevated rates in an exercising or resting patient are normal and expected. Resting rate should always be taken after recovery from exercise. Treatment depends upon the underlying cause but may include medications, implantable cardioverter defibrillators, ablation, or surgery.
Correlation Between Heart Rates and Cardiac Output
Initially, physiological conditions that cause HR to increase also trigger an increase in SV. During exercise, the rate of blood returning to the heart increases. However as the HR rises, there is less time spent in diastole and consequently less time for the ventricles to fill with blood. Even though there is less filling time, SV will initially remain high. However, as HR continues to increase, SV gradually decreases due to decreased filling time. CO will initially stabilize as the increasing HR compensates for the decreasing SV, but at very high rates, CO will eventually decrease as increasing rates are no longer able to compensate for the decreasing SV. Consider this phenomenon in a healthy young individual. Initially, as HR increases from resting to approximately 120 bpm, CO will rise. As HR increases from 120 to 160 bpm, CO remains stable, since the increase in rate is offset by decreasing ventricular filling time and, consequently, SV. As HR continues to rise above 160 bpm, CO actually decreases as SV falls faster than HR increases. So although aerobic exercises are critical to maintain the health of the heart, individuals are cautioned to monitor their HR to ensure they stay within the target heart rate range of between 120 and 160 bpm, so CO is maintained. The target HR is loosely defined as the range in which both the heart and lungs receive the maximum benefit from the aerobic workout and is dependent upon age.
Cardiovascular Centers
Nervous control over HR is centralized within the two paired cardiovascular centers of the medulla oblongata (Figure \(\PageIndex{2}\)). The cardioaccelerator regions stimulate activity via sympathetic stimulation of the cardioaccelerator nerves, and the cardioinhibitory centers decrease heart activity via parasympathetic stimulation as one component of the vagus nerve, cranial nerve X. During rest, both centers provide slight stimulation to the heart, contributing to autonomic tone . This is a similar concept to tone in skeletal muscles. Normally, vagal stimulation predominates as, left unregulated, the SA node would initiate a sinus rhythm of approximately 100 bpm.
Both sympathetic and parasympathetic stimulations flow through a paired complex network of nerve fibers known as the cardiac plexus near the base of the heart. The cardioaccelerator center also sends additional fibers, forming the cardiac nerves via sympathetic ganglia (the cervical ganglia plus superior thoracic ganglia T1–T4) to both the SA and AV nodes, plus additional fibers to the atria and ventricles. The ventricles are more richly innervated by sympathetic fibers than parasympathetic fibers. Sympathetic stimulation causes the release of the neurotransmitter norepinephrine (NE) at the neuromuscular junction of the cardiac nerves. NE shortens the repolarization period, thus speeding the rate of depolarization and contraction, which results in an increase in HR. It opens chemical- or ligand-gated sodium and calcium ion channels, allowing an influx of positively charged ions.
NE binds to the beta-1 receptor. Some cardiac medications (for example, beta blockers) work by blocking these receptors, thereby slowing HR and are one possible treatment for hypertension. Overprescription of these drugs may lead to bradycardia and even stoppage of the heart.
Parasympathetic stimulation originates from the cardioinhibitory region with impulses traveling via the vagus nerve (cranial nerve X). The vagus nerve sends branches to both the SA and AV nodes, and to portions of both the atria and ventricles. Parasympathetic stimulation releases the neurotransmitter acetylcholine (ACh) at the neuromuscular junction. ACh slows HR by opening chemical- or ligand-gated potassium ion channels to slow the rate of spontaneous depolarization, which extends repolarization and increases the time before the next spontaneous depolarization occurs. Without any nervous stimulation, the SA node would establish a sinus rhythm of approximately 100 bpm. Since resting rates are considerably less than this, it becomes evident that parasympathetic stimulation normally slows HR. This is similar to an individual driving a car with one foot on the brake pedal. To speed up, one need merely remove one’s foot from the break and let the engine increase speed. In the case of the heart, decreasing parasympathetic stimulation decreases the release of ACh, which allows HR to increase up to approximately 100 bpm. Any increases beyond this rate would require sympathetic stimulation. Figure \(\PageIndex{3}\): illustrates the effects of parasympathetic and sympathetic stimulation on the normal sinus rhythm.
Input to the Cardiovascular Center
The cardiovascular center receives input from a series of visceral receptors with impulses traveling through visceral sensory fibers within the vagus and sympathetic nerves via the cardiac plexus. Among these receptors are various proprioreceptors, baroreceptors, and chemoreceptors, plus stimuli from the limbic system. Collectively, these inputs normally enable the cardiovascular centers to regulate heart function precisely, a process known as cardiac reflexes . Increased physical activity results in increased rates of firing by various proprioreceptors located in muscles, joint capsules, and tendons. Any such increase in physical activity would logically warrant increased blood flow. The cardiac centers monitor these increased rates of firing, and suppress parasympathetic stimulation and increase sympathetic stimulation as needed in order to increase blood flow.
Similarly, baroreceptors are stretch receptors located in the aortic sinus, carotid bodies, the venae cavae, and other locations, including pulmonary vessels and the right side of the heart itself. Rates of firing from the baroreceptors represent blood pressure, level of physical activity, and the relative distribution of blood. The cardiac centers monitor baroreceptor firing to maintain cardiac homeostasis, a mechanism called the baroreceptor reflex . With increased pressure and stretch, the rate of baroreceptor firing increases, and the cardiac centers decrease sympathetic stimulation and increase parasympathetic stimulation. As pressure and stretch decrease, the rate of baroreceptor firing decreases, and the cardiac centers increase sympathetic stimulation and decrease parasympathetic stimulation.
There is a similar reflex, called the atrial reflex or Bainbridge reflex , associated with varying rates of blood flow to the atria. Increased venous return stretches the walls of the atria where specialized baroreceptors are located. However, as the atrial baroreceptors increase their rate of firing and as they stretch due to the increased blood pressure, the cardiac center responds by increasing sympathetic stimulation and inhibiting parasympathetic stimulation to increase HR. The opposite is also true.
Increased metabolic byproducts associated with increased activity, such as carbon dioxide, hydrogen ions, and lactic acid, plus falling oxygen levels, are detected by a suite of chemoreceptors innervated by the glossopharyngeal and vagus nerves. These chemoreceptors provide feedback to the cardiovascular centers about the need for increased or decreased blood flow, based on the relative levels of these substances.
The limbic system can also significantly impact HR related to emotional state. During periods of stress, it is not unusual to identify higher than normal HRs, often accompanied by a surge in the stress hormone cortisol. Individuals experiencing extreme anxiety may manifest panic attacks with symptoms that resemble those of heart attacks. These events are typically transient and treatable. Meditation techniques have been developed to ease anxiety and have been shown to lower HR effectively. Doing simple deep and slow breathing exercises with one’s eyes closed can also significantly reduce this anxiety and HR.
Disorders of the...
Heart: Broken Heart Syndrome
Extreme stress from such life events as the death of a loved one, an emotional break up, loss of income, or foreclosure of a home may lead to a condition commonly referred to as broken heart syndrome. This condition may also be called Takotsubo cardiomyopathy, transient apical ballooning syndrome, apical ballooning cardiomyopathy, stress-induced cardiomyopathy, Gebrochenes-Herz syndrome, and stress cardiomyopathy. The recognized effects on the heart include congestive heart failure due to a profound weakening of the myocardium not related to lack of oxygen. This may lead to acute heart failure, lethal arrhythmias, or even the rupture of a ventricle. The exact etiology is not known, but several factors have been suggested, including transient vasospasm, dysfunction of the cardiac capillaries, or thickening of the myocardium—particularly in the left ventricle—that may lead to the critical circulation of blood to this region. While many patients survive the initial acute event with treatment to restore normal function, there is a strong correlation with death. Careful statistical analysis by the Jaap Spreeuw and Iqbal Owadally of Bayes Business School in London, revealed that within one year of the death of a loved one, women are more than twice as likely to die and men are six times as likely to die as would otherwise be expected. The same study indicated that longer relationships also had a greater incidence of a person dying soon after the death of their spouse.
Other Factors Influencing Heart Rate
Using a combination of autorhythmicity and innervation, the cardiovascular center is able to provide relatively precise control over HR. However, there are a number of other factors that have an impact on HR as well, including epinephrine, NE, and thyroid hormones; levels of various ions including calcium, potassium, and sodium; body temperature; hypoxia; and pH balance (Table \(\PageIndex{1}\) and Table \(\PageIndex{2}\)). After reading this section, the importance of maintaining homeostasis should become even more apparent.
| Factor | Effect |
|---|---|
| Cardioaccelerator nerves | Release of norepinephrine by cardioaccelerator nerves |
| Proprioreceptors | Increased firing rates of proprioreceptors (e.g. during exercise) |
| Chemoreceptors | Chemoreceptors sensing decreased levels of O 2 or increased levels of H + , CO 2 and lactic acid |
| Baroreceptors | Decreased firing rates of baroreceptors (indicating falling blood volume/pressure) |
| Limbic system | Anticipation of physical exercise or strong emotions by the limbic system |
| Catecholamines | Decreased epinephrine and norepinephrine release by the adrenal glands |
| Thyroid hormones | Increased T 3 and T 4 in the blood (released by thyroid) |
| Calcium | Decrease in calcium ions in the blood |
| Potassium | Decrease in potassium ions in the blood |
| Sodium | Decrease in sodium ions in the blood |
| Body temperature | Increase in body temperature |
| Nicotine and caffeine | Presence of nicotine, caffeine or other stimulants |
| Factor | Effect |
|---|---|
| Cardioinhibitor nerves (vagus) | Release of acetylcholine by cardioinhibitor nerves |
| Proprioreceptors | Decreased firing rates of proprioreceptors (e.g. during rest) |
| Chemoreceptors | Chemoreceptors sensing increased levels of O 2 or decreased levels of H + , CO 2 and lactic acid |
| Baroreceptors | Increased firing rates of baroreceptors (indicating rising blood volume/pressure) |
| Limbic system | Anticipation of relaxation by the limbic system |
| Catecholamines | Decreased epinephrine and norepinephrine release by the adrenal glands |
| Thyroid hormones | Decreased T 3 and T 4 in the blood (released by thyroid) |
| Calcium | Decrease in calcium ions in the blood |
| Potassium | Increase in potassium ions in the blood |
| Sodium | Increase in sodium ions in the blood |
| Body temperature | Decrease in body temperature |
| Opiates and tranquilizers | Presence of opiates (heroin), tranquilizers or other depressants |
Epinephrine and Norepinephrine
The catecholamines, epinephrine and NE, secreted by the adrenal medulla form one component of the extended fight-or-flight mechanism. The other component is sympathetic stimulation. Epinephrine and NE have similar effects: binding to the beta-1 receptors, and opening sodium and calcium ion chemical- or ligand-gated channels. The rate of depolarization is increased by this additional influx of positively charged ions, so the threshold is reached more quickly and the period of repolarization is shortened. However, massive releases of these hormones coupled with sympathetic stimulation may actually lead to arrhythmias. There is no parasympathetic stimulation to the adrenal medulla.
Thyroid Hormones
In general, increased levels of thyroid hormone, or thyroxin, increase cardiac rate and contractility. The impact of thyroid hormone is typically of a much longer duration than that of the catecholamines. The physiologically active form of thyroid hormone, T 3 or triiodothyronine, has been shown to directly enter cardiomyocytes and alter activity at the level of the genome. It also impacts the beta adrenergic response similar to epinephrine and NE described above. Excessive levels of thyroxin may trigger tachycardia.
Calcium
Calcium ion levels have great impacts upon both HR and contractility; as the levels of calcium ions increase, so do HR and contractility. High levels of calcium ions (hypercalcemia) may be implicated in a short QT interval and a widened T wave in the ECG. The QT interval represents the time from the start of depolarization to repolarization of the ventricles, and includes the period of ventricular systole. Extremely high levels of calcium may induce cardiac arrest. Drugs known as calcium channel blockers slow HR by binding to these channels and blocking or slowing the inward movement of calcium ions.
Caffeine and Nicotine
Caffeine and nicotine are not found naturally within the body. Both of these nonregulated drugs have an excitatory effect on membranes of neurons in general and have a stimulatory effect on the cardiac centers specifically, causing an increase in HR. Caffeine works by increasing the rates of depolarization at the SA node, whereas nicotine stimulates the activity of the sympathetic neurons that deliver impulses to the heart.
Although it is the world’s most widely consumed psychoactive drug, caffeine is legal and not regulated. While precise quantities have not been established, “normal” consumption is not considered harmful to most people, although it may cause disruptions to sleep and acts as a diuretic. Its consumption by pregnant people is cautioned against, although no evidence of negative effects has been confirmed.
Nicotine, too, is a stimulant and produces addiction. While legal and nonregulated, concerns about nicotine’s safety and documented links to respiratory and cardiac disease have resulted in warning labels on cigarette packages.
Factors Decreasing Heart Rate
HR can be slowed when a person experiences altered sodium and potassium levels, hypoxia, acidosis, alkalosis, and hypothermia (see Table 19.1). The relationship between electrolytes and HR is complex, but maintaining electrolyte balance is critical to the normal wave of depolarization. Of the two ions, potassium has the greater clinical significance. Initially, both hyponatremia (low sodium levels) and hypernatremia (high sodium levels) may lead to tachycardia. Severely high hypernatremia may lead to fibrillation, which may cause CO to cease. Severe hyponatremia leads to both bradycardia and other arrhythmias. Hypokalemia (low potassium levels) also leads to arrhythmias, whereas hyperkalemia (high potassium levels) causes the heart to become weak and flaccid, and ultimately to fail.
Acidosis is a condition in which excess hydrogen ions are present, and the patient’s blood expresses a low pH value. Alkalosis is a condition in which there are too few hydrogen ions, and the patient’s blood has an elevated pH. Normal blood pH falls in the range of 7.35–7.45, so a number lower than this range represents acidosis and a higher number represents alkalosis. Recall that enzymes are the regulators or catalysts of virtually all biochemical reactions; they are sensitive to pH and will change shape slightly with values outside their normal range. These variations in pH and accompanying slight physical changes to the active site on the enzyme decrease the rate of formation of the enzyme-substrate complex, subsequently decreasing the rate of many enzymatic reactions, which can have complex effects on HR. Severe changes in pH will lead to denaturation of the enzyme.
The last variable is body temperature. Elevated body temperature is called hyperthermia, and suppressed body temperature is called hypothermia. Slight hyperthermia results in increasing HR and strength of contraction. Hypothermia slows the rate and strength of heart contractions. This distinct slowing of the heart is one component of the larger diving reflex that diverts blood to essential organs while submerged. If sufficiently chilled, the heart will stop beating, a technique that may be employed during open heart surgery. In this case, the patient’s blood is normally diverted to an artificial heart-lung machine to maintain the body’s blood supply and gas exchange until the surgery is complete, and sinus rhythm can be restored. Excessive hyperthermia and hypothermia will both result in death, as enzymes drive the body systems to cease normal function, beginning with the central nervous system.
Stroke Volume
Many of the same factors that regulate HR also impact cardiac function by altering SV. While a number of variables are involved, SV is ultimately dependent upon the difference between EDV and ESV. The three primary factors to consider are preload, or the stretch on the ventricles prior to contraction; the contractility, or the force or strength of the contraction itself; and afterload, the force the ventricles must generate to pump blood against the resistance in the vessels. These factors are summarized in Table 19.1 and Table 19.2.
Preload
Preload is another way of expressing EDV. Therefore, the greater the EDV is, the greater the preload is. One of the primary factors to consider is filling time , or the duration of ventricular diastole during which filling occurs. The more rapidly the heart contracts, the shorter the filling time becomes, and the lower the EDV and preload are. This effect can be partially overcome by increasing the second variable, contractility, and raising SV, but over time, the heart is unable to compensate for decreased filling time, and preload also decreases.
With increasing ventricular filling, both EDV or preload increase, and the cardiac muscle itself is stretched to a greater degree. At rest, there is little stretch of the ventricular muscle, and the sarcomeres remain short. With increased ventricular filling, the ventricular muscle is increasingly stretched and the sarcomere length increases. As the sarcomeres reach their optimal lengths, they will contract more powerfully, because more of the myosin heads can bind to the actin on the thin filaments, forming cross bridges and increasing the strength of contraction and SV. If this process were to continue and the sarcomeres stretched beyond their optimal lengths, the force of contraction would decrease. However, due to the physical constraints of the location of the heart, this excessive stretch is not a concern.
The relationship between ventricular stretch and contraction has been stated in the well-known Frank-Starling mechanism or simply Starling’s Law of the Heart. This principle states that, within physiological limits, the force of heart contraction is directly proportional to the initial length of the muscle fiber. This means that the greater the stretch of the ventricular muscle (within limits), the more powerful the contraction is, which in turn increases SV. Therefore, by increasing preload, you increase the second variable, contractility.
Otto Frank (1865–1944) was a German physiologist; among his many published works are detailed studies of this important heart relationship. Ernest Starling (1866–1927) was an important English physiologist who also studied the heart. Although they worked largely independently, their combined efforts and similar conclusions have been recognized in the name “Frank-Starling mechanism.”
Any sympathetic stimulation to the venous system will increase venous return to the heart, which contributes to ventricular filling, and EDV and preload. While much of the ventricular filling occurs while both atria and ventricles are in diastole, the contraction of the atria, the atrial kick, plays a crucial role by providing the last 20–30 percent of ventricular filling.
Contractility
It is virtually impossible to consider preload or ESV without including an early mention of the concept of contractility. Indeed, the two parameters are intimately linked. Contractility refers to the force of the contraction of the heart muscle, which controls SV, and is the primary parameter for impacting ESV. The more forceful the contraction is, the greater the SV and smaller the ESV are. Less forceful contractions result in smaller SVs and larger ESVs. Factors that increase contractility are described as positive inotropic factors , and those that decrease contractility are described as negative inotropic factors (ino- = “fiber;” -tropic = “turning toward”).
Not surprisingly, sympathetic stimulation is a positive inotrope, whereas parasympathetic stimulation is a negative inotrope. Sympathetic stimulation triggers the release of NE at the neuromuscular junction from the cardiac nerves and also stimulates the adrenal cortex to secrete epinephrine and NE. In addition to their stimulatory effects on HR, they also bind to both alpha and beta receptors on the cardiac muscle cell membrane to increase metabolic rate and the force of contraction. This combination of actions has the net effect of increasing SV and leaving a smaller residual ESV in the ventricles. In comparison, parasympathetic stimulation releases ACh at the neuromuscular junction from the vagus nerve. The membrane hyperpolarizes and inhibits contraction to decrease the strength of contraction and SV, and to raise ESV. Since parasympathetic fibers are more widespread in the atria than in the ventricles, the primary site of action is in the upper chambers. Parasympathetic stimulation in the atria decreases the atrial kick and reduces EDV, which decreases ventricular stretch and preload, thereby further limiting the force of ventricular contraction. Stronger parasympathetic stimulation also directly decreases the force of contraction of the ventricles.
Several synthetic drugs, including dopamine and isoproterenol, have been developed that mimic the effects of epinephrine and NE by stimulating the influx of calcium ions from the extracellular fluid. Higher concentrations of intracellular calcium ions increase the strength of contraction. Excess calcium (hypercalcemia) also acts as a positive inotropic agent. The drug digitalis lowers HR and increases the strength of the contraction, acting as a positive inotropic agent by blocking the sequestering of calcium ions into the sarcoplasmic reticulum. This leads to higher intracellular calcium levels and greater strength of contraction. In addition to the catecholamines from the adrenal medulla, other hormones also demonstrate positive inotropic effects. These include thyroid hormones and glucagon from the pancreas.
Negative inotropic agents include hypoxia, acidosis, hyperkalemia, and a variety of synthetic drugs. These include numerous beta blockers and calcium channel blockers. Early beta blocker drugs include propranolol and pronethalol, and are credited with revolutionizing treatment of cardiac patients experiencing angina pectoris. There is also a large class of dihydropyridine, phenylalkylamine, and benzothiazepine calcium channel blockers that may be administered decreasing the strength of contraction and SV.
Afterload
Afterload refers to the tension that the ventricles must develop to pump blood effectively against the resistance in the vascular system. Any condition that increases resistance requires a greater afterload to force open the semilunar valves and pump the blood. Damage to the valves, such as stenosis, which makes them harder to open will also increase afterload. Any decrease in resistance decreases the afterload. Figure \(\PageIndex{4}\) summarizes the major factors influencing SV, Figure \(\PageIndex{5}\) summarizes the major factors influencing CO, and Table \(\PageIndex{3}\) and Table \(\PageIndex{4}\) summarize cardiac responses to increased and decreased blood flow and pressure in order to restore homeostasis.
| Baroreceptors (aorta, carotid arteries, venae cavae, and atria) | Chemoreceptors (both central nervous system and in proximity to baroreceptors) | |
|---|---|---|
| Sensitive to | Decreasing stretch | Decreasing O 2 and increasing CO 2 , H + , and lactic acid |
| Target | Parasympathetic stimulation suppressed | Sympathetic stimulation increased |
| Response of heart | Increasing heart rate and increasing stroke volume | Increasing heart rate and increasing stroke volume |
| Overall effect | Increasing blood flow and pressure due to increasing cardiac output; homeostasis restored | Increasing blood flow and pressure due to increasing cardiac output; homeostasis restored |
| Baroreceptors (aorta, carotid arteries, venae cavae, and atria) | Chemoreceptors (both central nervous system and in proximity to baroreceptors) | |
|---|---|---|
| Sensitive to | Increasing stretch | Increasing O 2 and decreasing CO 2 , H + , and lactic acid |
| Target | Parasympathetic stimulation increased | Sympathetic stimulation suppressed |
| Response of heart | Decreasing heart rate and decreasing stroke volume | Decreasing heart rate and decreasing stroke volume |
| Overall effect | Decreasing blood flow and pressure due to decreasing cardiac output; homeostasis restored | Decreasing blood flow and pressure due to decreasing cardiac output; homeostasis restored |


30.7: Invention Expositions
30.7 Invention Expositions
J.S. Bach’s Two-Part Inventions were not only intended as instructional keyboard pieces but also as examples of how to compose. In this section we will wed our species counterpoint knowledge with our knowledge of harmony in order to write a four-measure invention exposition using Bach’s C major and E minor inventions as models. Note the I–I–V–V–I harmonic progression.
30.7.1 How to Write an Invention Exposition
In the majority of examples, you will be given an invention theme in the first measure, which you will write an octave lower in the bass part in the second measure, then a perfect fifth higher than the first measure in the third measure, the fourth measure will be the third measure transposed down one octave, and, finally, you will end on the interval of a 10th—tonic in the bass part and 3^ in the upper part. Please see the example below.
30.7.2 Altering Themes to Fit the Harmonies
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You may find that a theme that leads naturally from I to V needs its contour altered
when progressing from V to I in measures 4 to 5
in order to emphasize the V chord and the dominant-to-tonic harmonic motion. Modify the contour to emphasize the V chord at the end of the measure, altering as few notes as possible.
-
Additionally, you may be given themes that land on 3^ on the downbeat of the second measure, not the 5^ that occurred in the Bach C major and E minor invention expositions. You will need to alter the theme at the end of measure 2 to emphasize the I chord as well as make the first bass note of the third measure the 5^ scale degree.
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Finally, when a theme in minor containing the 2^ and 3^ scale degrees in the first two measures is transposed up a fifth in the succeeding two measures, scale degrees 2^ and ♭3^
will be transposed to ↑6^ and ↑7^
. The reasoning is that ↑7^ is the 3rd of the V chord, and ↑6^ is a step below, typically a passing tone. Another way to think of this is to
use the melodic minor scale in the 3rd and 4th measures
.
30.7.3 Adding Counterpoint to the Theme Statements
After copying and transposing the theme throughout this four-measure invention exposition, you will need to add counterpoint to accompany the statements of the theme in the second, third, and fourth measures. Unlike species counterpoint, your counterpoint in these invention expositions must emphasize the harmony in each measure.
You may find the need to create “microharmonies” within a measure with some themes, such as a I–IV–I or I–viivii∘6–I progression even though the overriding harmony of the bar is the I chord. Examples are found in the first two measures of the C major invention.
Given these pieces of information, you are prepared to begin writing tonal counterpoint in two parts. Go to the Invention Exposition Practice Exercises .


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1.21: Multiplying and Dividing Fractions
Learning Objectives
- Use multiplication and division when evaluating expressions with fractions
Fraction Multiplication
A model may help you understand multiplication of fractions. We will use fraction tiles to model \(\frac{1}{2}\cdot \frac{3}{4}\).
To multiply \(\frac{1}{2}\) and \(\frac{3}{4}\), think “I need to find \(\frac{1}{2}\) of \(\frac{3}{4}\).”
Start with fraction tiles for three-fourths. To find one-half of three-fourths, we need to divide them into two equal groups. Since we cannot divide the three \(\frac{1}{4}\) tiles evenly into two parts, we exchange them for smaller tiles.
We see \(\frac{6}{8}\) is equivalent to \(\frac{3}{4}\). Taking half of the six \(\frac{1}{8}\) tiles gives us three \(\frac{1}{8}\) tiles, which is \(\frac{3}{8}\).
Therefore, \(\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}\)
Example
Use a diagram to model \(\frac{1}{3}\cdot \frac{2}{5}\)
Solution:
You want to find one-third of two-fifths.
First shade in \(\frac{2}{5}\) of the rectangle.
We will take \(\frac{1}{3}\) of this \(\frac{2}{5}\), so we heavily shade \(\frac{1}{3}\) of the shaded region.
Notice that \(2\) out of the \(15\) pieces are heavily shaded. This means that \(\frac{2}{15}\) of the rectangle is heavily shaded.
Therefore, \(\frac{1}{3}\) of \(\frac{2}{15}\) is \(\frac{2}{15}\), or \(\frac{1}{3}\cdot \frac{2}{5}=\frac{2}{15}\)
Try it
[ohm_question height=”270″]146020[/ohm_question]
Look at the result we got from the examples above. We found that \(\frac{1}{2}\cdot \frac{3}{4}=\frac{3}{8}\) and \(\frac{1}{3}\cdot \frac{2}{5}=\frac{2}{15}\). Do you notice that we could have gotten the same answers by multiplying the numerators and multiplying the denominators?
| \(\frac{1}{2}\cdot \frac{3}{4}\) | \(\frac{1}{3}\cdot \frac{2}{5}\) | |
| Multiply the numerators, and multiply the denominators. | \(\frac{1}{2}\cdot \frac{3}{4}=\frac{1\cdot 3}{2\cdot 4}\) | \(\frac{1}{3}\cdot \frac{2}{5}=\frac{1\cdot 2}{3\cdot 5}\) |
| Simplify. | \(\frac{3}{8}\) | \(\frac{2}{15}\) |
This leads to the definition of fraction multiplication. To multiply fractions, we multiply the numerators and multiply the denominators. Then we write the fraction in simplified form.
Fraction Multiplication
If \(a,b,c,\text{ and }d\) are numbers where \(b\ne 0\text{ and }d\ne 0\), then \(\Large\frac{a}{b}\cdot \Large\frac{c}{d}=\Large\frac{ac}{bd}\)
Example
Multiply, and write the answer in simplified form: \(\frac{3}{4}\cdot \frac{1}{5}\)
[reveal-answer q=”56385″]Show Answer[/reveal-answer]
[hidden-answer a=”56385″]
Solution:
| \(\frac{3}{4}\cdot \frac{1}{5}\) | |
| Multiply the numerators; multiply the denominators. | \(\frac{3\cdot 1}{4\cdot 5}\) |
| Simplify. | \(\frac{3}{20}\) |
There are no common factors, so the fraction is simplified.
[/hidden-answer]
Try It
[ohm_question]146021[/ohm_question]
Note that when multiplying fractions, the properties of positive and negative numbers still apply. It is a good idea to determine the sign of the product as the first step.
The following video provides more examples of how to multiply fractions, and simplify the result.
When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, \(a\), can be written as \(\large\frac{a}{1}\). So, \(3=\frac{3}{1}\), for example.
example
Multiply, and write the answer in simplified form:
- \(\frac{1}{7}\cdot 56\)
- \((-20)(\large\frac{12}{5})\)
[reveal-answer q=”597781″]Show Answer[/reveal-answer]
[hidden-answer a=”597781″]
Solution:
| 1. | |
| \(\frac{1}{7}\cdot 56\) | |
| Write \(56\) as a fraction. | \(\frac{1}{7}\cdot \frac{56}{1}\) |
| Determine the sign of the product; multiply. | \(\frac{56}{7}\) |
| Simplify. | \(8\) |
| 2. | |
| \((-20)(\frac{12}{5})\) | |
| Write \(−20\) as a fraction. | \(\frac{-20}{1}(\frac{12}{5})\) |
| Determine the sign of the product; multiply. | \(-\frac{20\cdot 12\cdot}{1\cdot 5}\) |
| Multiply and simplify. | \(-\frac{240}{5}=-\frac{2\cdot\color{red}{5}\cdot24}{\color{red}{5}}\) |
| \(-48\) |
[/hidden-answer]
Try it
[ohm_question]156968[/ohm_question]
Watch the following video to see more examples of how to multiply a fraction and a whole number,
Reciprocals
The fractions \(\frac{2}{3}\) and \(\frac{3}{2}\) are related to each other in a special way. So are \(-\frac{10}{7}\) and \(-\frac{7}{10}\). Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be \(1\).
\(\frac{2}{3}\cdot \frac{3}{2}=1\text{ and }-\frac{10}{7}\left(-\frac{7}{10}\right)=1\)
Such pairs of numbers are called reciprocals.
Reciprocal
The reciprocal of the fraction \(\frac{a}{b}\) is \(\frac{b}{a}\), where \(a\ne 0\) and \(b\ne 0\). A number and its reciprocal have a product of \(1\).
\(\frac{a}{b}\cdot \frac{b}{a}=1\)
To find the reciprocal of a fraction, we invert the fraction. This means that we place the numerator in the denominator and the denominator in the numerator. To get a positive result when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.
To find the reciprocal, keep the same sign and invert the fraction. The number zero does not have a reciprocal. Why? A number and its reciprocal multiply to \(1\). Is there any number \(r\) so that \(0\cdot r=1?\) No. So, the number \(0\) does not have a reciprocal.
Example
Find the reciprocal of each number. Then check that the product of each number and its reciprocal is \(1\).
- \(\frac{4}{9}\)
- \(-\frac{1}{6}\)
- \(-\frac{14}{5}\)
- \(7\)
Solution:
To find the reciprocals, we keep the sign and invert the fractions.
| 1. | |
| Find the reciprocal of \(\frac{4}{9}\). | The reciprocal of \(\frac{4}{9}\) is \(\frac{9}{4}\). |
| Check: | |
| Multiply the number and its reciprocal. | \(\frac{4}{9}\cdot \frac{9}{4}\) |
| Multiply numerators and denominators. | \(\frac{36}{36}\) |
| Simplify. | \(1\quad\checkmark\) |
| 2. | |
| Find the reciprocal of \(-\frac{1}{6}\). | \(-\frac{6}{1}\) |
| Simplify. | \(-6\) |
| Check: | \(-\frac{1}{6}\cdot \left(-6\right)\) |
| \(1\quad\checkmark\) |
| 3. | |
| Find the reciprocal of \(-\frac{14}{5}\). | \(-\frac{5}{14}\) |
| Check: | \(-\frac{14}{5}\cdot \left(-\frac{5}{14}\right)\) |
| \(\frac{70}{70}\) | |
| \(1\quad\checkmark\) |
| 4. | |
| Find the reciprocal of \(7\). | |
| Write \(7\) as a fraction. | \(\frac{7}{1}\) |
| Write the reciprocal of \(\frac{7}{1}\). | \(\frac{1}{7}\) |
| Check: | \(7\cdot \left(\frac{1}{7}\right)\) |
| \(1\quad\checkmark\) |
Try It
[ohm_question]141842[/ohm_question]
In the following video we will show more examples of how to find the reciprocal of integers, fractions and mixed numbers.
Dividing Fractions
Why is \(12\div 3=4?\) We previously modeled this with counters. How many groups of \(3\) counters can be made from a group of \(12\) counters?
There are \(4\) groups of \(3\) counters. In other words, there are four \(3\text{s}\) in \(12\). So, \(12\div 3=4\).
What about dividing fractions? Suppose we want to find the quotient: \(\frac{1}{2}\div \frac{1}{6}\). We need to figure out how many \(\frac{1}{6}\text{s}\) there are in \(\frac{1}{2}\). We can use fraction tiles to model this division. We start by lining up the half and sixth fraction tiles as shown below. Notice, there are three \(\frac{1}{6}\) tiles in \(\frac{1}{2}\), so \(\frac{1}{2}\div \frac{1}{6}=3\).
Example
Model: \(\frac{1}{4}\div \frac{1}{8}\)
Solution:
We want to determine how many \(\frac{1}{8}\text{s}\) are in \(\frac{1}{4}\). Start with one \(\frac{1}{4}\) tile. Line up \(\frac{1}{8}\) tiles underneath the \(\frac{1}{4}\) tile.
There are two \(\frac{1}{8}\text{s}\) in \(\frac{1}{4}\).
So, \(\frac{1}{4}\div \frac{1}{8}=2\).
The following video shows a whole number being divided by a fraction using a slightly different method.
Example
Model: \(2\div \frac{1}{4}\)
[reveal-answer q=”391699″]Show Answer[/reveal-answer]
[hidden-answer a=”391699″]
Solution:
We are trying to determine how many \(\frac{1}{4}\text{s}\) there are in \(2\). We can model this as shown.
Because there are eight \(\frac{1}{4}\text{s}\) in \(2,2\div \frac{1}{4}=8\).
[/hidden-answer]
Try It
Model: \(2\div \frac{1}{3}\)
[reveal-answer q=”73567″]Show Answer[/reveal-answer]
[hidden-answer a=”73567″]
[/hidden-answer]
Model: \(3\div \frac{1}{2}\)
[reveal-answer q=”354856″]Show Answer[/reveal-answer]
[hidden-answer a=”354856″]
[/hidden-answer]
[ohm_question height=”270″]117216[/ohm_question]
Let’s use money to model \(2\div \frac{1}{4}\) in another way. We often read \(\frac{1}{4}\) as a ‘quarter’, and we know that a quarter is one-fourth of a dollar as shown in the image below. So we can think of \(2\div \frac{1}{4}\) as, “How many quarters are there in two dollars?” One dollar is \(4\) quarters, so \(2\) dollars would be \(8\) quarters. So again, \(2\div \frac{1}{4}=8\).
Using fraction tiles, we showed that \(\frac{1}{2}\div \frac{1}{6}=3\). Notice that \(\frac{1}{2}\cdot \frac{6}{1}=3\) also. How are \(\frac{1}{6}\) and \(\frac{6}{1}\) related? They are reciprocals. This leads us to the procedure for fraction division.
Fraction Division
If \(a,b,c,\text{ and }d\) are numbers where \(b\ne 0,c\ne 0,\text{ and }d\ne 0\), then \(\frac{a}{b}\div \frac{c}{d}=\frac{a}{b}\cdot \frac{d}{c}\)
To divide fractions, multiply the first fraction by the reciprocal of the second.
We need to say \(b\ne 0,c\ne 0\text{ and }d\ne 0\) to be sure we don’t divide by zero.
Example
Divide, and write the answer in simplified form: \(\frac{2}{5}\div \left(-\frac{3}{7}\right)\)
[reveal-answer q=”261121″]Show Answer[/reveal-answer]
[hidden-answer a=”261121″]
Solution:
| \(\frac{2}{5}\div \left(-\frac{3}{7}\right)\) | |
| Multiply the first fraction by the reciprocal of the second. | \(\frac{2}{5}\left(-\frac{7}{3}\right)\) |
| Multiply. The product is negative. | \(-\frac{14}{15}\) |
[/hidden-answer]
Try It
[ohm_question height=”270″]146066[/ohm_question]
[ohm_question height=”270″]146067[/ohm_question]
Watch this video for more examples of dividing fractions using a reciprocal.
Example
Divide, and write the answer in simplified form: \(\frac{7}{18}\div \frac{14}{27}\)
[reveal-answer q=”987562″]Show Answer[/reveal-answer]
[hidden-answer a=”987562″]
Solution:
| \(\frac{7}{18}\div \frac{14}{27}\) | |
| Multiply the first fraction by the reciprocal of the second. | \(\frac{7}{18}\cdot \frac{27}{14}\) |
| Multiply. | \(\frac{7\cdot 27}{18\cdot 14}\) |
| Rewrite showing common factors. | \(\frac{\color{red}{7}\cdot\color{blue}{9}\cdot3}{\color{blue}{9}\cdot2\cdot\color{red}{7}\cdot2}\) |
| Remove common factors. | \(\frac{3}{2\cdot 2}\) |
| Simplify. | \(\frac{3}{4}\) |
[/hidden-answer]
Try It
[ohm_question height=”270″]146091[/ohm_question]


5.2: Interactionism and Views of Self
-
- Last updated
- Save as PDF
Early childhood is a time of forming an initial sense of self. A self-concept or idea of who we are, what we are capable of doing, and how we think and feel is a social process that involves taking into consideration how others view us. So, in order to develop a sense of self, you must have interaction with others. Interactionist theorists, Cooley and Mead offer two interesting explanations of how a sense of self develops.
Cooley
Charles Horton Cooley (1964) suggests that our self-concept comes from looking at how others respond to us. This process, known as the looking-glass self involves looking at how others seem to view us and interpreting this as we make judgments about whether we are good or bad, strong or weak, beautiful or ugly, and so on. Of course, we do not always interpret their responses accurately so our self-concept is not simply a mirror reflection of the views of others. After forming an initial self-concept, we may use it as a mental filter screening out those responses that do not seem to fit our ideas of who we are. Some compliments may be negated, for example. The process of the looking-glass self is pronounced when we are preschoolers, or perhaps when we are in a new school or job or are taking on a new role in our personal lives and are trying to gauge our own performances. When we feel more sure of who we are, we focus less on how we appear to others. 2
Mead
Herbert Mead (1967) offers an explanation of how we develop a social sense of self by being able to see ourselves through the eyes of others. There are two parts of the self: the “I” which is the part of the self that is spontaneous, creative, innate, and is not concerned with how others view us and the “me” or the social definition of who we are.
When we are born, we are all “I” and act without concern about how others view us. But the socialized self begins when we are able to consider how one important person views us. This initial stage is called “taking the role of the significant other”. For example, a child may pull a cat’s tail and be told by his mother, “No! Don’t do that, that’s bad” while receiving a slight slap on the hand. Later, the child may mimic the same behavior toward the self and say aloud, “No, that’s bad” while patting his own hand. What has happened? The child is able to see himself through the eyes of the mother. As the child grows and is exposed to many situations and rules of culture, he begins to view the self in the eyes of many others through these cultural norms or rules. This is referred to as “taking the role of the generalized other” and results in a sense of self with many dimensions. The child comes to have a sense of self as student, as friend, as son, and so on.
Exaggerated Sense of Self
One of the ways to gain a clearer sense of self is to exaggerate those qualities that are to be incorporated into the self. Preschoolers often like to exaggerate their own qualities or to seek validation as the biggest, smartest, or child who can jump the highest. This exaggeration tends to be replaced by a more realistic sense of self in middle childhood.
Self-Esteem
Early childhood is a time of forming an initial sense of self. Self-concept is our self-description according to various categories, such as our external and internal qualities. In contrast, self-esteem is an evaluative judgment about who we are. The emergence of cognitive skills in this age group results in improved perceptions of the self, but they tend to focus on external qualities, which are referred to as the categorical self . When researchers ask young children to describe themselves, their descriptions tend to include physical descriptors, preferred activities, and favorite possessions. Thus, the self-description of a 3-year-old might be a 3-year-old girl with red hair, who likes to play with blocks. However, even children as young as three know there is more to themselves than these external characteristics.
Harter and Pike (1984) challenged the method of measuring personality with an open-ended question as they felt that language limitations were hindering the ability of young children to express their self-knowledge. They suggested a change to the method of measuring self-concept in young children, whereby researchers provide statements that ask whether something is true of the child (e.g., “I like to boss people around”, “I am grumpy most of the time”). They discovered that in early childhood, children answer these statements in an internally consistent manner, especially after the age of four (Goodvin, Meyer, Thompson & Hayes, 2008) and often give similar responses to what others (parents and teachers) say about the child (Brown, Mangelsdorf, Agathen, & Ho, 2008; Colwell & Lindsey, 2003).
Young children tend to have a generally positive self-image. This optimism is often the result of a lack of social comparison when making self-evaluations (Ruble, Boggiano, Feldman, & Loeble, 1980), and with comparison between what the child once could do to what they can do now (Kemple, 1995). However, this does not mean that preschool children are exempt from negative self-evaluations. Preschool children with insecure attachments to their caregivers tend to have lower self-esteem at age four (Goodvin et al., 2008). Maternal negative affect (emotional state) was also found by Goodwin and her colleagues to produce more negative self-evaluations in preschool children.
Self-Control
Self-control is not a single phenomenon, but is multi-facetted. It includes response initiation , the ability to not initiate a behavior before you have evaluated all of the information, response inhibition , the ability to stop a behavior that has already begun, and delayed gratification, the ability to hold out for a larger reward by forgoing a smaller immediate reward (Dougherty, Marsh, Mathias, & Swann, 2005). It is in early childhood that we see the start of self-control, a process that takes many years to fully develop. In the now classic “Marshmallow Test” (Mischel, Ebbesen, & Zeiss, 1972) children are confronted with the choice of a small immediate reward (a marshmallow) and a larger delayed reward (more marshmallows). Walter Mischel and his colleagues over the years have found that the ability to delay gratification at the age of four predicted better academic performance and health later in life (Mischel, et al., 2011). Self-control is related to executive function, discussed earlier in the chapter. As executive function improves, children become less impulsive (Traverso, Viterbori, & Usai, 2015). 5
Self-Control and Play
Thanks to the new Centre for Research on Play in Education, Development and Learning (PEDaL), Whitebread, Baker, Gibson and a team of researchers hope to provide evidence on the role played by play in how a child develops.
“A strong possibility is that play supports the early development of children’s self-control,” explains Baker. “These are our abilities to develop awareness of our own thinking processes – they influence how effectively we go about undertaking challenging activities.”
In a study carried out by Baker with toddlers and young preschoolers, she found that children with greater self-control solved problems quicker when exploring an unfamiliar set-up requiring scientific reasoning, regardless of their IQ. “This sort of evidence makes us think that giving children the chance to play will make them more successful and creative problem-solvers in the long run.”
If playful experiences do facilitate this aspect of development, say the researchers, it could be extremely significant for educational practices because the ability to self-regulate has been shown to be a key predictor of academic performance.
Gibson adds: “Playful behavior is also an important indicator of healthy social and emotional development. In my previous research, I investigated how observing children at play can give us important clues about their well being and can even be useful in the diagnosis of neurodevelopmental disorders like autism.”6
Contributors and Attributions
2. Children’s Development by Ana R. Leon is licensed under CC BY 4.0
5. Lifespan Development: A Psychological Perspective by Martha Lally and Suzanne Valentine-French is licensed under CC BY-NC-SA 3.0
6. Play’s the Thing by the University of Cambridge is licensed under CC BY 4.0


8.1: Introduction to the Eigenvalue Problem
Introduction
Harking back to our previous discussion of The Laplace Transform we labeled the complex number \(\lambda\) an eigenvalue of \(B\) if \(\lambda I-B\) was not invertible. In order to find such \(\lambda\) one has only to find those \(s\) for which \((sI-B)^{-1}\) is not defined. To take a concrete example we note that if
\[B = \begin{pmatrix} {1}&{0}&{0}\\ {1}&{1}&{0}\\ {0}&{0}&{2} \end{pmatrix} \nonumber\]
then
\[(sI-B)^{-1} = \frac{1}{(s-1)^{2} (s-2)} \begin{pmatrix} {(s-1)(s-2)}&{s-2}&{0}\\ {0}&{(s-1)(s-2)}&{0}\\ {0}&{0}&{(s-1)^2} \end{pmatrix} \nonumber\]
and so \(\lambda_{1} = 1\) and \(\lambda_{2} = 2\) are the two eigenvalues of \(B\). Now, to say that \(\lambda_{j}I-B\) is not invertible is to say that its columns are linearly dependent, or, equivalently, that the null space \(\mathscr{N} (\lambda_{j}I-B)\) contains more than just the zero vector. We call \(\mathscr{N} (\lambda_{j}I-B)\) the jth eigenspace and call each of its nonzero members a jth eigenvector . The dimension of \(\mathscr{N} (\lambda_{j}I-B)\) is referred to as the geometric multiplicity of \(\lambda_{j}\). With respect to \(B\) above, we compute \(\mathscr{N} (\lambda_{j}I-B)\) by solving \((I-B)x = 0\) i.e.,
\[\begin{pmatrix} {0}&{-1}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{1} \end{pmatrix} \begin{pmatrix} {x_{1}}\\ {x_{2}}\\ {x_{3}} \end{pmatrix} = \begin{pmatrix} {0}\\ {0}\\ {0} \end{pmatrix} \nonumber\]
Clearly,
\[\mathscr{N} (\lambda_{1}I-B) = \{c \begin{pmatrix} {1}&{0}&{0} \end{pmatrix}^{T} | c \in \mathbb{R}\} \nonumber\]
Arguing along the same lines we also find
\[\mathscr{N} (\lambda_{2}I-B) = \{c \begin{pmatrix} {0}&{0}&{1} \end{pmatrix}^{T} | c \in \mathbb{R}\} \nonumber\]
That \(B\) is \(3 \times 3\) but possesses only 2 linearly eigenvectors leads us to speak of \(B\) as defective. The cause of its defect is most likely the fact that \(\lambda_{1}\) is a double pole of \((sI-B)^{-1}\). In order to flesh out that remark and uncover the missing eigenvector we must take a much closer look at the transfer function
\[R(s) \equiv (sI-B)^{-1} \nonumber\]
In the mathematical literature this quantity is typically referred to as the Resolvent of \(B\).


6.1: Policing in Ancient Times
kin policing .  The blood feuds would go on for a long period of time. [1]
urban cohorts . The urban cohorts were men from the Praetorian Guard (Augustus’ army), charged with ensuring peace in the city. As crime rose and became more violent, Augustus formed the vigils , which were not affiliated with the Praetorian Guard, but were charged with fighting crime and fires. The vigils were given the power to protect and arrest. [4]
posse comitatus to go after wanted felons. Constables, a police officer with limited authority, assisted the sheriffs with serving summons and warrants. Because young volunteers did the policing work, there were several problems, such as corruption and drunkenness. Victims who had the means to hire private police or bodyguards to so for protection, but unfortunately, that meant those who were poor, had neither help nor protection. [5]
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Berg, B., (1999).
Policing in a modern society.
Oxford: Elsevier.
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Berg, B., (1999).
Policing in a modern society.
Oxford: Elsevier.
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Berg, B., (1999).
Policing in a modern society.
Oxford: Elsevier.
↵
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Berg, B., (1999).
Policing in a modern society.
Oxford: Elsevier.
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Berg, B., (1999).
Policing in a modern society.
Oxford: Elsevier.
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15.4: Undamped Harmonic Forced Vibrations
Often, mechanical systems are not undergoing free vibration, but are subject to some applied force that causes the system to vibrate. In this section, we will consider only harmonic (that is, sine and cosine) forces, but any changing force can produce vibration.
When we consider the free body diagram of the system, we now have an additional force to add: namely, the external harmonic excitation.
The equation of motion of the system above will be:
\[ m \ddot{x} + kx = F, \]
where \(F\) is a force of the form:
\[ F = F_0 \sin (\omega_0 t). \]
This equation of motion for the system can be re-written in standard form:
\[ \ddot{x} + \frac{k}{m} x = \frac{F_0}{m} \sin (\omega_0 t). \]
The solution to this system consists of the superposition of two solutions: a particular solution, \(x_p\) (related to the forcing function), and a complementary solution, \(x_c\) (which is the solution to the system without forcing).
As we saw previously , the complementary solution is the solution to the undamped free system: \[x_c = C \sin (\omega_n t + \phi). \]
We can obtain the particular solution by assuming a solution of the form: \[x_p = D \sin (\omega_0 t), \]
where \(\omega_0\) is the frequency of the harmonic forcing function. We differentiate this form of the solution, and then sub into the above equation of motion:
\[ \ddot{x}_p = - \omega_0^2 D \sin (\omega_0 t) \]
\[ - m \omega_0^2 D \sin (\omega_0 t) + k D \sin (\omega_0 t) = F_0 \sin (\omega_0 t) \]
After solving for \(D\), we can then use it to find the particular solution, \(x_p\):
\[ D = \frac{ \dfrac{F_0}{k}}{1 - \left( \dfrac{\omega_0}{\omega_n} \right) ^2} \]
\[ x_p = \frac{ \dfrac{F_0}{k}}{1 - \left( \dfrac{\omega_0}{\omega_n} \right) ^2} \ \sin (\omega_0 t). \]
Thus, the general solution for a forced, undamped system is:
\[ x_G (t) = \frac{ \dfrac{F_0}{k}}{1- \left( \dfrac{\omega_0}{\omega_n} \right) ^2} \ \sin (\omega_0 t) + C \sin (\omega_n t + \phi) \]
The above figures show the two responses at different frequencies. Recall that the value of \(\omega_n\) comes from the physical characteristics of the system (\(m, \, k\)) and \(\omega_0\) comes from the force being applied to the system. These responses are summed, to achieve the blue response (general solution) in Figure \(\PageIndex{4}\) above.
Steady-State Response:
In reality, this superimposed response does not last long. Every real system has some damping, and the natural response of the system will be damped out. As long as the external harmonic force is applied, however, the response to it will remain. When evaluating the response of the system to a harmonic forcing function, we will typically consider the steady-state response, when the natural response has been damped out and the response to the forcing function remains.
Amplitude of Forced Vibration
The amplitude of the steady-state forced vibration depends on the ratio of the forced frequency to the natural frequency. As \(\omega_0\) approaches \(\omega_n\) (the ratio approaches 1), the magnitude \(D\) becomes very large. We can define a magnification factor:
\[ MF = \dfrac{ \dfrac{ \dfrac{F_0}{k}}{1 - \left( \dfrac{\omega_0}{\omega_n} \right) ^2}} {\dfrac{F_0}{k}} = \dfrac{1}{1 - \left( \dfrac{\omega_0}{\omega_n} \right) ^2} \]
From the figure above, we can discuss various cases:
- \(\bf{\omega_0 = \omega_n}\): Resonance occurs. This results in very large-amplitude vibrations, and is associated with high stress and failure to the system.
- \(\bf{\omega_0 \sim 0, MF \sim 1}\): The forcing function is nearly static, leaving essentially the static deflection and limited natural vibration.
- \(\bf{\omega_0 < \omega_n}\): Magnification is positive and greater than 1, meaning the vibrations are in phase (when the force acts to the left, the system displaces to the left) and the amplitude of vibration is larger than the static deflection.
- \(\bf{\omega_0 > \omega_n}\): Magnification is negative and the absolute value is typically smaller than 1, meaning the vibration is out of phase with the motion of the forcing function (when the force acts to the left, the system displaces to the right) and the amplitude of vibration is smaller than the static deflection.
- \(\bf{\omega_0 >> \omega_n}\): The force is changing direction too fast for the block's motion to respond.
Rotating Unbalance:
One common cause of harmonic forced vibration in mechanical systems is rotating unbalance. This occurs when the axis of rotation does not pass through the center of mass, meaning that the center of mass experiences some acceleration instead of remaining stationary. This causes a force on the axle that changes direction as the center of mass rotates. We can represent this as a small mass, \(m\), rotating about the axis of rotation at some distance, called an eccentricity, \(e\). The forced angular frequency, \(\omega_0\), in this case is the angular frequency of the rotating system.
A 10-kg fan is fixed to a lightweight beam. The static weight of the fan deflects the beam by 20 mm. If the blade is designed to spin at \(\omega\) = 15 rad/s, and the blade is mounted off-center (equivalent to a 1.5 kg mass at 50 mm from the axis of rotation), determine the steady-state amplitude of vibration.
- Solution:
You are designing a stylish fan that uses only one blade. Approximate that blade as a narrow plate with a density per unit length of 20 g/cm. The base weight of the rest of the device (except for the blade) is 4 kg, and the whole thing is mounted on a lightweight beam. If the spring constant of the beam is \(k\) = 1000 N/m, find the length of blade than will cause resonance if the fan is designed to spin at \(\omega\) = 15 rad/s.
- Solution: