Calculating Torque With Examples

Calculating Torque With Examples When studying how objects rotate, it quickly becomes necessary to figure out how a given force results in a change in the rotational motion. The tendency of a force to cause or change rotational motion is called torque, and its one of the most important concepts to understand in resolving rotational motion situations. The Meaning of Torque Torque (also called moment - mostly by engineers) is calculated by multiplying force and distance. The SI units of torque are newton-meters, or N*m (even though these units are the same as Joules, torque isnt work or energy, so should just be newton-meters). In calculations, torque is represented by the Greek letter tau: Ï„. Torque is a vector quantity, meaning it has both a direction and a magnitude. This is honestly one of the trickiest parts of working with torque because it is calculated using a vector product, which means you have to apply the right-hand rule. In this case, take your right hand and curl the fingers of your hand in the direction of rotation caused by the force. The thumb of your right hand now points in the direction of the torque vector. (This can occasionally feel slightly silly, as youre holding your hand up and pantomiming in order to figure out the result of a mathematical equation, but its the best way to visualize the direction of the vector.) The vector formula that yields the torque vector Ï„ is: Ï„ r Ãâ€" F The vector r is the position vector with respect to an origin on the axis of rotation (This axis is the Ï„ on the graphic). This is a vector with a magnitude of the distance from where the force is applied to the axis of rotation. It points from the axis of rotation toward the point where the force is applied. The magnitude of the vector is calculated based upon ÃŽ ¸, which is the angle difference between r and F, using the formula: Ï„ rFsin(ÃŽ ¸) Special Cases of Torque A couple of key points about the above equation, with some benchmark values of ÃŽ ¸: ÃŽ ¸ 0 ° (or 0 radians) - The force vector is pointing out in the same direction as r. As you might guess, this is a situation where the force will not cause any rotation around the axis ... and the mathematics bears this out. Since sin(0) 0, this situation results in Ï„ 0.ÃŽ ¸ 180 ° (or Ï€ radians) - This is a situation where the force vector points directly into r. Again, shoving toward the axis of rotation isnt going to cause any rotation either and, once again, the mathematics supports this intuition. Since sin(180 °) 0, the value of the torque is once again Ï„ 0.ÃŽ ¸ 90 ° (or Ï€/2 radians) - Here, the force vector is perpendicular to the position vector. This seems like the most effective way that you could push on the object to get an increase in rotation, but does the mathematics support this? Well, sin(90 °) 1, which is the maximum value that the sine function can reach, yielding a result of Ï„ rF. In other words, a force applied at any other angle would provide less torque than when it is applied at 90 degrees. The same argument as above applies to cases of ÃŽ ¸ -90 ° (or -Ï€/2 radians), but with a value of sin(-90 °) -1 resulting in the maximum torque in the opposite direction. Torque Example Lets consider an example where youre applying a vertical force downward, such as when trying to loosen the lug nuts on a flat tire by stepping on the lug wrench. In this situation, the ideal situation is to have the lug wrench perfectly horizontal, so that you can step on the end of it and get the maximum torque. Unfortunately, that doesnt work. Instead, the lug wrench fits onto the lug nuts so that it is at a 15% incline to the horizontal. The lug wrench is 0.60 m long until the end, where you apply your full weight of 900 N. What is the magnitude of the torque? What about direction?: Applying the lefty-loosey, righty-tighty rule, you will want to have the lug nut rotating to the left - counter-clockwise - in order to loosen it. Using your right hand and curling your fingers in the counter-clockwise direction, the thumb sticks out. So the direction of the torque is away from the tires ... which is also direction you want the lug nuts to ultimately go. To begin calculating the value of the torque, you have to realize that theres a slightly misleading point in the above set-up. (This is a common problem in these situations.) Note that the 15% mentioned above is the incline from the horizontal, but thats not the angle ÃŽ ¸. The angle between r and F has to be calculated. Theres a 15 ° incline from the horizontal plus a 90 ° distance from the horizontal to the downward force vector, resulting in a total of 105 ° as the value of ÃŽ ¸. Thats the only variable that requires set-up, so with that in place we just assign the other variable values: ÃŽ ¸ 105 °r 0.60 mF 900 N Ï„ rF sin(ÃŽ ¸) (0.60 m)(900 N)sin(105 °) 540 Ãâ€" 0.097 Nm 520 Nm Note that the above answer involved maintaining only two significant figures, so it is rounded. Torque and Angular Acceleration The above equations are particularly helpful when there is a single known force acting on an object, but there are many situations where a rotation can be caused by a force that cannot easily be measured (or perhaps many such forces). Here, the torque often isnt calculated directly, but can instead be calculated in reference to the total angular acceleration, ÃŽ ±, that the object undergoes. This relationship is given by the following equation: ÃŽ £Ãâ€ž - The net sum of all torque acting on the objectI - the moment of inertia, which represents the objects resistance to a change in angular velocityÃŽ ± - angular acceleration

American foreign policy Essay Example | Topics and Well Written Essays - 3500 words

American foreign policy - Essay Example heir own spheres of influence, thus playing a role in shaping the final outcome within the policy making process. Keeping in line with what Robert Divine had once remarked, â€Å"From the Revolution to the Cold War, Americans have been willing to fight for their interests, their beliefs, and their ambitions† (Divine, 1965, 172), there are clear evidences which suggest that the American foreign policies after the WWII, have roots in its domestic issues, concerns, and interests. Various researches on the subject of US international relations have distinguished 3 specific factors that appear to exert their influences on the foreign policies of the country, especially after the end of the WWII. ... Discussion The official goals of US foreign policies as stated by the government, are "to create a more secure, democratic, and prosperous world for the benefit of the American people and the international community" (US department of state and U.S. Agency for International Development, 2007). While examining the factors that work towards achieving these goals, the first potent question that arises in one’s mind is the fundamental query as to â€Å"who governs?†(Dahl, 1961) This question has been the subject of many studies, related to American politics, and has given rise to many debates within the realms of international relations. The question in the arena of international relations relates to the general attributes of the so called ‘domestic roots’ that help to shape US foreign policies; the democratic aspects in the behaviour of US towards other countries, globally; and whether its national concerns and interests aid in shaping its foreign policies. By seeking an answer to the basic question as to which of these factors actually influence the US government officials the most, would help the reader to understand the determinants of the country’s foreign policies. In this context, we will first examine the 3 main factors considered to have a strong influence in the shaping of US foreign policies. Theories on the nature of the influence of public opinion: Median voter theory contends that the voter preferences for any government policy have a strong influence on the state officials, as these preferences often turn out to be the deciding factor during the electoral results. Various researches and the subsequent

Historical and Theoretical Perspectives in Teaching Reading Research Paper

Historical and Theoretical Perspectives in Teaching Reading - Research Paper Example The learner has to be taught visual identification, so that meaningful recognition can occur. Prodded to the habit of reading, the learner gains ease in perceiving meaning, relationships and messages of the printed word.   In the Traditional Theory, the reader is a passive receiver of information, and so the bottoms-up view approach to teaching reading is adopted and this consists in giving prime importance to building pre-cognitive skills which lead to the end processes of construction of meaning or comprehension (Dole et al. (1991).  The second Cognitive Theory is the psycho-logistic model which recognizes the fact that the   heart of the reading process is the reader, not the text. Under this theory, the learner of reading is introduced to the sampling of   the text, making   guesses or a series of hypothesis about the meaning of the text, and proceeding   with the process of fusion, interpretation and comprehension.   The cognitive experts highlight the importance o f observation, data judgment, recall, imagination, problem recognition, and analysis in teaching reading.   They also train learners to identify levels of comprehension namely: literal, interpretative, applied and evaluative (Nunan, 1991). .   Thirdly, the Metacognitive Theory centers on training the reader to have control in executing his ability to interpret and comprehend the text.   In this theory,   meta-reading is   characterized by self-awareness, careful reading and greater retention.... Yes, there had been schooling in reading especially so among ancient civilizations, but these were most rudimentary and confined to the more precocious in knowledge and love of the written word. Only since the 1980s can there be claimed theoretical perspectives in teaching and their benefits still have to be widely and deeply felt by modern learners. Theoretical perspective There are three main theories in the turns and trends in scientific approaches to teaching reading. The first Traditional Theory has been practiced for ages. This theory is founded on the neuro-physiological process which gives importance to the textual material . The learner has to be taught visual identification, so that meaningful recognition can occur. Prodded to the habit of reading, the learner gains ease in perceiving meaning, relationships and messages of the printed word. In the Traditional Theory, the reader is a passive receiver of information, and so the bottoms-up view approach to teaching reading is adopted and this consists in giving prime importance to building pre-cognitive skills which lead to the end processes of construction of meaning or comprehension (Dole et al. (1991). The second Cognitive Theory is the psycho-logistic model which recognizes the fact that the heart of the reading process is the reader, not the text. Under this theory, the learner of reading is introduced to the sampling of the text, making guesses or a series of hypothesis about the meaning of the text, and proceeding with the process of fusion, interpretation and comprehension. The cognitive experts highlight the importance of observation, data judgment, recall, imagination, problem

The freedom of the children and the sins of the fathers Essay

The freedom of the children and the sins of the fathers - Essay Example Given this dual nature, some good advice may generate bad results, and vice-versa. The point being that one truly never knows the outcome of what one does until one has done it, and given this uncertainty, one should hold individual freedom in a high regard, and furthermore, one should be cautious in the advice that one dispenses, because good intentions don't necessarily translate in good results. I think that Michael's father was thoroughly cautious in his advice. The essence of his message was of respect for free will, as well as of a warning at the entanglement of perspectives, ideas and half-truths that human interaction often consists of. A half truth is what happens not when one lies, but when one does not know the whole of the truth. And I believe that the nature of his advice implied that he knew more than his boisterous and passionate youth of a son. It's basically a matter of committed relativism. Maybe his father realized something that Michael could never have, being part of a younger generation, who never witnessed first-hand the horrors of Nazi Germany. Maybe his insight allowed him to fathom that maybe Hanna was seeking redemption trough punishment.

Western expansion and development Essay Example | Topics and Well Written Essays - 1000 words

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Adaptive Supply Chain Co-Management manual Essay

Adaptive Supply Chain Co-Management manual - Essay Example Collaboration and competition have often been regarded as incompatible concepts especially as evolutionary biology became well-known in the late nineteenth and twentieth centuries. It was presumed that only the fittest can survive if those with the most desirable traits are naturally selected over those that did not possess these ideas. This mentality spilled over into business literature and led to the manifestation of certain laws such as the Sherman and Clayton Anti-trust Acts (Zeiltin, 2007). However, time illustrated that sometimes certain kinds of cooperation and competition did not necessarily have to replace each other as seen through the formation of professional associations, cartels and business groups. Individuals may participate in a cartel in order to enhance the quality of output or their safety; this may also be done for reasons of efficiency. Some forms of collaboration such as the inter-firm network came about due to rising complexities of organisations. However, they took relatively long periods of time before they came to be accepted as cultural norms; a lot of ambiguity exists on the need for collaboration or cooperation; all of it depends in the nature of acceptance in the concerned nation (Zeiltin, 2007). Businesses today have now moved from the anti-trust mindset of yester years; in fact, it is not uncommon to find some form of collaboration between competitors in almost all industries of the economy. New technologies are becoming more complex and require huge capital investment to develop; markets are getting more turbulent, and the time needed to deliver products to the market is also increasing. Additionally, time and geographical boundaries have shrunk thus indicating that operating business internationally is a critical factor. Government policies are not particularly favourable in conducting business and so are other issues within

Western expansion and development Essay Example | Topics and Well Written Essays - 1000 words

Western expansion and development - Essay Example The immigrants from China faced a lot of discrimination upon arriving in the United States which was evident in the manner of treatment that they received in the mines. In 1850s, the U.S government limited the right of naturalization to the white immigrants only which made it difficult for Chinese immigrants to pursue independent livelihoods. The Missouri Compromise allowed for slavery to continue in the newly acquired states which meant that immigrants of African descent were to be treated as slaves and not as people who came in search for greener pastures. The Mexican War saw the expansion of the United States to parts of territories that initially belonged to Mexico. This made it easy for immigrants from Mexico to move to the country with most of them settling in the states of Texas and Oregon regions. The United States played a critical role in ending the Second World War. The nation initially tried to avoid entering the war even as pressure from the Britain mounted. They finally agreed to take part in it as it was seen as a threat to its security. During the Holocaust, the United States did not lower its tight stance on immigration and it is believed many Jews would have been saved if the laws were lessened to allow Jews immigrants into the country. The American Jewish community had undermined the impact of the Holocaust and they failed to consider advocating for the uplifting of more Jews from the affected areas. During the World War II period the American Jewish community was bitterly and deeply divided, and was unable to form a common front. Proponents of a Jewish state and Jewish army agitated, but many leaders were so fearful of an anti-Semitic backlash inside the U.S. that they demanded that all Jews keep a low public profile. The Bracero Program which was formed in the year 1942 saw the increase of immigrants from Mexico who mainly came into the United States to offer cheap labor. At the onset of World War II, the