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127-514: The term torque (from Latin torquēre , 'to twist') is said to have been suggested by James Thomson and appeared in print in April, 1884. Usage is attested the same year by Silvanus P. Thompson in the first edition of Dynamo-Electric Machinery . Thompson motivates the term as follows: Just as the Newtonian definition of force is that which produces or tends to produce motion (along

254-1054: A {\displaystyle \mathbf {F} =m\mathbf {a} } the torque equation becomes: τ = r × F = r × ( m α × r ) = m ( ( r ⋅ r ) α − ( r ⋅ α ) r ) = m r 2 α = I α k ^ , {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}&=\mathbf {r} \times \mathbf {F} =\mathbf {r} \times (m{\boldsymbol {\alpha }}\times \mathbf {r} )\\&=m\left(\left(\mathbf {r} \cdot \mathbf {r} \right){\boldsymbol {\alpha }}-\left(\mathbf {r} \cdot {\boldsymbol {\alpha }}\right)\mathbf {r} \right)\\&=mr^{2}{\boldsymbol {\alpha }}=I\alpha \mathbf {\hat {k}} ,\end{aligned}}} where k ^ {\displaystyle \mathbf {\hat {k}} }

381-445: A d / s ) 2 ≈ 0.99   m . {\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}\approx {\frac {9.81\ \mathrm {m/s^{2}} }{(3.14\ \mathrm {rad/s} )^{2}}}\approx 0.99\ \mathrm {m} .} Notice that the distance to the center of oscillation of the seconds pendulum must be adjusted to accommodate different values for the local acceleration of gravity. Kater's pendulum

508-534: A force is allowed to act through a distance, it is doing mechanical work . Similarly, if torque is allowed to act through an angular displacement, it is doing work. Mathematically, for rotation about a fixed axis through the center of mass , the work W can be expressed as W = ∫ θ 1 θ 2 τ   d θ , {\displaystyle W=\int _{\theta _{1}}^{\theta _{2}}\tau \ \mathrm {d} \theta ,} where τ

635-456: A body hanging from a pivot, known as a compound pendulum . The term moment of inertia ("momentum inertiae" in Latin ) was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Euler's second law . The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on

762-442: A body to the angular acceleration α around a principal axis, that is τ = I α . {\displaystyle \tau =I\alpha .} For a simple pendulum , this definition yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as, I = m r 2 . {\displaystyle I=mr^{2}.} Thus,

889-715: A certain leverage. Today, torque is referred to using different vocabulary depending on geographical location and field of study. This article follows the definition used in US physics in its usage of the word torque . In the UK and in US mechanical engineering , torque is referred to as moment of force , usually shortened to moment . This terminology can be traced back to at least 1811 in Siméon Denis Poisson 's Traité de mécanique . An English translation of Poisson's work appears in 1842. A force applied perpendicularly to

1016-404: A complex body as an assembly of simpler shaped bodies. The parallel axis theorem is used to shift the reference point of the individual bodies to the reference point of the assembly. As one more example, consider the moment of inertia of a solid sphere of constant density about an axis through its center of mass. This is determined by summing the moments of inertia of the thin discs that can form

1143-743: A few in German , Dutch , Norwegian , Danish and Swedish . Latin is still spoken in Vatican City, a city-state situated in Rome that is the seat of the Catholic Church . The works of several hundred ancient authors who wrote in Latin have survived in whole or in part, in substantial works or in fragments to be analyzed in philology . They are in part the subject matter of the field of classics . Their works were published in manuscript form before

1270-408: A lever multiplied by its distance from the lever's fulcrum (the length of the lever arm ) is its torque. Therefore, torque is defined as the product of the magnitude of the perpendicular component of the force and the distance of the line of action of a force from the point around which it is being determined. In three dimensions, the torque is a pseudovector ; for point particles , it is given by

1397-418: A line), so torque may be defined as that which produces or tends to produce torsion (around an axis). It is better to use a term which treats this action as a single definite entity than to use terms like " couple " and " moment ", which suggest more complex ideas. The single notion of a twist applied to turn a shaft is better than the more complex notion of applying a linear force (or a pair of forces) with

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1524-562: A new Classical Latin arose, a conscious creation of the orators, poets, historians and other literate men, who wrote the great works of classical literature , which were taught in grammar and rhetoric schools. Today's instructional grammars trace their roots to such schools , which served as a sort of informal language academy dedicated to maintaining and perpetuating educated speech. Philological analysis of Archaic Latin works, such as those of Plautus , which contain fragments of everyday speech, gives evidence of an informal register of

1651-407: A physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor. The moment of inertia of

1778-570: A pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of. The natural frequency ( ω n {\displaystyle \omega _{\text{n}}} ) of a compound pendulum depends on its moment of inertia, I P {\displaystyle I_{P}} , ω n = m g r I P , {\displaystyle \omega _{\text{n}}={\sqrt {\frac {mgr}{I_{P}}}},} where m {\displaystyle m}

1905-476: A remarkable unity in phonological forms and developments, bolstered by the stabilising influence of their common Christian (Roman Catholic) culture. It was not until the Muslim conquest of Spain in 711, cutting off communications between the major Romance regions, that the languages began to diverge seriously. The spoken Latin that would later become Romanian diverged somewhat more from the other varieties, as it

2032-423: A rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions in roll, pitch and yaw. The moment of inertia is defined as the product of mass of section and

2159-582: A similar derivation to the previous equation. Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield E K = 1 2 m v ⋅ v = 1 2 ( m r 2 ) ω 2 = 1 2 I ω 2 . {\displaystyle E_{\text{K}}={\frac {1}{2}}m\mathbf {v} \cdot \mathbf {v} ={\frac {1}{2}}\left(mr^{2}\right)\omega ^{2}={\frac {1}{2}}I\omega ^{2}.} This shows that

2286-511: A simple pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum this is found to be the product of the mass of the particle m {\displaystyle m} with the square of its distance r {\displaystyle r} to the pivot, that is I = m r 2 . {\displaystyle I=mr^{2}.} This can be shown as follows: The force of gravity on

2413-762: A single point particle is: L = r × p {\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} } where p is the particle's linear momentum and r is the position vector from the origin. The time-derivative of this is: d L d t = r × d p d t + d r d t × p . {\displaystyle {\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}=\mathbf {r} \times {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}+{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\times \mathbf {p} .} This result can easily be proven by splitting

2540-709: A small number of Latin services held in the Anglican church. These include an annual service in Oxford, delivered with a Latin sermon; a relic from the period when Latin was the normal spoken language of the university. In the Western world, many organizations, governments and schools use Latin for their mottos due to its association with formality, tradition, and the roots of Western culture . Canada's motto A mari usque ad mare ("from sea to sea") and most provincial mottos are also in Latin. The Canadian Victoria Cross

2667-411: Is Veritas ("truth"). Veritas was the goddess of truth, a daughter of Saturn, and the mother of Virtue. Switzerland has adopted the country's Latin short name Helvetia on coins and stamps, since there is no room to use all of the nation's four official languages . For a similar reason, it adopted the international vehicle and internet code CH , which stands for Confoederatio Helvetica ,

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2794-411: Is diagonal and torques around the axes act independently of each other. In mechanical engineering , simply "inertia" is often used to refer to " inertial mass " or "moment of inertia". When a body is free to rotate around an axis, torque must be applied to change its angular momentum . The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity )

2921-470: Is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a gravimeter . The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar pendulum . A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis. The period of oscillation of

3048-647: Is a general proof for point particles, but it can be generalized to a system of point particles by applying the above proof to each of the point particles and then summing over all the point particles. Similarly, the proof can be generalized to a continuous mass by applying the above proof to each point within the mass, and then integrating over the entire mass. In physics , rotatum is the derivative of torque with respect to time P = d τ d t , {\displaystyle \mathbf {P} ={\frac {\mathrm {d} {\boldsymbol {\tau }}}{\mathrm {d} t}},} where τ

3175-420: Is a kind of written Latin used in the 3rd to 6th centuries. This began to diverge from Classical forms at a faster pace. It is characterised by greater use of prepositions, and word order that is closer to modern Romance languages, for example, while grammatically retaining more or less the same formal rules as Classical Latin. Ultimately, Latin diverged into a distinct written form, where the commonly spoken form

3302-640: Is a reversal of the original phrase Non terrae plus ultra ("No land further beyond", "No further!"). According to legend , this phrase was inscribed as a warning on the Pillars of Hercules , the rocks on both sides of the Strait of Gibraltar and the western end of the known, Mediterranean world. Charles adopted the motto following the discovery of the New World by Columbus, and it also has metaphorical suggestions of taking risks and striving for excellence. In

3429-401: Is a unit vector perpendicular to the plane of the pendulum. (The second to last step uses the vector triple product expansion with the perpendicularity of α {\displaystyle {\boldsymbol {\alpha }}} and r {\displaystyle \mathbf {r} } .) The quantity I = m r 2 {\displaystyle I=mr^{2}} is

3556-414: Is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point ( x , y , z ) {\displaystyle (x,y,z)} in the solid, and the integration is evaluated over the volume V {\displaystyle V} of the body Q {\displaystyle Q} . The moment of inertia of a flat surface is similar with

3683-460: Is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their outstretched arms or divers curl their bodies into a tuck position during a dive, to spin faster. If the shape of the body does not change, then its moment of inertia appears in Newton's law of motion as the ratio of an applied torque τ on

3810-502: Is determined from the formula, ω n = g L = m g r I P , {\displaystyle \omega _{\text{n}}={\sqrt {\frac {g}{L}}}={\sqrt {\frac {mgr}{I_{P}}}},} or L = g ω n 2 = I P m r . {\displaystyle L={\frac {g}{\omega _{\text{n}}^{2}}}={\frac {I_{P}}{mr}}.} The seconds pendulum , which provides

3937-416: Is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by m r 2 {\displaystyle mr^{2}} , where r {\displaystyle r} is the distance of the point from the axis, and m {\displaystyle m} is the mass. For an extended rigid body,

Torque - Misplaced Pages Continue

4064-660: Is highly fusional , with classes of inflections for case , number , person , gender , tense , mood , voice , and aspect . The Latin alphabet is directly derived from the Etruscan and Greek alphabets . Latin remains the official language of the Holy See and the Roman Rite of the Catholic Church at the Vatican City . The church continues to adapt concepts from modern languages to Ecclesiastical Latin of

4191-689: Is modelled after the British Victoria Cross which has the inscription "For Valour". Because Canada is officially bilingual, the Canadian medal has replaced the English inscription with the Latin Pro Valore . Spain's motto Plus ultra , meaning "even further", or figuratively "Further!", is also Latin in origin. It is taken from the personal motto of Charles V , Holy Roman Emperor and King of Spain (as Charles I), and

4318-405: Is often presented as projected onto this ground plane so that the axis of rotation appears as a point. In this case, the angular velocity and angular acceleration of the body are scalars and the fact that they are vectors along the rotation axis is ignored. This is usually preferred for introductions to the topic. But in the case of moment of inertia, the combination of mass and geometry benefits from

4445-484: Is proportional to the moment of inertia of the body. Moments of inertia may be expressed in units of kilogram metre squared (kg·m ) in SI units and pound-foot-second squared (lbf·ft·s ) in imperial or US units. The moment of inertia plays the role in rotational kinetics that mass (inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass

4572-1011: Is taught at many high schools, especially in Europe and the Americas. It is most common in British public schools and grammar schools, the Italian liceo classico and liceo scientifico , the German Humanistisches Gymnasium and the Dutch gymnasium . Occasionally, some media outlets, targeting enthusiasts, broadcast in Latin. Notable examples include Radio Bremen in Germany, YLE radio in Finland (the Nuntii Latini broadcast from 1989 until it

4699-424: Is the moment of inertia of the body and ω is its angular speed . Power is the work per unit time , given by P = τ ⋅ ω , {\displaystyle P={\boldsymbol {\tau }}\cdot {\boldsymbol {\omega }},} where P is power, τ is torque, ω is the angular velocity , and ⋅ {\displaystyle \cdot } represents

4826-405: Is the newton-metre (N⋅m). For more on the units of torque, see § Units . The net torque on a body determines the rate of change of the body's angular momentum , τ = d L d t {\displaystyle {\boldsymbol {\tau }}={\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}} where L is the angular momentum vector and t

4953-487: Is the angular velocity of the system and V {\displaystyle \mathbf {V} } is the velocity of R {\displaystyle \mathbf {R} } . For planar movement the angular velocity vector is directed along the unit vector k {\displaystyle \mathbf {k} } which is perpendicular to the plane of movement. Introduce the unit vectors e i {\displaystyle \mathbf {e} _{i}} from

5080-419: Is the mass of the object, g {\displaystyle g} is local acceleration of gravity, and r {\displaystyle r} is the distance from the pivot point to the center of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body. Thus, to determine the moment of inertia of

5207-413: Is the mass of the sphere. If a mechanical system is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis k ^ {\displaystyle \mathbf {\hat {k}} } parallel to this plane. In this case, the moment of inertia of the mass in this system is a scalar known as the polar moment of inertia . The definition of

Torque - Misplaced Pages Continue

5334-476: Is the net force on the mass. Associated with this torque is an angular acceleration , α {\displaystyle {\boldsymbol {\alpha }}} , of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is a = α × r {\displaystyle \mathbf {a} ={\boldsymbol {\alpha }}\times \mathbf {r} } . Since F = m

5461-420: Is the period (duration) of oscillation (usually averaged over multiple periods). A simple pendulum that has the same natural frequency as a compound pendulum defines the length L {\displaystyle L} from the pivot to a point called the center of oscillation of the compound pendulum. This point also corresponds to the center of percussion . The length L {\displaystyle L}

5588-1748: Is time. For the motion of a point particle, L = I ω , {\displaystyle \mathbf {L} =I{\boldsymbol {\omega }},} where I = m r 2 {\textstyle I=mr^{2}} is the moment of inertia and ω is the orbital angular velocity pseudovector. It follows that τ n e t = I 1 ω 1 ˙ e 1 ^ + I 2 ω 2 ˙ e 2 ^ + I 3 ω 3 ˙ e 3 ^ + I 1 ω 1 d e 1 ^ d t + I 2 ω 2 d e 2 ^ d t + I 3 ω 3 d e 3 ^ d t = I ω ˙ + ω × ( I ω ) {\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }=I_{1}{\dot {\omega _{1}}}{\hat {\boldsymbol {e_{1}}}}+I_{2}{\dot {\omega _{2}}}{\hat {\boldsymbol {e_{2}}}}+I_{3}{\dot {\omega _{3}}}{\hat {\boldsymbol {e_{3}}}}+I_{1}\omega _{1}{\frac {d{\hat {\boldsymbol {e_{1}}}}}{dt}}+I_{2}\omega _{2}{\frac {d{\hat {\boldsymbol {e_{2}}}}}{dt}}+I_{3}\omega _{3}{\frac {d{\hat {\boldsymbol {e_{3}}}}}{dt}}=I{\boldsymbol {\dot {\omega }}}+{\boldsymbol {\omega }}\times (I{\boldsymbol {\omega }})} using

5715-450: Is torque, and θ 1 and θ 2 represent (respectively) the initial and final angular positions of the body. It follows from the work–energy principle that W also represents the change in the rotational kinetic energy E r of the body, given by E r = 1 2 I ω 2 , {\displaystyle E_{\mathrm {r} }={\tfrac {1}{2}}I\omega ^{2},} where I

5842-767: Is torque. This word is derived from the Latin word rotātus meaning 'to rotate', but the term rotatum is not universally recognized but is commonly used. There is not a universally accepted lexicon to indicate the successive derivatives of rotatum, even if sometimes various proposals have been made. Using the cross product definition of torque, an alternative expression for rotatum is: P = r × d F d t + d r d t × F . {\displaystyle \mathbf {P} =\mathbf {r} \times {\frac {\mathrm {d} \mathbf {F} }{\mathrm {d} t}}+{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}\times \mathbf {F} .} Because

5969-506: Is valid for any type of trajectory. In some simple cases like a rotating disc, where only the moment of inertia on rotating axis is, the rotational Newton's second law can be τ = I α {\displaystyle {\boldsymbol {\tau }}=I{\boldsymbol {\alpha }}} where α = ω ˙ {\displaystyle {\boldsymbol {\alpha }}={\dot {\boldsymbol {\omega }}}} . The definition of angular momentum for

6096-624: Is zero because velocity and momentum are parallel, so the second term vanishes. Therefore, torque on a particle is equal to the first derivative of its angular momentum with respect to time. If multiple forces are applied, according Newton's second law it follows that d L d t = r × F n e t = τ n e t . {\displaystyle {\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}=\mathbf {r} \times \mathbf {F} _{\mathrm {net} }={\boldsymbol {\tau }}_{\mathrm {net} }.} This

6223-1878: The z {\displaystyle z} -axis is r ( z ) 2 = x 2 + y 2 = R 2 − z 2 . {\displaystyle r(z)^{2}=x^{2}+y^{2}=R^{2}-z^{2}.} Therefore, the moment of inertia of the sphere is the sum of the moments of inertia of the discs along the z {\displaystyle z} -axis, I C , sphere = ∫ − R R 1 2 π ρ r ( z ) 4 d z = ∫ − R R 1 2 π ρ ( R 2 − z 2 ) 2 d z = 1 2 π ρ [ R 4 z − 2 3 R 2 z 3 + 1 5 z 5 ] − R R = π ρ ( 1 − 2 3 + 1 5 ) R 5 = 2 5 m R 2 , {\displaystyle {\begin{aligned}I_{C,{\text{sphere}}}&=\int _{-R}^{R}{\tfrac {1}{2}}\pi \rho r(z)^{4}\,dz=\int _{-R}^{R}{\tfrac {1}{2}}\pi \rho \left(R^{2}-z^{2}\right)^{2}\,dz\\[1ex]&={\tfrac {1}{2}}\pi \rho \left[R^{4}z-{\tfrac {2}{3}}R^{2}z^{3}+{\tfrac {1}{5}}z^{5}\right]_{-R}^{R}\\[1ex]&=\pi \rho \left(1-{\tfrac {2}{3}}+{\tfrac {1}{5}}\right)R^{5}\\[1ex]&={\tfrac {2}{5}}mR^{2},\end{aligned}}} where m = 4 3 π R 3 ρ {\textstyle m={\frac {4}{3}}\pi R^{3}\rho }

6350-543: The Corpus Inscriptionum Latinarum (CIL). Authors and publishers vary, but the format is about the same: volumes detailing inscriptions with a critical apparatus stating the provenance and relevant information. The reading and interpretation of these inscriptions is the subject matter of the field of epigraphy . About 270,000 inscriptions are known. The Latin influence in English has been significant at all stages of its insular development. In

6477-528: The Holy See , the primary language of its public journal , the Acta Apostolicae Sedis , and the working language of the Roman Rota . Vatican City is also home to the world's only automatic teller machine that gives instructions in Latin. In the pontifical universities postgraduate courses of Canon law are taught in Latin, and papers are written in the same language. There are

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6604-502: The Late Latin period, language changes reflecting spoken (non-classical) norms tend to be found in greater quantities in texts. As it was free to develop on its own, there is no reason to suppose that the speech was uniform either diachronically or geographically. On the contrary, Romanised European populations developed their own dialects of the language, which eventually led to the differentiation of Romance languages . Late Latin

6731-607: The Middle Ages as a working and literary language from the 9th century to the Renaissance , which then developed a classicizing form, called Renaissance Latin . This was the basis for Neo-Latin which evolved during the early modern period . In these periods Latin was used productively and generally taught to be written and spoken, at least until the late seventeenth century, when spoken skills began to erode. It then became increasingly taught only to be read. Latin grammar

6858-574: The Middle Ages , borrowing from Latin occurred from ecclesiastical usage established by Saint Augustine of Canterbury in the 6th century or indirectly after the Norman Conquest , through the Anglo-Norman language . From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed " inkhorn terms ", as if they had spilled from a pot of ink. Many of these words were used once by

6985-407: The common language of international communication , science, scholarship and academia in Europe until well into the early 19th century, by which time modern languages had supplanted it in common academic and political usage. Late Latin is the literary language from the 3rd century AD onward. No longer spoken as a native language, Medieval Latin was used across Western and Catholic Europe during

7112-426: The cross product of the displacement vector and the force vector. The direction of the torque can be determined by using the right hand grip rule : if the fingers of the right hand are curled from the direction of the lever arm to the direction of the force, then the thumb points in the direction of the torque. It follows that the torque vector is perpendicular to both the position and force vectors and defines

7239-569: The geometrical theorem of the same name) states that the resultant torques due to several forces applied to about a point is equal to the sum of the contributing torques: Latin Latin ( lingua Latina , pronounced [ˈlɪŋɡʷa ɫaˈtiːna] , or Latinum [ɫaˈtiːnʊ̃] ) is a classical language belonging to the Italic branch of the Indo-European languages . Latin

7366-430: The mass moment of inertia , angular/rotational mass , second moment of mass , or most accurately, rotational inertia , of a rigid body is defined relative to a rotational axis. It is the ratio between the torque applied and the resulting angular acceleration about that axis. It plays the same role in rotational motion as mass does in linear motion. A body's moment of inertia about a particular axis depends both on

7493-413: The moment of inertia of this single mass around the pivot point. The quantity I = m r 2 {\displaystyle I=mr^{2}} also appears in the angular momentum of a simple pendulum, which is calculated from the velocity v = ω × r {\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} } of

7620-414: The scalar product . Algebraically, the equation may be rearranged to compute torque for a given angular speed and power output. The power injected by the torque depends only on the instantaneous angular speed – not on whether the angular speed increases, decreases, or remains constant while the torque is being applied (this is equivalent to the linear case where the power injected by a force depends only on

7747-583: The "tick" and "tock" of a grandfather clock, takes one second to swing from side-to-side. This is a period of two seconds, or a natural frequency of π   r a d / s {\displaystyle \pi \ \mathrm {rad/s} } for the pendulum. In this case, the distance to the center of oscillation, L {\displaystyle L} , can be computed to be L = g ω n 2 ≈ 9.81   m / s 2 ( 3.14   r

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7874-637: The British Crown. The motto is featured on all presently minted coinage and has been featured in most coinage throughout the nation's history. Several states of the United States have Latin mottos , such as: Many military organizations today have Latin mottos, such as: Some law governing bodies in the Philippines have Latin mottos, such as: Some colleges and universities have adopted Latin mottos, for example Harvard University 's motto

8001-703: The Germanic and Slavic nations. It became useful for international communication between the member states of the Holy Roman Empire and its allies. Without the institutions of the Roman Empire that had supported its uniformity, Medieval Latin was much more liberal in its linguistic cohesion: for example, in classical Latin sum and eram are used as auxiliary verbs in the perfect and pluperfect passive, which are compound tenses. Medieval Latin might use fui and fueram instead. Furthermore,

8128-599: The Grinch Stole Christmas! , The Cat in the Hat , and a book of fairy tales, " fabulae mirabiles ", are intended to garner popular interest in the language. Additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissner's Latin Phrasebook . Some inscriptions have been published in an internationally agreed, monumental, multivolume series,

8255-630: The Latin language. Contemporary Latin is more often studied to be read rather than spoken or actively used. Latin has greatly influenced the English language , along with a large number of others, and historically contributed many words to the English lexicon , particularly after the Christianization of the Anglo-Saxons and the Norman Conquest . Latin and Ancient Greek roots are heavily used in English vocabulary in theology ,

8382-467: The United States the unofficial national motto until 1956 was E pluribus unum meaning "Out of many, one". The motto continues to be featured on the Great Seal . It also appears on the flags and seals of both houses of congress and the flags of the states of Michigan, North Dakota, New York, and Wisconsin. The motto's 13 letters symbolically represent the original Thirteen Colonies which revolted from

8509-563: The University of Kentucky, the University of Oxford and also Princeton University. There are many websites and forums maintained in Latin by enthusiasts. The Latin Misplaced Pages has more than 130,000 articles. Italian , French , Portuguese , Spanish , Romanian , Catalan , Romansh , Sardinian and other Romance languages are direct descendants of Latin. There are also many Latin borrowings in English and Albanian , as well as

8636-721: The above expression for work, , gives W = ∫ s 1 s 2 F ⋅ d θ × r {\displaystyle W=\int _{s_{1}}^{s_{2}}\mathbf {F} \cdot \mathrm {d} {\boldsymbol {\theta }}\times \mathbf {r} } The expression inside the integral is a scalar triple product F ⋅ d θ × r = r × F ⋅ d θ {\displaystyle \mathbf {F} \cdot \mathrm {d} {\boldsymbol {\theta }}\times \mathbf {r} =\mathbf {r} \times \mathbf {F} \cdot \mathrm {d} {\boldsymbol {\theta }}} , but as per

8763-469: The author and then forgotten, but some useful ones survived, such as 'imbibe' and 'extrapolate'. Many of the most common polysyllabic English words are of Latin origin through the medium of Old French . Romance words make respectively 59%, 20% and 14% of English, German and Dutch vocabularies. Those figures can rise dramatically when only non-compound and non-derived words are included. Moment of inertia The moment of inertia , otherwise known as

8890-535: The beginning of the Renaissance . Petrarch for example saw Latin as a literary version of the spoken language. Medieval Latin is the written Latin in use during that portion of the post-classical period when no corresponding Latin vernacular existed, that is from around 700 to 1500 AD. The spoken language had developed into the various Romance languages; however, in the educated and official world, Latin continued without its natural spoken base. Moreover, this Latin spread into lands that had never spoken Latin, such as

9017-425: The benefit of those who do not understand Latin. There are also songs written with Latin lyrics . The libretto for the opera-oratorio Oedipus rex by Igor Stravinsky is in Latin. Parts of Carl Orff 's Carmina Burana are written in Latin. Enya has recorded several tracks with Latin lyrics. The continued instruction of Latin is seen by some as a highly valuable component of a liberal arts education. Latin

9144-655: The body, simply suspend it from a convenient pivot point P {\displaystyle P} so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation ( t {\displaystyle t} ), to obtain I P = m g r ω n 2 = m g r t 2 4 π 2 , {\displaystyle I_{P}={\frac {mgr}{\omega _{\text{n}}^{2}}}={\frac {mgrt^{2}}{4\pi ^{2}}},} where t {\displaystyle t}

9271-428: The body, where r {\displaystyle r} is the perpendicular distance to the specified axis. To see how moment of inertia arises in the study of the movement of an extended body, it is convenient to consider a rigid assembly of point masses. (This equation can be used for axes that are not principal axes provided that it is understood that this does not fully describe the moment of inertia. ) Consider

9398-430: The comic playwrights Plautus and Terence and the author Petronius . While often called a "dead language", Latin did not undergo language death . By the 6th to 9th centuries, natural language change eventually resulted in Latin as a vernacular language evolving into distinct Romance languages in the large areas where it had come to be natively spoken. However, even after the fall of Western Rome , Latin remained

9525-465: The country's full Latin name. Some film and television in ancient settings, such as Sebastiane , The Passion of the Christ and Barbarians (2020 TV series) , have been made with dialogue in Latin. Occasionally, Latin dialogue is used because of its association with religion or philosophy, in such film/television series as The Exorcist and Lost (" Jughead "). Subtitles are usually shown for

9652-474: The cross-section, weighted by its density. This is also called the polar moment of the area , and is the sum of the second moments about the x {\displaystyle x} - and y {\displaystyle y} -axes. The stresses in a beam are calculated using the second moment of the cross-sectional area around either the x {\displaystyle x} -axis or y {\displaystyle y} -axis depending on

9779-503: The decline in written Latin output. Despite having no native speakers, Latin is still used for a variety of purposes in the contemporary world. The largest organisation that retains Latin in official and quasi-official contexts is the Catholic Church . The Catholic Church required that Mass be carried out in Latin until the Second Vatican Council of 1962–1965 , which permitted the use of the vernacular . Latin remains

9906-439: The definition of torque, and since the parameter of integration has been changed from linear displacement to angular displacement, the equation becomes W = ∫ θ 1 θ 2 τ ⋅ d θ {\displaystyle W=\int _{\theta _{1}}^{\theta _{2}}{\boldsymbol {\tau }}\cdot \mathrm {d} {\boldsymbol {\theta }}} If

10033-411: The derivative of a vector is d e i ^ d t = ω × e i ^ {\displaystyle {d{\boldsymbol {\hat {e_{i}}}} \over dt}={\boldsymbol {\omega }}\times {\boldsymbol {\hat {e_{i}}}}} This equation is the rotational analogue of Newton's second law for point particles, and

10160-568: The development of European culture, religion and science. The vast majority of written Latin belongs to this period, but its full extent is unknown. The Renaissance reinforced the position of Latin as a spoken and written language by the scholarship by the Renaissance humanists . Petrarch and others began to change their usage of Latin as they explored the texts of the Classical Latin world. Skills of textual criticism evolved to create much more accurate versions of extant texts through

10287-413: The earliest extant Latin literary works, such as the comedies of Plautus and Terence . The Latin alphabet was devised from the Etruscan alphabet . The writing later changed from what was initially either a right-to-left or a boustrophedon script to what ultimately became a strictly left-to-right script. During the late republic and into the first years of the empire, from about 75 BC to AD 200,

10414-445: The fifteenth and sixteenth centuries, and some important texts were rediscovered. Comprehensive versions of authors' works were published by Isaac Casaubon , Joseph Scaliger and others. Nevertheless, despite the careful work of Petrarch, Politian and others, first the demand for manuscripts, and then the rush to bring works into print, led to the circulation of inaccurate copies for several centuries following. Neo-Latin literature

10541-546: The history of Latin, and the kind of informal Latin that had begun to move away from the written language significantly in the post-Imperial period, that led ultimately to the Romance languages . During the Classical period, informal language was rarely written, so philologists have been left with only individual words and phrases cited by classical authors, inscriptions such as Curse tablets and those found as graffiti . In

10668-758: The individual masses, E K = ∑ i = 1 N 1 2 m i v i ⋅ v i = ∑ i = 1 N 1 2 m i ( ω r i ) 2 = 1 2 ω 2 ∑ i = 1 N m i r i 2 . {\displaystyle E_{\text{K}}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\mathbf {v} _{i}\cdot \mathbf {v} _{i}=\sum _{i=1}^{N}{\frac {1}{2}}\,m_{i}\left(\omega r_{i}\right)^{2}={\frac {1}{2}}\,\omega ^{2}\sum _{i=1}^{N}m_{i}r_{i}^{2}.} This shows that

10795-568: The infinitesimal linear displacement d s {\displaystyle \mathrm {d} \mathbf {s} } is related to a corresponding angular displacement d θ {\displaystyle \mathrm {d} {\boldsymbol {\theta }}} and the radius vector r {\displaystyle \mathbf {r} } as d s = d θ × r {\displaystyle \mathrm {d} \mathbf {s} =\mathrm {d} {\boldsymbol {\theta }}\times \mathbf {r} } Substitution in

10922-586: The instantaneous speed – not on the resulting acceleration, if any). The work done by a variable force acting over a finite linear displacement s {\displaystyle s} is given by integrating the force with respect to an elemental linear displacement d s {\displaystyle \mathrm {d} \mathbf {s} } W = ∫ s 1 s 2 F ⋅ d s {\displaystyle W=\int _{s_{1}}^{s_{2}}\mathbf {F} \cdot \mathrm {d} \mathbf {s} } However,

11049-703: The invention of printing and are now published in carefully annotated printed editions, such as the Loeb Classical Library , published by Harvard University Press , or the Oxford Classical Texts , published by Oxford University Press . Latin translations of modern literature such as: The Hobbit , Treasure Island , Robinson Crusoe , Paddington Bear , Winnie the Pooh , The Adventures of Tintin , Asterix , Harry Potter , Le Petit Prince , Max and Moritz , How

11176-400: The kinetic energy of an assembly of N {\displaystyle N} masses m i {\displaystyle m_{i}} that lie at the distances r i {\displaystyle r_{i}} from the pivot point P {\displaystyle P} , which is the nearest point on the axis of rotation. It is the sum of the kinetic energy of

11303-704: The language of the Roman Rite . The Tridentine Mass (also known as the Extraordinary Form or Traditional Latin Mass) is celebrated in Latin. Although the Mass of Paul VI (also known as the Ordinary Form or the Novus Ordo) is usually celebrated in the local vernacular language, it can be and often is said in Latin, in part or in whole, especially at multilingual gatherings. It is the official language of

11430-405: The language, Vulgar Latin (termed sermo vulgi , "the speech of the masses", by Cicero ). Some linguists, particularly in the nineteenth century, believed this to be a separate language, existing more or less in parallel with the literary or educated Latin, but this is now widely dismissed. The term 'Vulgar Latin' remains difficult to define, referring both to informal speech at any time within

11557-421: The load. The moment of inertia of a compound pendulum constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass. A list of moments of inertia formulas for standard body shapes provides a way to obtain the moment of inertia of

11684-585: The mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia. The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, except with infinitely many point particles. Thus the limits of summation are removed, and the sum is written as follows: I P = ∑ i m i r i 2 {\displaystyle I_{P}=\sum _{i}m_{i}r_{i}^{2}} Another expression replaces

11811-412: The mass and its distribution relative to the axis, increasing with mass & distance from the axis. It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about

11938-443: The mass density being replaced by its areal mass density with the integral evaluated over its area. Note on second moment of area : The moment of inertia of a body moving in a plane and the second moment of area of a beam's cross-section are often confused. The moment of inertia of a body with the shape of the cross-section is the second moment of this area about the z {\displaystyle z} -axis perpendicular to

12065-462: The mass of a simple pendulum generates a torque τ = r × F {\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} } around the axis perpendicular to the plane of the pendulum movement. Here r {\displaystyle \mathbf {r} } is the distance vector from the torque axis to the pendulum center of mass, and F {\displaystyle \mathbf {F} }

12192-399: The mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. The moment of inertia also appears in momentum , kinetic energy , and in Newton's laws of motion for a rigid body as

12319-431: The meanings of many words were changed and new words were introduced, often under influence from the vernacular. Identifiable individual styles of classically incorrect Latin prevail. Renaissance Latin, 1300 to 1500, and the classicised Latin that followed through to the present are often grouped together as Neo-Latin , or New Latin, which have in recent decades become a focus of renewed study , given their importance for

12446-416: The moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object. In 1673, Christiaan Huygens introduced this parameter in his study of the oscillation of

12573-399: The moment of inertia of the body is the sum of each of the m r 2 {\displaystyle mr^{2}} terms, that is I P = ∑ i = 1 N m i r i 2 . {\displaystyle I_{P}=\sum _{i=1}^{N}m_{i}r_{i}^{2}.} Thus, moment of inertia is a physical property that combines

12700-441: The moment of inertia of the pendulum depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation. This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses dm each multiplied by the square of its perpendicular distance r to an axis k . An arbitrary object's moment of inertia thus depends on

12827-1099: The pendulum mass around the pivot, where ω {\displaystyle {\boldsymbol {\omega }}} is the angular velocity of the mass about the pivot point. This angular momentum is given by L = r × p = r × ( m ω × r ) = m ( ( r ⋅ r ) ω − ( r ⋅ ω ) r ) = m r 2 ω = I ω k ^ , {\displaystyle {\begin{aligned}\mathbf {L} &=\mathbf {r} \times \mathbf {p} =\mathbf {r} \times \left(m{\boldsymbol {\omega }}\times \mathbf {r} \right)\\&=m\left(\left(\mathbf {r} \cdot \mathbf {r} \right){\boldsymbol {\omega }}-\left(\mathbf {r} \cdot {\boldsymbol {\omega }}\right)\mathbf {r} \right)\\&=mr^{2}{\boldsymbol {\omega }}=I\omega \mathbf {\hat {k}} ,\end{aligned}}} using

12954-402: The plane in which the two vectors lie. The resulting torque vector direction is determined by the right-hand rule. Therefore any force directed parallel to the particle's position vector does not produce a torque. The magnitude of torque applied to a rigid body depends on three quantities: the force applied, the lever arm vector connecting the point about which the torque is being measured to

13081-436: The point of force application, and the angle between the force and lever arm vectors. In symbols: τ = r × F ⟹ τ = r F ⊥ = r F sin ⁡ θ {\displaystyle {\boldsymbol {\tau }}=\mathbf {r} \times \mathbf {F} \implies \tau =rF_{\perp }=rF\sin \theta } where The SI unit for torque

13208-414: The polar moment of inertia can be obtained by considering momentum, kinetic energy and Newton's laws for the planar movement of a rigid system of particles. If a system of n {\displaystyle n} particles, P i , i = 1 , … , n {\displaystyle P_{i},i=1,\dots ,n} , are assembled into a rigid body, then the momentum of

13335-479: The quantity I = m r 2 {\displaystyle I=mr^{2}} is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values m r 2 {\displaystyle mr^{2}} for all of the elements of mass in the body. A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around

13462-528: The rate of change of force is yank Y {\textstyle \mathbf {Y} } and the rate of change of position is velocity v {\textstyle \mathbf {v} } , the expression can be further simplified to: P = r × Y + v × F . {\displaystyle \mathbf {P} =\mathbf {r} \times \mathbf {Y} +\mathbf {v} \times \mathbf {F} .} The law of conservation of energy can also be used to understand torque. If

13589-1734: The reference point R {\displaystyle \mathbf {R} } to a point r i {\displaystyle \mathbf {r} _{i}} , and the unit vector t ^ i = k ^ × e ^ i {\displaystyle \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i}} , so e ^ i = Δ r i Δ r i , k ^ = ω ω , t ^ i = k ^ × e ^ i , v i = ω × Δ r i + V = ω k ^ × Δ r i e ^ i + V = ω Δ r i t ^ i + V {\displaystyle {\begin{aligned}\mathbf {\hat {e}} _{i}&={\frac {\Delta \mathbf {r} _{i}}{\Delta r_{i}}},\quad \mathbf {\hat {k}} ={\frac {\boldsymbol {\omega }}{\omega }},\quad \mathbf {\hat {t}} _{i}=\mathbf {\hat {k}} \times \mathbf {\hat {e}} _{i},\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} =\omega \mathbf {\hat {k}} \times \Delta r_{i}\mathbf {\hat {e}} _{i}+\mathbf {V} =\omega \,\Delta r_{i}\mathbf {\hat {t}} _{i}+\mathbf {V} \end{aligned}}} This defines

13716-399: The relative position vector and the velocity vector for the rigid system of the particles moving in a plane. Note on the cross product : When a body moves parallel to a ground plane, the trajectories of all the points in the body lie in planes parallel to this ground plane. This means that any rotation that the body undergoes must be around an axis perpendicular to this plane. Planar movement

13843-442: The same axis). Its simplest definition is the second moment of mass with respect to distance from an axis . For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3-by-3 matrix, with a set of mutually perpendicular principal axes for which this matrix

13970-421: The sciences , medicine , and law . A number of phases of the language have been recognized, each distinguished by subtle differences in vocabulary, usage, spelling, and syntax. There are no hard and fast rules of classification; different scholars emphasize different features. As a result, the list has variants, as well as alternative names. In addition to the historical phases, Ecclesiastical Latin refers to

14097-421: The spatial distribution of its mass. In general, given an object of mass m , an effective radius k can be defined, dependent on a particular axis of rotation, with such a value that its moment of inertia around the axis is I = m k 2 , {\displaystyle I=mk^{2},} where k is known as the radius of gyration around the axis. Mathematically, the moment of inertia of

14224-445: The sphere whose centers are along the axis chosen for consideration. If the surface of the sphere is defined by the equation x 2 + y 2 + z 2 = R 2 , {\displaystyle x^{2}+y^{2}+z^{2}=R^{2},} then the square of the radius r {\displaystyle r} of the disc at the cross-section z {\displaystyle z} along

14351-400: The square of the distance between the reference axis and the centroid of the section. The moment of inertia I is also defined as the ratio of the net angular momentum L of a system to its angular velocity ω around a principal axis, that is I = L ω . {\displaystyle I={\frac {L}{\omega }}.} If the angular momentum of a system

14478-641: The styles used by the writers of the Roman Catholic Church from late antiquity onward, as well as by Protestant scholars. The earliest known form of Latin is Old Latin, also called Archaic or Early Latin, which was spoken from the Roman Kingdom , traditionally founded in 753 BC, through the later part of the Roman Republic , up to 75 BC, i.e. before the age of Classical Latin . It is attested both in inscriptions and in some of

14605-545: The summation with an integral , I P = ∭ Q ρ ( x , y , z ) ‖ r ‖ 2 d V {\displaystyle I_{P}=\iiint _{Q}\rho (x,y,z)\left\|\mathbf {r} \right\|^{2}dV} Here, the function ρ {\displaystyle \rho } gives the mass density at each point ( x , y , z ) {\displaystyle (x,y,z)} , r {\displaystyle \mathbf {r} }

14732-976: The system can be written in terms of positions relative to a reference point R {\displaystyle \mathbf {R} } , and absolute velocities v i {\displaystyle \mathbf {v} _{i}} : Δ r i = r i − R , v i = ω × ( r i − R ) + V = ω × Δ r i + V , {\displaystyle {\begin{aligned}\Delta \mathbf {r} _{i}&=\mathbf {r} _{i}-\mathbf {R} ,\\\mathbf {v} _{i}&={\boldsymbol {\omega }}\times \left(\mathbf {r} _{i}-\mathbf {R} \right)+\mathbf {V} ={\boldsymbol {\omega }}\times \Delta \mathbf {r} _{i}+\mathbf {V} ,\end{aligned}}} where ω {\displaystyle {\boldsymbol {\omega }}}

14859-875: The torque and the angular displacement are in the same direction, then the scalar product reduces to a product of magnitudes; i.e., τ ⋅ d θ = | τ | | d θ | cos ⁡ 0 = τ d θ {\displaystyle {\boldsymbol {\tau }}\cdot \mathrm {d} {\boldsymbol {\theta }}=\left|{\boldsymbol {\tau }}\right|\left|\mathrm {d} {\boldsymbol {\theta }}\right|\cos 0=\tau \,\mathrm {d} \theta } giving W = ∫ θ 1 θ 2 τ d θ {\displaystyle W=\int _{\theta _{1}}^{\theta _{2}}\tau \,\mathrm {d} \theta } The principle of moments, also known as Varignon's theorem (not to be confused with

14986-484: The trifilar pendulum yields the moment of inertia of the system. Moment of inertia of area is also known as the second moment of area and its physical meaning is completely different from the mass moment of inertia. These calculations are commonly used in civil engineering for structural design of beams and columns. Cross-sectional areas calculated for vertical moment of the x-axis I x x {\displaystyle I_{xx}} and horizontal moment of

15113-733: The vectors into components and applying the product rule . But because the rate of change of linear momentum is force F {\textstyle \mathbf {F} } and the rate of change of position is velocity v {\textstyle \mathbf {v} } , d L d t = r × F + v × p {\displaystyle {\frac {\mathrm {d} \mathbf {L} }{\mathrm {d} t}}=\mathbf {r} \times \mathbf {F} +\mathbf {v} \times \mathbf {p} } The cross product of momentum p {\displaystyle \mathbf {p} } with its associated velocity v {\displaystyle \mathbf {v} }

15240-422: The written form of Latin was increasingly standardized into a fixed form, the spoken forms began to diverge more greatly. Currently, the five most widely spoken Romance languages by number of native speakers are Spanish , Portuguese , French , Italian , and Romanian . Despite dialectal variation, which is found in any widespread language, the languages of Spain, France, Portugal, and Italy have retained

15367-413: The y-axis I y y {\displaystyle I_{yy}} . Height ( h ) and breadth ( b ) are the linear measures, except for circles, which are effectively half-breadth derived, r {\displaystyle r} The moment of inertia about an axis of a body is calculated by summing m r 2 {\displaystyle mr^{2}} for every particle in

15494-783: Was also used as a convenient medium for translations of important works first written in a vernacular, such as those of Descartes . Latin education underwent a process of reform to classicise written and spoken Latin. Schooling remained largely Latin medium until approximately 1700. Until the end of the 17th century, the majority of books and almost all diplomatic documents were written in Latin. Afterwards, most diplomatic documents were written in French (a Romance language ) and later native or other languages. Education methods gradually shifted towards written Latin, and eventually concentrating solely on reading skills. The decline of Latin education took several centuries and proceeded much more slowly than

15621-491: Was extensive and prolific, but less well known or understood today. Works covered poetry, prose stories and early novels, occasional pieces and collections of letters, to name a few. Famous and well regarded writers included Petrarch, Erasmus, Salutati , Celtis , George Buchanan and Thomas More . Non fiction works were long produced in many subjects, including the sciences, law, philosophy, historiography and theology. Famous examples include Isaac Newton 's Principia . Latin

15748-503: Was largely separated from the unifying influences in the western part of the Empire. Spoken Latin began to diverge into distinct languages by the 9th century at the latest, when the earliest extant Romance writings begin to appear. They were, throughout the period, confined to everyday speech, as Medieval Latin was used for writing. For many Italians using Latin, though, there was no complete separation between Italian and Latin, even into

15875-732: Was originally spoken by the Latins in Latium (now known as Lazio ), the lower Tiber area around Rome , Italy. Through the expansion of the Roman Republic it became the dominant language in the Italian Peninsula and subsequently throughout the Roman Empire . By the late Roman Republic , Old Latin had evolved into standardized Classical Latin . Vulgar Latin refers to the less prestigious colloquial registers , attested in inscriptions and some literary works such as those of

16002-529: Was perceived as a separate language, for instance early French or Italian dialects, that could be transcribed differently. It took some time for these to be viewed as wholly different from Latin however. After the Western Roman Empire fell in 476 and Germanic kingdoms took its place, the Germanic people adopted Latin as a language more suitable for legal and other, more formal uses. While

16129-482: Was shut down in June 2019), and Vatican Radio & Television, all of which broadcast news segments and other material in Latin. A variety of organisations, as well as informal Latin 'circuli' ('circles'), have been founded in more recent times to support the use of spoken Latin. Moreover, a number of university classics departments have begun incorporating communicative pedagogies in their Latin courses. These include

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