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Minority group
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A minority group refers to a category of people who experience relative disadvantage as compared to members of a dominant social group.[1] Minority group membership is typically based on differences in observable characteristics or practices, such as: sex, ethnicity, race, religion, disability, sexual orientation, or gender identity.[2] Utilizing the framework of intersectionality, it is important to recognize that an individual may simultaneously hold membership in multiple minority groups (e.g. both a racial and religious minority).[3] Likewise, individuals may also be part of a minority group in regard to some characteristics, but part of a dominant group in regard to others.[3]
The term "minority group" often occurs within the discourse of civil rights and collective rights, as members of minority groups are prone to differential treatment in the countries and societies in which they live.[4] Minority group members often face discrimination in multiple areas of social life, including housing, employment, healthcare, and education, among others.[5][6] While discrimination may be committed by individuals, it may also occur through structural inequalities, in which rights and opportunities are not equally accessible to all.[7] The language of minority rights, is often used to discuss laws designed to protect minority groups from discrimination and afford them equal social status to the dominant group.[8]
Contents
1 Definitions
1.1 Sociological
1.1.1 Criticisms
1.2 Political
2 Examples of minority groups
2.1 Age minorities
2.2 Educational minorities
2.3 Involuntary minorities
2.4 Voluntary minorities
2.5 Gender and sexuality minorities
2.6 People with disabilities
2.7 Political minorities
2.8 Religious minorities
2.9 Regional minorities
2.10 Women as minorities
3 Law and government
4 See also
5 References
6 External links
Definitions[edit]
Sociological[edit]
Louis Wirth defined a minority group as "a group of people who, because of their physical or cultural characteristics, are singled out from the others in the society in which they live for differential and unequal treatment, and who therefore regard themselves as objects of collective discrimination".[9] The definition includes both objective and subjective criteria: membership of a minority group is objectively ascribed by society, based on an individual's physical or behavioral characteristics; it is also subjectively applied by its members, who may use their status as the basis of group identity or solidarity.[10] Thus, minority group status is categorical in nature: an individual who exhibits the physical or behavioral characteristics of a given minority group is accorded the status of that group and is subject to the same treatment as other members of that group.[9]
Joe Feagin, states that a minority group has five characteristics: (1) suffering discrimination and subordination, (2) physical and/or cultural traits that set them apart, and which are disapproved by the dominant group, (3) a shared sense of collective identity and common burdens, (4) socially shared rules about who belongs and who does not determine minority status, and (5) tendency to marry within the group.[11]
Criticisms[edit]
There is a controversy with the use of the word minority, as it has a common and an academic usage.[12] Common usage of the term indicates a statistical minority; however, academics refer to power differences among groups rather than differences in population size among groups.[13]
Some sociologists have criticised the concept of "minority/majority", arguing this language excludes or neglects changing or unstable cultural identities, as well as cultural affiliations across national boundaries.[14] As such, the term historically excluded groups (HEGs) is often similarly used to highlight the role of historical oppression and domination, and how this results in the underrepresentation of particular groups in various areas of social life.[15]
Political[edit]
The term national minority is often used to discuss minority groups in international and national politics.[16] All countries contain some degree of racial, ethnic, or linguistic diversity.[1] In addition, minorities may also be minorities they may be migrant, indigenous or landless nomadic communities.[17] This often results in variations in language, culture, beliefs, practices, that set some groups apart from the dominant grop. As these differences are usually perceived negatively by, this results in loss of social and political power for members of minority groups.[18]
There is no legal definition of national minorities in international law, though protection of minority groups is outlined by the United Nations Declaration on the Rights of Persons Belonging to National or Ethnic, Religious and Linguistic Minorities. International criminal law can protect the rights of racial or ethnic minorities in a number of ways.[19] The right to self-determination is a key issue.
The formal level of protection of national minorities is highest in European countries.[16] The Council of Europe proposes a definition of national minorities in the European Charter for Regional or Minority Languages and by the Framework Convention for the Protection of National Minorities; however these definitions are not binding upon member states.[20] Using this framework, a national minority can be theoretically defined as a group of people within a given nation state:
which is numerically smaller than the rest of population of the state or a part of the state,
which is not in a dominant position,
which has culture, language, religion, race etc. distinct from that of the majority of the population,
whose members have a will to preserve their group identity,
whose members are citizens of the state where they have the status of a minority, and
whose members have had long-term presence in the territory.
In some places, subordinate ethnic groups may constitute a numerical majority, such as Blacks in South Africa under apartheid.[21] In the United States, for example, non-Hispanic Whites constitute the majority (63.4%) and all other racial and ethnic groups (Hispanic or Latino, African Americans, Asian Americans, American Indian, and Native Hawaiians) are classified as "minorities".[22] If the non-Hispanic White population falls below 50% the group will only be the plurality, not the majority.
Examples of minority groups[edit]
Age minorities[edit]
The elderly, while traditionally influential or even (in a gerontocracy) dominant in the past, are now usually reduced to the minority role of economically 'non-active' groups.[citation needed] Children can also be understood as a minority group in these terms, and the discrimination faced by the young is known as adultism. Discrimination against the elderly is known as ageism.
Various local and international statutes are in place to mitigate the exploitation of children, such as the Convention on the Rights of the Child, as well as a number of organizations that make up the children's rights movement. The youth rights movement campaigns for social empowerment for young people, and against the legal and social restrictions placed on legal minors. Groups that advocate the interests of senior citizens range from the charitable (Help the Aged) to grass-roots activism (Gray Panthers), and often overlap with disability rights issues.
Educational minorities[edit]
Involuntary minorities[edit]
Also known as "castelike minorities," involuntary minorities are a term for people who were originally brought into any society against their will. In the United States, for instance, it includes but is not limited to Native Americans, Puerto Ricans, African Americans,[23] and native-born Mexican Americans.[24] For reasons of cultural differences, involuntary minorities may experience difficulties in school more than members of other (voluntary) minority groups. Social capital helps children engage with different age groups that share a common goal.[25]
Voluntary minorities[edit]
Immigrants take on minority status in their new country, usually in hopes of a better future economically, educationally, and politically than in their homeland. Because of their focus on success, voluntary minorities are more likely to do better in school than other migrating minorities.[23] Adapting to a very different culture and language make difficulties in the early stages of life in the new country. Voluntary immigrants do not experience a sense of divided identity as much as involuntary minorities, and are often rich in social capital because of their educational ambitions.[25] Major immigrant groups in the United States include Mexicans, Central and South Americans, Cubans, Africans, and Indians.[24]
Gender and sexuality minorities[edit]
Main article: Sexual minority
The term sexual minority is frequently used by public health researchers to recognize a wide variety of individuals who engage in same-sex sexual behavior, including those who do not identify under the LGBTQ umbrella. For example, men who have sex with men (MSM), but do not identify as gay. In addition, the term gender minorities can include many types of gender variant people, such as intersex people, transgender people, or gender non-conforming individuals. However, the terms sexual and gender minority are often not preferred by LGBTQ people, as they represent clinical categories rather than individual identity.[26]
Though lesbian, gay, bisexual, transgender, and queer (LGBTQ) people have existed throughout human history, LGBT rights movements across many western countries led to the recognition of LGBTQ people as members of a minority group.[27] LGBTQ people represent a numerical and social minority. They experience numerous social inequalities stemming from their group membership as LGBTQ people. These inequalities include social discrimination and isolation, unequal access to healthcare, employment, and housing, and experience negative mental and physical health outcomes due to these experiences.[26]
People with disabilities[edit]
The disability rights movement has contributed to an understanding of people with disabilities (including not to be called 'disabled') as a minority or a coalition of minorities who are disadvantaged by society, not just as people who are disadvantaged by their impairments. Advocates of disability rights emphasize difference in physical or psychological functioning, rather than inferiority. For example, some people with autism argue for acceptance of neurodiversity, much as opponents of racism argue for acceptance of ethnic diversity. The deaf community is often regarded as a linguistic and cultural minority rather than a group with disabilities, and some deaf people do not see themselves as having a disability at all. Rather, they are disadvantaged by technologies and social institutions that are designed to cater for the dominant group. (See the Convention on the Rights of Persons with Disabilities.)
Photo of the Rosenbergs in jail
Political minorities[edit]
One of the most controversial minorities in the United States and other countries has been communists. Along with the Red Scare and execution of Julius and Ethel Rosenberg, the United States ran open campaigns to fight, contain and promote fear of communism in the country. Some were persecuted as communist even when they were not actually so: for example, many activists for civil rights were portrayed as inspired by a communist agenda. Communists in the United States, as in many European countries, are often afraid to proclaim their politics, fearing abuse and discrimination from the political majority.
Religious minorities[edit]
Main article: Religious minorities
People belonging to religious minorities have a faith which is different from that held by the majority. Most countries of the world have religious minorities. It is now widely accepted in the west that people should have the freedom to choose their own religion, including not having any religion (atheism and/or agnosticism), and including the right to convert from one religion to another. However, in many countries this freedom is constricted. For example, in Egypt, a new system of identity cards[28] requires all citizens to state their religion—and the only choices are Islam, Christianity, or Judaism (See Egyptian identification card controversy).
Regional minorities[edit]
Authors have pointed out that many coal workers would be unwilling to move for work or were not likely to be able to be retrained as Appalachians are an "ethnic minority".[29]
Women as minorities[edit]
While in most societies, numbers of men and women are roughly equal, the status of women as a subordinate group has led to many social scientists to study them as a minority group.[30] Though women's legal rights and status vary widely across countries, women experience social inequalities relative to men in most societies.[31] Women are often denied access to education, subject to violence, and lack access to the same economic opportunities as men.[32]
Law and government[edit]
In the politics of some countries, a "minority" is an ethnic group recognized by law, and having specified rights. Speakers of a legally recognized minority language, for instance, might have the right to education or communication with the government in their mother tongue. Countries with special provisions[which?] for minorities include Canada, China, Ethiopia, Germany, India, the Netherlands, Poland, Romania, Russia, Croatia, and the United Kingdom.[citation needed]
The various minority groups in a country are often not given equal treatment. Some groups are too small or indistinct to obtain minority protections. For example, a member of a particularly small ethnic group might be forced to check "Other" on a checklist of different backgrounds and so might receive fewer privileges than a member of a more defined group.
Many contemporary governments prefer to assume the people they rule all belong to the same nationality rather than separate ones based on ethnicity. The United States asks for race and ethnicity on its official census forms, which thus breaks up and organizes its population into sub-groups, primarily racial rather than national. Spain does not divide its nationals by ethnic group, although it does maintain an official notion of minority languages.
Some especially significant or powerful minorities receive comprehensive protection and political representation. For example, the former Yugoslav republic of Bosnia and Herzegovina recognizes the three constitutive nations, none of which constitutes a numerical majority (see nations of Bosnia and Herzegovina). However, other minorities such as Romani[33] and Jews, are officially labelled "foreign" and are excluded from many of these protections. For example, they may be excluded from political positions, including the presidency.[34]
There is debate over recognizing minority groups and their privileges. One view[35] is that the application of special rights to minority groups may harm some countries, such as new states in Africa or Latin America not founded on the European nation-state model, since minority recognition may interfere with establishing a national identity. It may hamper the integration of the minority into mainstream society, perhaps leading to separatism or supremacism. In Canada, some[who?] feel that the failure of the dominant English-speaking majority to integrate French Canadians has provoked Quebec separatism.
Others assert that minorities require specific protections to ensure that they are not marginalised: for example, bilingual education may be needed to allow linguistic minorities to fully integrate into the school system and compete equally in society. In this view, rights for minorities strengthen the nation-building project, as members of minorities see their interests well served, and willingly accept the legitimacy of the nation and their integration (not assimilation) within it.[36]
See also[edit]
Sociology portal
Dominant minority
Ethnic minorities in Northern Ireland
Ethnic penalty
Intangible cultural heritage
Interminority racism
List of active NGOs of national minorities
List of minority political parties
Middleman minority
Minority influence
Minority language
Minority (philosophy)
Minority religion
Minority Rights Group International
Model minority
Serge Moscovici
Social exclusion
Social stratification
Tokenism
References[edit]
Jump up ^ 1945-, Healey, Joseph F., (2018-03-02). Race, ethnicity, gender, & class : the sociology of group conflict and change. Stepnick, Andi,, O'Brien, Eileen, 1972- (Eighth ed.). Thousand Oaks, California. ISBN 9781506346946. OCLC 1006532841.
Jump up ^ George,, Ritzer, (2014-01-15). Essentials of sociology. Los Angeles. ISBN 9781483340173. OCLC 871004576.
^ Jump up to: a b Laurie, Timothy; Khan, Rimi (2017), "The Concept of Minority for the Study of Culture", Continuum: Journal for Media and Cultural Studies, 31 (1): 3
Jump up ^ Johnson, Kevin. "The Struggle for Civil Rights: The Need for, and Impediments to, Political Coalitions among and within Minority Groups". heinonline.org. Louisiana Law Review. Retrieved 2018-08-14.
Jump up ^ 1930-2014., Becker, Gary S. (Gary Stanley), (1971). The economics of discrimination (2nd ed.). Chicago: University of Chicago Press. ISBN 9780226041049. OCLC 658199810.
Jump up ^ WILLIAMS, DAVID R. (1999). "Race, Socioeconomic Status, and Health The Added Effects of Racism and Discrimination". Annals of the New York Academy of Sciences. 896 (1): 173–188. doi:10.1111/j.1749-6632.1999.tb08114.x. ISSN 0077-8923.
Jump up ^ Verloo, Mieke (2006). "Multiple Inequalities, Intersectionality and the European Union". European Journal of Women's Studies. 13 (3): 211–228. doi:10.1177/1350506806065753. ISSN 1350-5068.
Jump up ^ David., Skrentny, John (2002). The minority rights revolution. Cambridge, Mass.: Belknap Press of Harvard University Press. ISBN 9780674043732. OCLC 431342257.
^ Jump up to: a b Wirth, L. (1945). "The Problem of Minority Groups". In Linton, Ralph. The Science of Man in the World Crisis. New York: Columbia University Press. p. 347. The political scientist and law professor, Gad Barzilai, has offered a theoretical definition of non-ruling communities that conceptualizes groups that do not rule and are excluded from resources of political power. Barzilai, G. Communities and Law: Politics and Cultures of Legal Identities. Ann Arbor: University of Michigan Press.
Jump up ^ Wagley, Charles; Harris, Marvin (1958). Minorities in the new world : six case studies. New York : Columbia University Press.
Jump up ^ Joe R. Feagin (1984). Racial and Ethnic Relations (2nd ed.). Prentice-Hall. p. 10. ISBN 978-0-13-750125-0.
Jump up ^ Diversity Training University International (2008). Cultural Diversity Glossary of Terms. Diversity Training University International Publications Division. p. 4.
Jump up ^ Barzilai, Gad (2010). Communities and Law: Politics and Cultures of Legal Identities. University of Michigan Press.
Jump up ^ Laurie, Timothy; Khan, Rimi (2017), "The Concept of Minority for the Study of Culture", Continuum: Journal for Media and Cultural Studies, 31 (1): 1–12
Jump up ^ Konrad, Alison M.; Linnehan, Frank (1999), "Handbook of Gender & Work Handbook of gender & work", Handbook of Gender & Work, SAGE Publications, Inc., pp. 429–452, doi:10.4135/9781452231365.n22, ISBN 9780761913559, retrieved 2018-08-15
^ Jump up to: a b Daniel Šmihula (2008). "National Minorities in the Law of the EC/EU" (PDF). Romanian Journal of European Affairs. 8 (3): 51–81. Archived from the original (PDF) on 2011-08-23.
Jump up ^ Oleh., Protsyk, (2010). The representation of minorities and indigenous peoples in parliament : a global overview. Inter-parliamentary Union. Geneva: Inter-parliamentary Union. ISBN 9789291424627. OCLC 754152959.
Jump up ^ Verkuyten, Maykel (2005). "Ethnic Group Identification and Group Evaluation Among Minority and Majority Groups: Testing the Multiculturalism Hypothesis". Journal of Personality and Social Psychology. 88 (1): 121–138. CiteSeerX 10.1.1.595.7633. doi:10.1037/0022-3514.88.1.121. ISSN 1939-1315. PMID 15631579.
Jump up ^ Lyal S. Sunga (2004). International Criminal Law: Protection of Minority Rights, Beyond a One-Dimensional State: An Emerging Right to Autonomy? ed. Zelim Skurbaty. (2004) 255–275.
Jump up ^ "Factsheet on the Framework Convention for the Protection of National Minorities". National Minorities (FCNM). Retrieved 2018-08-17.
Jump up ^ du Toit, Pierre; Theron, François (1988). "Ethnic and minority groups, and constitutional change in South Africa". Journal of Contemporary African Studies. 7 (1–2): 133–147. doi:10.1080/02589008808729481. ISSN 0258-9001.
Jump up ^ "U.S. Census Bureau QuickFacts: UNITED STATES". Census Bureau QuickFacts. Retrieved 2018-08-17.
^ Jump up to: a b Ogbu, John U. "Understanding Cultural Diversity and Learning" (PDF).
^ Jump up to: a b Ogbu and Simons (1998). "Voluntary and Involuntary Minorities: A Cultural-Ecological Theory of School Performance with Some Implications for Education" (PDF). Anthropology and Education Quarterly.
^ Jump up to: a b Valenzuela, Angela. Subtractive Schooling. pp. 116–118.
^ Jump up to: a b Mayer, Kenneth H.; Bradford, Judith B.; Makadon, Harvey J.; Stall, Ron; Goldhammer, Hilary; Landers, Stewart (2008). "Sexual and Gender Minority Health: What We Know and What Needs to Be Done". American Journal of Public Health. 98 (6): 989–995. doi:10.2105/ajph.2007.127811. ISSN 0090-0036. PMC 2377288. PMID 18445789.
Jump up ^ Mayer, Kenneth H.; Bradford, Judith B.; Makadon, Harvey J.; Stall, Ron; Goldhammer, Hilary; Landers, Stewart (2008). "Sexual and Gender Minority Health: What We Know and What Needs to Be Done". American Journal of Public Health. 98 (6): 989–995. doi:10.2105/ajph.2007.127811. ISSN 0090-0036. PMC 2377288. PMID 18445789.
Jump up ^ See "The Situation of the Bahá'í Community of Egypt" and "Religion Today: Bahais' struggle for recognition reveals a less tolerant face of Egypt", Bahai.org, DWB.sacbee.com Archived 2007-10-14 at the Wayback Machine.
Jump up ^ http://www.washingtontimes.com, The Washington Times (June 14, 2008). "OPINION: America's other minority?".
Jump up ^ Hacker, Helen Mayer (1951). "Women as a Minority Group". Social Forces. 30 (1): 60–69. doi:10.2307/2571742. JSTOR 2571742.
Jump up ^ 1966-, Shachar, Ayelet, (2001). Multicultural jurisdictions : cultural differences and women's rights. Cambridge, UK: Cambridge University Press. ISBN 978-0511040801. OCLC 56216656.
Jump up ^ Women, U. N. (2018). Annual Report 2017–2018.
Jump up ^ "Political recognition of Roma People in Spain. [Social Impact]. WORKALÓ. The creation of new occupational patterns for cultural minorities: the Gypsy Case (2001-2004). Framework Programme 5 (FP5)". SIOR, Social Impact Open Repository.
Jump up ^ Opinion of the Council of Europe's Advisory Committee on the Framework Convention for the Protection of National Minorities, in particular paragraphs 37–43 Archived 2007-06-16 at the Wayback Machine.
Jump up ^ For example, J.A. Lindgren-Alves, member of the United Nations Committee on the Elimination of Racial Discrimination, speaking at the Committee's 67th Session (Summary Record of the 1724th Meeting, 23 August 2005, CERD/C/SR.1724)
Jump up ^ See Henrard, K. (2000). Devising an Adequate System of Minority Protection: Individual Human Rights, Minority Rights and the Right to Self-Determination. Martinus Nijhoff. pp. 218–224. ISBN 978-9041113597.
External links[edit]
Union of Minority Shareholders
ECMI - European Centre for Minority Issues
Minority language tool
What is a Minority Group? definitions from Dayton Law School.
From Paris to Cairo: Resistance of the Unacculturated
Minorities at Risk project at the University of Maryland
MINELRES - Minority Electronic Resources
European Academy Bozen/Bolzano (EURAC)
Eurominority - Stateless and national minorities portal
State of the World's Minorities, an annual report by Minority Rights Group International
American Psychological Association's Office of Ethnic Minority Affairs
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text:
Coefficient of restitution
From Wikipedia, the free encyclopedia
Jump to: navigation, search
A bouncing basketball captured with a stroboscopic flash at 25 images per second: Ignoring air resistance, the square root of the ratio of the height of one bounce to that of the preceding bounce gives the coefficient of restitution for the ball/surface impact.
The coefficient of restitution (COR) is the ratio of the final to initial relative velocity between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectly inelastic collision has a coefficient of 0, but a 0 value does not have to be perfectly inelastic. It is measured in the Leeb rebound hardness test, expressed as 1000 times the COR, but it is only a valid COR for the test, not as a universal COR for the material being tested.
The value is almost always less than one due to initial translational kinetic energy being lost to rotational kinetic energy, plastic deformation, and heat. It can be more than 1 if there is an energy gain during the collision from a chemical reaction, a reduction in rotational energy, or another internal energy decrease that contributes to the post-collision velocity.
Coefficient of restitution ( e ) = Relative velocity after collision Relative velocity before collision {\displaystyle {\text{Coefficient of restitution }}(e)={\frac {\text{Relative velocity after collision}}{\text{ Relative velocity before collision}}}}
The coefficient is related to energy by e = K E (after collision) K E (before collision) {\displaystyle e={\sqrt {\frac {KE_{\text{(after collision)}}}{KE_{\text{(before collision)}}}}}}
The mathematics were developed by Sir Isaac Newton[1] in 1687. It is also known as Newton's experimental law.
Contents
[hide]
1 Further details
1.1 Range of values for e – treated as a constant
1.2 Paired objects
1.3 Relationship with conservation of energy and momentum
1.4 Sports equipment
2 Equations
3 Speeds after impact
3.1 Derivation
3.2 COR variation due to object shape and off-center collisions
3.3 Colliding different materials and practical measurement
3.4 Predicting from material properties
4 See also
5 References
6 External links
Further details[edit]
Line of impact – It is the line along which e is defined or in absence of tangential reaction force between colliding surfaces, force of impact is shared along this line between bodies. During physical contact between bodies during impact its line along common normal to pair of surfaces in contact of colliding bodies. Hence e is defined as a dimensionless one-dimensional parameter.
Range of values for e – treated as a constant[edit]
e is usually a positive, real number between 0 and 1:
e = 0: This is a perfectly inelastic collision. The objects do not move apart after the collision, but instead they coalesce. Kinetic energy is converted to heat or work done in deforming the objects.
0 < e < 1: This is a real-world inelastic collision, in which some kinetic energy is dissipated.
e = 1: This is a perfectly elastic collision, in which no kinetic energy is dissipated, and the objects rebound from one another with the same relative speed with which they approached.
e < 0: A COR less than zero would represent a collision in which the separation velocity of the objects has the same direction (sign) as the closing velocity, implying the objects passed through one another without fully engaging. This may also be thought of as an incomplete transfer of momentum. An example of this might be a small, dense object passing through a large, less dense one – e.g., a bullet passing through a target, or a motorcycle passing through a motor home or a wave tearing through a dam.
e > 1: This would represent a collision in which energy is released, for example, nitrocellulose billiard balls can literally explode at the point of impact. Also, some recent articles have described superelastic collisions in which it is argued that the COR can take a value greater than one in a special case of oblique collisions.[2][3][4] These phenomena are due to the change of rebound trajectory caused by friction. In such collision kinetic energy is increased in a manner energy is released in some sort of explosion. It is possible that e = ∞ {\displaystyle e=\infty } for a perfect explosion of a rigid system.
Maximum deformation phase – In any collision for 0 < e ≤ 1, there is a condition when for short moment along line of impact colliding bodies have same velocity when its condition of kinetic energy is lost in maximum fraction as heat, sound and light with deformation potential energy. For this short duration this collision e=0 and may be referred as inelastic phase.
Paired objects[edit]
The COR is a property of a pair of objects in a collision, not a single object. If a given object collides with two different objects, each collision would have its own COR. When an object is described as having a coefficient of restitution, as if it were an intrinsic property without reference to a second object, it is assumed to be between identical spheres or against a perfectly rigid wall.
A perfectly rigid wall is not possible but can be approximated by a steel block if investigating the COR of spheres with a much smaller modulus of elasticity. Otherwise, the COR will rise and then fall based on collision velocity in a more complicated manner.[5]
Relationship with conservation of energy and momentum[edit]
In a one-dimensional collision, the two key principles are: conservation of energy (conservation of kinetic energy if the collision is perfectly elastic) and conservation of (linear) momentum. A third equation can be derived[6] from these two, which is the restitution equation as stated above. When solving problems, any two of the three equations can be used. The advantage of using the restitution equation is that it sometimes provides a more convenient way to approach the problem.
Let m 1 {\displaystyle m_{1}} , m 2 {\displaystyle m_{2}} be the mass of object 1 and object 2 respectively. Let u 1 {\displaystyle u_{1}} , u 2 {\displaystyle u_{2}} be the initial velocity of object 1 and object 2 respectively. Let v 1 {\displaystyle v_{1}} , v 2 {\displaystyle v_{2}} be the final velocity of object 1 and object 2 respectively.
{ 1 2 m 1 u 1 2 + 1 2 m 2 u 2 2 = 1 2 m 1 v 1 2 + 1 2 m 2 v 2 2 m 1 u 1 + m 2 u 2 = m 1 v 1 + m 2 v 2 {\displaystyle {\begin{cases}{\frac {1}{2}}m_{1}u_{1}^{2}+{\frac {1}{2}}m_{2}u_{2}^{2}={\frac {1}{2}}m_{1}v_{1}^{2}+{\frac {1}{2}}m_{2}v_{2}^{2}\\m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}\end{cases}}}
From the first equation,
m 1 ( u 1 2 − v 1 2 ) = m 2 ( v 2 2 − u 2 2 ) {\displaystyle m_{1}(u_{1}^{2}-v_{1}^{2})=m_{2}(v_{2}^{2}-u_{2}^{2})}
m 1 ( u 1 + v 1 ) ( u 1 − v 1 ) = m 2 ( v 2 + u 2 ) ( v 2 − u 2 ) {\displaystyle m_{1}(u_{1}+v_{1})(u_{1}-v_{1})=m_{2}(v_{2}+u_{2})(v_{2}-u_{2})}
From the second equation,
m 1 ( u 1 − v 1 ) = m 2 ( v 2 − u 2 ) {\displaystyle m_{1}(u_{1}-v_{1})=m_{2}(v_{2}-u_{2})}
After division,
u 1 + v 1 = v 2 + u 2 {\displaystyle u_{1}+v_{1}=v_{2}+u_{2}}
u 1 − u 2 = − ( v 1 − v 2 ) {\displaystyle u_{1}-u_{2}=-(v_{1}-v_{2})}
− v 1 − v 2 u 1 − u 2 = 1 {\displaystyle -{\frac {v_{1}-v_{2}}{u_{1}-u_{2}}}=1}
The equation above is the restitution equation, and the coefficient of restitution is 1, which is a perfectly elastic collision.
Sports equipment[edit]
See also: Bouncing ball § Sport regulations
The coefficient of restitution entered the common vocabulary, among golfers at least, when golf club manufacturers began making thin-faced drivers with a so-called "trampoline effect" that creates drives of a greater distance as a result of the flexing and subsequent release of stored energy, imparting greater impulse to the ball. The USGA (America's governing golfing body) has started testing drivers for COR and has placed the upper limit at 0.83. According to one article (addressing COR in tennis racquets), "[f]or the Benchmark Conditions, the coefficient of restitution used is 0.85 for all racquets, eliminating the variables of string tension and frame stiffness which could add or subtract from the coefficient of restitution."[7]
The International Table Tennis Federation specifies that the ball shall bounce up 24–26 cm when dropped from a height of 30.5 cm on to a standard steel block thereby having a COR of 0.89 to 0.92.[8] For a hard linoleum floor with concrete underneath, a leather basketball has a COR around 0.81–0.85.[9]
Equations[edit]
In the case of a one-dimensional collision involving two objects, object A and object B, the coefficient of restitution is given by:
C R = v b − v a u a − u b {\displaystyle C_{R}={\frac {v_{\text{b}}-v_{\text{a}}}{u_{\text{a}}-u_{\text{b}}}}} , where:
v a {\displaystyle v_{\text{a}}} is the final velocity of object A after impact
v b {\displaystyle v_{\text{b}}} is the final velocity of object B after impact
u a {\displaystyle u_{\text{a}}} is the initial velocity of object A before impact
u b {\displaystyle u_{\text{b}}} is the initial velocity of object B before impact
Though C R {\displaystyle C_{R}} does not explicitly depend on the masses of the objects, it is important to note that the final velocities are mass-dependent. For two- and three-dimensional collisions of rigid bodies, the velocities used are the components perpendicular to the tangent line/plane at the point of contact, i.e. along the line of impact.
For an object bouncing off a stationary target, C R {\displaystyle C_{R}} is defined as the ratio of the object's speed after the impact to that of prior to impact:
C R = v u {\displaystyle C_{R}={\frac {v}{u}}} , where
v {\displaystyle v} is the speed of the object after impact
u {\displaystyle u} is the speed of the object before impact
In a case where frictional forces can be neglected and the object is dropped from rest onto a horizontal surface, this is equivalent to:
C R = h H {\displaystyle C_{R}={\sqrt {\frac {h}{H}}}} , where
h {\displaystyle h} is the bounce height
H {\displaystyle H} is the drop height
The coefficient of restitution can be thought of as a measure of the extent to which mechanical energy is conserved when an object bounces off a surface. In the case of an object bouncing off a stationary target, the change in gravitational potential energy, PE, during the course of the impact is essentially zero; thus, C R {\displaystyle C_{R}} is a comparison between the kinetic energy, KE, of the object immediately before impact with that immediately after impact:
C R = K E (after impact) K E (before impact) = 1 2 m v 2 1 2 m u 2 = v 2 u 2 = v u {\displaystyle C_{R}={\sqrt {\frac {KE_{\text{(after impact)}}}{KE_{\text{(before impact)}}}}}={\sqrt {\frac {{\frac {1}{2}}mv^{2}}{{\frac {1}{2}}mu^{2}}}}={\sqrt {\frac {v^{2}}{u^{2}}}}={\frac {v}{u}}}
In a cases where frictional forces can be neglected (nearly every student laboratory on this subject[10]) and the object is dropped from rest onto a horizontal surface, the above is equivalent to a comparison between the PE of the object at the drop height with that at the bounce height. In this case, the change in KE is zero (the object is essentially at rest during the course of the impact and is also at rest at the apex of the bounce); thus:
C R = P E (at bounce height) P E (at drop height) = m g h m g H = h H {\displaystyle C_{R}={\sqrt {\frac {PE_{\text{(at bounce height)}}}{PE_{\text{(at drop height)}}}}}={\sqrt {\frac {mgh}{mgH}}}={\sqrt {\frac {h}{H}}}}
Speeds after impact[edit]
The equations for collisions between elastic particles can be modified to use the COR, thus becoming applicable to inelastic collisions, as well, and every possibility in between.
v a = m a u a + m b u b + m b C R ( u b − u a ) m a + m b {\displaystyle v_{a}={\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}+m_{\text{b}}C_{R}(u_{\text{b}}-u_{\text{a}})}{m_{\text{a}}+m_{\text{b}}}}}
and
v b = m a u a + m b u b + m a C R ( u a − u b ) m a + m b {\displaystyle v_{\text{b}}={\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}+m_{\text{a}}C_{R}(u_{\text{a}}-u_{\text{b}})}{m_{\text{a}}+m_{\text{b}}}}}
where
v a {\displaystyle v_{\text{a}}} is the final velocity of the first object after impact
v b {\displaystyle v_{\text{b}}} is the final velocity of the second object after impact
u a {\displaystyle u_{\text{a}}} is the initial velocity of the first object before impact
u b {\displaystyle u_{\text{b}}} is the initial velocity of the second object before impact
m a {\displaystyle m_{\text{a}}} is the mass of the first object
m b {\displaystyle m_{\text{b}}} is the mass of the second object
Derivation[edit]
The above equations can be derived from the analytical solution to the system of equations formed by the definition of the COR and the law of the conservation of momentum (which holds for all collisions). Using the notation from above where u {\displaystyle u} represents the velocity before the collision and v {\displaystyle v} after, yields:
m a u a + m b u b = m a v a + m b v b C R = v b − v a u a − u b {\displaystyle {\begin{aligned}&m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}=m_{\text{a}}v_{\text{a}}+m_{\text{b}}v_{\text{b}}\\&C_{R}={\frac {v_{\text{b}}-v_{\text{a}}}{u_{\text{a}}-u_{\text{b}}}}\\\end{aligned}}}
Solving the momentum conservation equation for v a {\displaystyle v_{\text{a}}} and the definition of the coefficient of restitution for v b {\displaystyle v_{\text{b}}} yields:
m a u a + m b u b − m b v b m a = v a v b = C R ( u a − u b ) + v a {\displaystyle {\begin{aligned}&{\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}-m_{\text{b}}v_{\text{b}}}{m_{\text{a}}}}=v_{\text{a}}\\&v_{\text{b}}=C_{R}(u_{\text{a}}-u_{\text{b}})+v_{\text{a}}\\\end{aligned}}}
Next, substitution into the first equation for v b {\displaystyle v_{\text{b}}} and then resolving for v a {\displaystyle v_{\text{a}}} gives:
m a u a + m b u b − m b C R ( u a − u b ) − m b v a m a = v a m a u a + m b u b + m b C R ( u b − u a ) m a = v a [ 1 + m b m a ] m a u a + m b u b + m b C R ( u b − u a ) m a + m b = v a {\displaystyle {\begin{aligned}&{\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}-m_{\text{b}}C_{R}(u_{\text{a}}-u_{\text{b}})-m_{\text{b}}v_{\text{a}}}{m_{\text{a}}}}=v_{\text{a}}\\&\\&{\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}+m_{\text{b}}C_{R}(u_{\text{b}}-u_{\text{a}})}{m_{\text{a}}}}=v_{\text{a}}\left[1+{\frac {m_{\text{b}}}{m_{\text{a}}}}\right]\\&\\&{\frac {m_{\text{a}}u_{\text{a}}+m_{\text{b}}u_{\text{b}}+m_{\text{b}}C_{R}(u_{\text{b}}-u_{\text{a}})}{m_{\text{a}}+m_{\text{b}}}}=v_{\text{a}}\\\end{aligned}}}
A similar derivation yields the formula for v b {\displaystyle v_{\text{b}}} .
COR variation due to object shape and off-center collisions[edit]
When colliding objects do not have a direction of motion that is in-line with their centers of gravity and point of impact, or if their contact surfaces at that point are not perpendicular to that line, some energy that would have been available for the post-collision velocity difference will be lost to rotation and friction. Energy losses to vibration and the resulting sound are usually negligible.
Colliding different materials and practical measurement[edit]
When a soft object strikes a harder object, most of the energy available for the post-collision velocity will be stored in the soft object. The COR will depend on how efficient the soft object is at storing the energy in compression without losing it to heat and plastic deformation. A rubber ball will bounce better off concrete than a glass ball, but the COR of glass-on-glass is a lot higher than rubber-on-rubber because some of the energy in rubber is lost to heat when it is compressed. When a rubber ball collides with a glass ball, the COR will depend entirely on the rubber. For this reason, determining the COR of a material when there is not identical material for collision is best done by using a much harder material.
Since there is no perfectly rigid material, hard materials such as metals and ceramics have their COR theoretically determined by considering the collision between identical spheres. In practice, a 2-ball Newton's cradle may be employed but such a set up is not conducive to quickly testing samples.
The Leeb rebound hardness test is the only commonly-available test related to determining the COR. It uses a tip of tungsten carbide, one of the hardest substances available, dropped onto test samples from a specific height. But the shape of the tip, the velocity of impact, and the tungsten carbide are all variables that affect the result that is expressed in terms of 1000*COR. It does not give an objective COR for the material that is independent from the test.
Predicting from material properties[edit]
The COR is not a material property because it changes with the shape of the material and the specifics of the collision, but it can be predicted from material properties and the velocity of impact when the specifics of the collision are simplified. To avoid the complications of rotational and frictional losses, we can consider the ideal case of an identical pair of spherical objects, colliding so that their centers of mass and relative velocity are all in-line.
Many materials like metals and ceramics (but not rubbers and plastics) are assumed to be perfectly elastic when their yield strength is not approached during impact. The impact energy is theoretically stored only in the spring-effect of elastic compression and results in e = 1. But this applies only at velocities less than about 0.1 m/s to 1 m/s. The elastic range can be exceeded at higher velocities because all the kinetic energy is concentrated at the point of impact. Specifically, the yield strength is usually exceeded in part of the contact area, losing energy to plastic deformation by not remaining in the elastic region. To account for this, the following estimates the COR by estimating the percent of the initial impact energy that did not get lost to plastic deformation. Approximately, it divides how easy a volume of the material can store energy in compression ( 1 / elastic modulus {\displaystyle 1/{\text{elastic modulus}}} ) by how well it can stay in the elastic range ( 1 / yield strength {\displaystyle 1/{\text{yield strength}}} ):
% impact energy available for restitution ∝ yield strength elastic modulus {\displaystyle \%{\text{impact energy available for restitution}}\propto {\frac {\text{yield strength}}{\text{elastic modulus}}}}
For a given material density and velocity this results in:
coefficient of restitution ∝ yield strength elastic modulus {\displaystyle {\text{coefficient of restitution}}\propto {\sqrt {\frac {\text{yield strength}}{\text{elastic modulus}}}}}
A high yield strength allows more of the "contact volume" of the material to stay in the elastic region at higher energies. A lower elastic modulus allows a larger contact area to develop during impact so the energy is distributed to a larger volume beneath the surface at the contact point. This helps prevent the yield strength from being exceeded.
A more precise theoretical development[11] shows the velocity and density of the material to also be important when predicting the COR at moderate velocities faster than elastic collision (greater than 0.1 m/s for metals) and slower than large permanent plastic deformation (less than 100 m/s). A lower velocity increases the coefficient by needing less energy to be absorbed. A lower density also means less initial energy needs to be absorbed. The density instead of mass is used because the volume of the sphere cancels out with the volume of the affected volume at the contact area. In this way, the radius of the sphere does not affect the coefficient. A pair of colliding spheres of different sizes but of the same material have the same coefficient as below, but multiplied by ( R 1 R 2 ) 3 8 {\displaystyle \left({\frac {R_{1}}{R_{2}}}\right)^{\frac {3}{8}}}
Combining these four variables, a theoretical estimation of the coefficient of restitution can be made when a ball is dropped onto a surface of the same material.[12]
e = coefficient of restitution
Sy = dynamic yield strength (dynamic "elastic limit")
E′ = effective elastic modulus
ρ = density
v = velocity at impact
μ = Poisson's ratio
e = 3.1 ( S y 1 ) 5 8 ( 1 E ′ ) 1 2 ( 1 v ) 1 4 ( 1 ρ ) 1 8 {\displaystyle e=3.1\left({\frac {S_{\text{y}}}{1}}\right)^{\frac {5}{8}}\left({\frac {1}{E'}}\right)^{\frac {1}{2}}\left({\frac {1}{v}}\right)^{\frac {1}{4}}\left({\frac {1}{\rho }}\right)^{\frac {1}{8}}}
E ′ = E 1 − μ 2 {\displaystyle E'={\frac {E}{1-\mu ^{2}}}}
This equation overestimates the actual COR. For metals, it applies when v is approximately between 0.1 m/s and 100 m/s and in general when:
0.001 < ρ v 2 S y < 0.1 {\displaystyle 0.001<{\frac {\rho v^{2}}{S_{\text{y}}}}<0.1}
At slower velocities the COR is higher than the above equation predicts, theoretically reaching e=1 when the above fraction is less than 10 − 6 {\displaystyle 10^{-6}} m/s. It gives the following theoretical coefficient of restitution for solid spheres dropped 1 meter (v = 4.5 m/s). Values greater than 1 indicate that the equation has errors. Yield strength instead of dynamic yield strength was used.
Metals and Ceramics: Predicted COR, e
silicon 1.79
Alumina 0.45 to 1.63
silicon nitride 0.38 to 1.63
silicon carbide 0.47 to 1.31
highest amorphous metal 1.27
tungsten carbide 0.73 to 1.13
stainless steel 0.63 to 0.93
magnesium alloys 0.5 to 0.89
titanium alloy grade 5 0.84
aluminum alloy 7075-T6 0.75
glass (soda-lime) 0.69
glass (borosilicate) 0.66
nickel alloys 0.15 to 0.70
zinc alloys 0.21 to 0.62
cast iron 0.3 to 0.6
copper alloys 0.15 to 0.55
titanium grade 2 0.46
tungsten 0.37
aluminum alloys 3003 6061, 7075-0 0.35
zinc 0.21
nickel 0.15
copper 0.15
aluminum 0.1
lead 0.08
The COR for plastics and rubbers are greater than their actual values because they do not behave as ideally elastic as metals, glasses, and ceramics due to heating during compression. So the following is only a guide to ranking of polymers.
Polymers (overestimated compared to metals and ceramics):
polybutadiene (golf balls shell) 11.8
butyl rubber 6.24
EVA 4.85
silicone elastomers 2.80
polycarbonate 1.46
nylon 1.28
polyethylene 1.24
Teflon 1.21
polypropylene 1.14
ABS 1.12
acrylic 1.06
PET 0.95
polystyrene 0.87
PVC 0.86
For metals the range of speeds to which this theory can apply is about 0.1 to 5 m/s which is a drop of 0.5 mm to 1.25 meters (page 366[13]).
See also[edit]
Bouncing ball
Collision
Resilience
References[edit]
Jump up ^ "'A' level Revision:Newton's Law of Restitution". Retrieved 12 March 2013.
Jump up ^ Louge, Michel; Adams, Michael (2002). "Anomalous behavior of normal kinematic restitution in the oblique impacts of a hard sphere on an elastoplastic plate". Physical Review E. 65 (2). Bibcode:2002PhRvE..65b1303L. doi:10.1103/PhysRevE.65.021303.
Jump up ^ Kuninaka, Hiroto; Hayakawa, Hisao (2004). "Anomalous Behavior of the Coefficient of Normal Restitution in Oblique Impact". Physical Review Letters. 93 (15): 154301. arXiv:cond-mat/0310058  . Bibcode:2004PhRvL..93o4301K. doi:10.1103/PhysRevLett.93.154301. PMID 15524884.
Jump up ^ Calsamiglia, J.; Kennedy, S. W.; Chatterjee, A.; Ruina, A.; Jenkins, J. T. (1999). "Anomalous Frictional Behavior in Collisions of Thin Disks". Journal of Applied Mechanics. 66 (1): 146. Bibcode:1999JAM....66..146C. doi:10.1115/1.2789141.
Jump up ^ "IMPACT STUDIES ON PURE METALS" (PDF). Archived from the original (PDF) on March 19, 2015.
Jump up ^ "Impulse and momentum. Conservation of momentum. Elastic and inelastic collisions. Coefficient of Restitution".
Jump up ^ "Coefficient of Restitution". Archived from the original on 2016-11-23.
Jump up ^ "ITTF Technical Leaflet T3: The Ball" (PDF). ITTF. December 2009. p. 4. Archived from the original (PDF) on 4 March 2011. Retrieved 28 July 2010.
Jump up ^ "UT Arlington Physicists Question New Synthetic NBA Basketball". Retrieved May 8, 2011.
Jump up ^ Mohazzabi, Pirooz. "When Does Air Resistance Become Significant in Free Fall?".
Jump up ^ http://www-mdp.eng.cam.ac.uk/web/library/enginfo/cueddatabooks/materials.pdf
Jump up ^ http://itzhak.green.gatech.edu/rotordynamics/Predicting%20the%20coefficient%20of%20restitution%20of%20impacting%20spheres.pdf
Jump up ^ http://www.ewp.rpi.edu/hartford/~ernesto/S2015/FWLM/Books_Links/Books/Johnson-CONTACTMECHANICS.pdf
Works cited
Cross, Rod (2006). "The bounce of a ball" (PDF). Physics Department, University of Sydney, Australia. Retrieved 2008-01-16.
Walker, Jearl (2011). Fundamentals Of Physics (9th ed.). David Halliday, Robert Resnick, Jearl Walker. ISBN 978-0-470-56473-8.
External links[edit]
Wolfram Article on COR
Bennett & Meepagala (2006). "Coefficients of Restitution". The Physics Factbook.
Chris Hecker's physics introduction
"Getting an extra bounce" by Chelsea Wald
FIFA Quality Concepts for Footballs – Uniform Rebound
Bowley, Roger (2009). "Coefficient of Restitution". Sixty Symbols. Brady Haran for the University of Nottingham.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Coefficient_of_restitution&oldid=829777410"
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values: {\displaystyle C_{R}={\sqrt {\frac {h}{H}}}}

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values: C_R =\sqrt{\frac{KE_\text{(after impact)}}{KE_\text{(before impact)}}} =\sqrt{\frac{\frac{1}{2}mv^2}{\frac{1}{2}mu^2}} =\sqrt{\frac{v^2}{u^2}} =\frac{v}{u}

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values: C_R =\sqrt{\frac{PE_\text{(at bounce height)}}{PE_\text{(at drop height)}}} =\sqrt{\frac{mgh}{mgH}} =\sqrt{\frac{h}{H}}

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values: {\displaystyle E'={\frac {E}{1-\mu ^{2}}}}

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