Banach-Tarski Paradox Talk Essay Example

Banach Banach-Tarski Paradox Talk Essay Banach-Tarski Paradox Talk Essay Essay Topic: Talk The catch, of course, is that each of these four pieces are so complex that they do not have any measure (i.e., their respective volumes are not well-defined), and that we do not know how to mathematically describe them other than the fact that they exist and exhibit the strange re-assembly property. In fact, it is quite possible that each of those pieces consists of isolated points spread out throughout the entire volume of the original sphere S. Now, accepting this is really pretty tough, but its possible to attempt to make an intuitive rationalization of it. Ill do it by way of an analogy with a physical sphere.  Lets forget for the moment the mathematical sphere S, which has infinite density. Lets consider a real, physical sphere B (for ball), also of radius 1. B is identical to S except that it consists of a finite number of atoms. These atoms are lain out in a crystalline structure. Usually, the crystalline structure of an object is a simple geometric relationship between neighbouring atoms.  Notice that although the geometric relationship between atoms define its crystalline structure, the precise distance between atoms in different molecules may vary. This leads to materials of different densities. Now, we perform the equivalent of a Banach-Tarski decomposition on our physical sphere B: we atomize B into four spherical clouds of atoms, lets call them C1, C2, C3, and C4. (Well ignore the central atom in B, just as in the mathematical version of this decomposition.) Lets assume that each of these clouds are sparse enough that they are gaseous, no longer solid by themselves (to try and imitate the immeasuarability of the four pieces of S ). Furthermore, lets say that the atoms in each of these clouds are laid out in a regular pattern, so that if we rotate C1 by some angle G, and put it together with C2 in the same spherical region, the atoms in both clouds line up into the same crystalline structure as B, except that now the distance between atoms is greater (to account for the missing atoms now in clouds C3 and C4). Similarly, assume we can do the same with C3 and C4: we just translate them away from the original spherical region of B so that they dont interfere with C1 and C2, and reassemble them into another sphere. Now, we have successfully built two (physical!) spheres with the same radius as B, using only material from B itself. Each of the two spheres have the same crystalline structure as B. The only difference between these spheres and B is that they each have only half the density of B.  To bring this analogy back to the mathematical sphere S: we can think of the infinitely complex pieces A1, A4 as the equivalent of atom clouds, which are non-solid (immeasurable). For those that know something of topology, which excludes myself, you can think of the crystalline structure of S as the topological structure of points in R3. These clouds lack this crystalline structure (i.e., they are unmeasurable); but by suitable rearrangement of them, we can form them into two identical spheres, with half the density of the original, so that they do have the same crystalline structure (i.e., the resultant two spheres are well-behaved, measurable sets). These two spheres are identical to S, except for having only half the density of S. However, S is infinitely dense, and so are its pieces A1, A4. This means the two resultant spheres are still infinitely dense. That is to say, they are identical to S. Et voila!There is no paradox here after all. We are merely seeing the logical consequence of mathematical sets like S being infinitely dense. In fact, if you think about it, this is not any stranger than how we managed to duplicate the set of all integers, by splitting it up into two halves, and renaming the members in each half so they each become identical to the original set again. It is only logical that we can continually extract more volume out of an infinitely dense, mathematical sphere S. Epilogue Now, having convinced you that the Banach-Tarski Paradox isnt really that strange after all, Id like to mention that the derivation of this paradox depends on the Axiom of Choice, and although most mathematicians accept the Axiom of Choice, not all agree with it. There has been much debate over the merit of adopting this axiom, as well as research into the consequences of choosing either way: it does simplify a lot of mathematical proofs, but it also introduces strange results like the Banach-Tarski paradox which we just discussed. If youre unfamiliar with the Axiom of Choice, it basically goes like this: if you have a collection of sets C (which may potentially contain an uncountably large number of sets), then there exists a set H, called the choice set, which contains precisely one element from each (non-empty) set in C. H is called the choice set because you are essentially going through each set in C and choosing one element from it. One feature of the Axiom of Choice is that H is simply assumed to exist; there is no algorithm given which might tell you how to construct an example of H. In the case of the Banach-Tarski paradox, each of the infinitely complex pieces of the sphere S is built from these choice sets. Since we do not know of any algorithm to actually construct these sets, we can only indirectly infer some of the properties of the pieces, such as their not having a (Lebesgue-) measure (i.e., they have intractible geometric complexity). Some of the debate surrounding the Axiom of Choice revolves around whether these non-constructible sets are mathematically admissible. The reader is encouraged to make good use to Google for more information about this debate; it is too vast a topic to explore in this article. It suffices to say that most mathematicians adopt the Axiom of Choice, simply because of the usefulness of results that can be derived. One might wonder, then, about what would happen if we didnt assume the Axiom of Choice. We do know that we would likely be unable to derive the Banach-Tarski paradox; however, we also know that paradoxical sets do exist even without the Axiom of Choice. These paradoxical sets exhibit the same weird behaviour of the Banach-Tarski spheres, in that you can decompose these sets into a finite number of parts, and reassemble them into multiple copies of the original.

How to Use the Perl Array Push() Function

How to Use the Perl Array Push() Function The Perl  push()  function is used to push a value or values onto the end of an array, which increases the number of elements. The new  values  then become the  last elements  in the array. It returns the new total number of elements in the array. Its easy to confuse this function with  the  unshift()  function, which adds elements to the  beginning  of an array. Heres an  example of the Perl push() function: myNames (Larry, Curly);push myNames, Moe;print myNames\n; When this code is executed, it delivers: Larry Curly Moe Picture a row of numbered boxes, going from left to right. The push() function pushes the new value or values onto the right side of the array  and increases the elements.   The array can also be thought of as a stack. Picture a stack of numbered boxes, starting with 0 at the top and increasing as it goes down. The push() function pushes the value onto the bottom of the stack  and increases the elements, like this: myNames (Larry,Curly);push myNames, Moe; You can also push multiple values onto the array directly ... myNames (Larry, Curly);push myNames, (Moe, Shemp); ... or by pushing on an array: myNames (Larry, Curly);moreNames (Moe, Shemp);push (myNames, moreNames); Note for beginning programmers:  Perl arrays begin with an symbol. Each complete line of code must end with a semicolon. If it doesnt, it wont execute. In the stacked example in this article, the lines without a semicolon are values contained in an array and enclosed in parentheses. This isnt an exception to the semicolon rule, as much as a result of the stack approach. The values in the array are not individual lines of code. It is easier to picture this in the horizontal approach to coding. Other Functions for Manipulating Arrays Other functions are also used to manipulate arrays. These make it easy and efficient to use a Perl array as a stack or as a queue. In addition to the push function, you can use: Pop function – removes and returns the last element of an arrayShift function – moves the whole array to the left. The element that is the first element of the array falls off the array and becomes the return value of the functionUnshift function – the opposite of the shift function, places a value at the beginning of an array and moves all the other element to the right.

Resistence to Change Case Study Example | Topics and Well Written Essays - 1250 words

Resistence to Change - Case Study Example The report further asserts that, lack of communication across the organization heightened resistance because majority of employees were unaware of impending changes that were being introduced (Griffin & Moorehead, 2012). In above connection, lack of single culture to integrate the whole organization was a significant source of resistance to change because almost every department at NASA had its own culture different from other departments (Griffin & Moorehead, 2012). Lack of proper risk mitigation measures in place to consolidated organization culture, was major factor that caused resistance to change (Columbia Accident Investigation Board, 2003). The report suggested that, termination of risky flights was not solution of reducing escalated cases of accidents but rather NASA should focus at establishing structural procedures that guarantees sustainability of a safety culture (Columbia Accident Investigation Board, 2003). Therefore, this analysis focuses at providing an insight of Col umbian mishap concerning resistance to change and how it contributes to accidents. The analysis will further focus on both individual and cultural resistance as well as the impact of the board recommendation on individual and structural barriers. Individual sources of resistance to change and their causes The Sources and causes of individual resistance could be attributed to numerous factors that may include; fear for unknown. People tend to fear uncertainties that may to occur as a result of change (Griffin, & Moorehead, 2012). This is because individuals find it difficult to cope with dynamics for instance, it can be scrutinized that lack of prior cultural integration within all departments at NASA brought great resistance (Columbia Accident Investigation Board, 2003). Additionally, Inability to adjust and start doing things using new ways may make people fear change and therefore, they may develop resistance. Inability to adjust may be overcome by ensuring that there is efficient and effective integration of NASA cultural goals and mission during the earliest juncture possible (Columbia Accident Investigation Board, 2003). In above connection, it can be scrutinized that individual developed resistance to change due to a lack of effective communication, whereby, NASA engineers could not effectively communicate safety measure to the management leading to individual resistance (Columbia Accident Investigation Board, 2003). On the other hand, NASA managers failed to communicate issues raised by engineers to all the departments leading to resistance among employees because changes were introduced without a prior notification (Columbia Accident Investigation Board, 2003). Lack of proper leadership within organization contributed significantly to individual resistance because managers were not involved in directing employees on what to do to ensure that safety standard were being observed. According to a report by Columbia Accident Investigation Board 2003, misund erstanding between managers and NASA engineers was significant source and cause of resistance. Structural sources of resistance to change and their causes The most common sources of resistance to change include; inertia within the organization structure (Griffin & Moorehead, 2012). Whereby, most organizations contain some mechanisms that enhance stability and sustainability making it difficult to

Marketing Plan Essay Example | Topics and Well Written Essays - 1500 words - 2

Marketing Plan - Essay Example This is a functional drink and can be made available in different flavors to suit different tastes. With the mission to reach the product to the vast majority of the people, it has been decided to undertake a marketing plan which includes stating the marketing objectives, the research to forecast the demand and then the marketing strategy. To determine the demand that could be generated for this new product, it is considered necessary to conduct a survey. To conduct the survey market segmentation is necessary so that the target market is studied rather than the whole market. Since this is a health drink and that too a milk product, focus would be on the children between 6 and 18 years and the women. The segmentation would further be based on psychographic details like lifestyle and social demographics like the income level. Thus to conduct the survey questionnaires have been designed (Appendix A). These questionnaires would be sent out via emails to housewives. The addresses would be procured from the local clubs and women’s organizations. Questionnaires would also be sent to the local clubs and organizations where a large number of women are employed. The marketing mix of any product would include the right product at the right time in the right place with the right sales and promotion strategy and at the right price. To have competitive positioning, these are the four P’s (Product, Price, Place and Promotion) very essential for marketing any product. To ensure good sales of our product and services, it is essential to understand the product, its need and importance, the price should be reasonable, if necessary dual pricing, the product should be easily available hence distribution has to be right and finally the right awareness of the product. The promotional strategies have to be taken into account depending on competitive factors. The product is new in the market with a focus on health and targeted at children and the women and hence success is

The Elderly and Malnutrition Essay -- Health Nutrition Pyramid Diet

The Elderly and Malnutrition      Ã‚  Ã‚  Ã‚  Ã‚   At any age, nutrition is vital to maintaining health and enhancing quality of life.   However, achieving good nutrition can be especially difficult for the elderly, the fastest growing portion of America's population.   Many factors, including physiological changes, changes in nutritional needs, illness and physical limitations, food-medication interactions, depression and loneliness, and food insecurity are common causes of malnutrition in the elderly.   This paper will explore factors affecting elderly nutrition, provide suggestions for obtaining and maintaining good nutrition after age 65, and describe the services available to help the elderly meet their nutritional needs.  Ã‚        Ã‚  Ã‚  Ã‚  Ã‚   A number of factors influence the nutritional status of the elderly.   First, several physiological changes impact elderly nutritional requirements.   Changes in taste and smell may decrease an individual's appetite or desire for food.   Gastro-intestinal motility decreases with age and may lead to constipation, decreased nutrient absorption, and a decrease in fat and protein absorption.   Metabolism also slows with age; this results in a decrease of lean body mass and an increase in body fat (8).   Lastly, bone mass decreases, leaving the elderly vulnerable to bone fractures from falls and osteoporosis.      Ã‚  Ã‚  Ã‚  Ã‚   Ultimately, these physiological changes result in different nutritional needs for the elderly.   The Food and Nutrition Board of the National Academy of Sciences issues the Recommended Daily Allowances for healthy people over the age of 51.   However, these RDAs are limited in that they have been derived from studies of younger, healthy populations and do not account ... ...;. 5) Good Nutrition is Ageless. Egg Nutrition Center. 22 Jan. 2002 <http://www.enc-online.org/elderly.htm>. 6) Good Nutrition for Seniors Caregiver Fact Sheet. 9 Aug. 2001. National Network for Health. 22 Jan. 2002 <http://www.nnh.org/products/gnfs.htm>. 7) Home page. Meals on Wheels Association of America. 22 Jan. 2000.   <http://www.projectmeal.org/mowaa.html>. 8) "Nutrition Index."   Gerontology and Geriatric Dentistry n.d.: n. pag. Columbia University Dental Education Software. N.d. 9) A Profile on Older Americans: 2001. 10 Jan. 2002. Administration on Aging. 22 Jan. 2002 <http://www.aoa.gov/default.htm>. 10) The Warning Signs. New York State Office for the Aging. 22 Jan. 2002 <http://agingwell.state.ny.us/eatwell/warning.htm>. 11) Wieand, Jane. Telephone Interview. 22 Jan. 2002.      

Challenges of Microfinance Banking in Nigeria

CHALLENGES OF MICROFINANCE BANKING IN NIGERIA-1 Nigerians, like many other Africans are generally known as their brother’s keepers in view of the extended family system. But when truly analyzed, there is a possibility that this may not be so? Perhaps a hypothetical deep may reveal that not up to 10% of Nigerians would invest in ventures or people that would yield them nothing in return and here I mean, â€Å"Returns in the short run or in the immediate†. The returns may be financial, emotional, spiritual, relational or social. But fact is, there is something the investors expect in return when they give and these could include being worshipped as personal heroes, fear of harassments, fear of being harmed, for family ties, religious ties, being perceived as a caring corporate organization with the intent of bigger business tickets etc. Hardly do the â€Å"Nigerian rich† set aside a portion of their riches for the â€Å"unknown poor† or for true charity except such gifts are meant to make them be seen by the Society or Divinity as good or circumvent some threatening harms etc. This anti-poor behavior is projected into organizations; corporate, social, religious and sometimes even Government and parastatals as people see only the short end of developments without heeding the social woes of economic imbalance and insecurity. Consequent upon my above beliefs, I may classify the challenges of Microfinance banking in Nigeria under five (5) main headings: 1. COMMUNICATION GAPS AND INADEQUATE AWARENESS As a background to the undeniable problem of communication for effective microfinance, listen to this quote by Stan Paris on his article on Microfinance As A Means of Reaching The People â€Å"Problems of communication are endemic in the industry, dating back to what could be considered the first micro-loan. In 1976, Muhammad Yunus, a young economics professor at Chittagong University, Bangladesh, took his students into a small village where he discovered a woman crafting beautiful stools out of bamboo. He inquired what the woman earned for her work after repaying the trader from hom she borrowed. She told him she earned the equivalent of 2 cents in net profit. Yunus was appalled. He wanted to help her find a means of financing that would allow her to make more net profit. But, first, Yunus had an enormous communication barrier to overcome. That was a time in Bangladesh when women didn't touch money and didn't talk to men, explains Sam Daley-Harris, director of the Microcredit Summi t Campaign, an important facilitator of dialogue in the industry. Yunus had to have a female student ask the woman a question, then return to tell him the answer. There were barriers of communication even in explaining the value proposition. † Truth is that today, communication barriers exists heavily in Nigeria particularly given that even within a State, there exist as many languages and dialects as exists ethnic groups. Also, high level of illiteracy even among the educated who fails to read important things that would enable them take positive steps towards personal, interpersonal and national development. I recall given many flyers of Elim Kit â€Å"n† Kin to a friend of mine who is well educated and is on a billion naira business locally and international. This guy was in a church service when a play-lets on Kit â€Å"n† Kin was delivered by a group of young people. But after the service, he approached me on how to go about empowering a friend he wants to help and when I suggested kit â€Å"n† Kin, he asked me what that was? Obviously, my educated friend neither read the flyer which is comprehensive and explanatory nor listened to the play†¦.. here lies the problem of many Nigerians of all classes, hence my concern for communication on micro financing. Lets now look at some of areas of specific communication challenges in Nigerian Microbanking In adequate awareness campaigns and supports by the regulators: The CBN and the NDIC as Federal Government’s engine of economic development need to do greater and continuous campaigns to all stake holders; investors, universal banks, the banking public, the micro targeted poor and low-medium scale entrepreneurs until the micro banking ideas sink in the society and gets well accepted. I hear and read the Governor of CBN assuring the public of safety of the Consolidated Banks. But the Similar truths and campaigns should also be carried for the microfinance industry Lack of trust by the oor themselves who believe that the microfinance banks are just like the unregulated community banks and unregulated finance houses. Going by the antecedents of the banking industry in general, no one would blame the people for lack of trust. Only effective and far-reaching campaigns and exhibited trust in the microfinance banks by the government and the regulators can achieve this desirable confi dence. An Uneducated Population: A critical ill of the populace in Nigeria is the bottom, bottom level of poor education of the people. A country where less than 10% of the populace pays attention to the print media, less than 20% listen or watch educative audio-visuals is a great challenge to information dissemination. Incidentally, the literally â€Å"very† educated people do not read handbills and communicative documents that deal with those things like microfinance banking since they have no interest in such activities. A survey of how many people would read this documentary would shock you even if the Guardian Newspapers circulate it free to just the middle and upper classers. This attitude is worse with the targeted microfinance clients. They are more interested in chasing their â€Å"kobos† than reading or listening to programs that would help them build up and improve on their standards of living The Active Poor’s Current Preference for Gifts Than Loans. Generally, a hindrance to economic development in Nigeria is the gift preference of the poor as a culture from political and religious biases. The politicians, the rich in religious circles, towns and villages all over Nigeria have cultured the poor to beggarliness and dependence rather than empowerment for productive and financial independence. Gifts are not usually regarded as re-investment treasures by the receivers. This is why givers ought to challenge the receivers to effectively put their gifts to work by rendering these helps through microfinance banks. Products such as ELIM Kit â€Å"n† Kin are meant for well intended givers to assist the poor live above perpetual begging. It is known internationally that micro-financing is not charity. Micro funds are meant for the productive activities of the beneficiaries. Consequently, only people who fall within the active poor or low income earners who are willing and able to utilize the loans for productive activities and repay both interest and capital are the bona-fide candidates for microfinance. 2. ANTI-POOR ATTITUDES OF NIGERIANS Dearth of quality investments in microfinance banks:. Most surplus spenders and investors in Nigeria, because of the lethargic attitudes to the poor, would not invest in Micro finance banks. They prefer to invest in the â€Å"billion naira profit making consolidated banks†. Again, Nigeria investors are coasted by the quick-return and trader-like attitudes which surely does not result in a meaningful development for the Nation. They are therefore unable to go through the gestation period required in the young microfinance industry for their investments to start yielding good dividends. The Grameen Bank of Mohammad Yunus did not start making profit in year one, but today, that bank is one of the most profitable financial institutions in Asia and yet, it is a financial institution with classic recognition for adding the greatest value to humanity. Snobbish and selfish attitudes by the financially well to do Nigerians. Hopefully, good campaigns and direct involvement by the who is who in Nigeria would solve the current snub by the rich on poverty alleviation programs and enable these categories of Nigerians see that Microfinance is both a good and rewarding investment and a good development project that directly and indirectly impact their lives. 3. INSUFFICIENT SUPPORT FROM THE REGULATORS AND GOVERNMENTS Poor buy-ins and competitive approaches by the State Governments and Local Governments. The policy direction towards the millennium development goals is that one percent of the annual votes of the State and Local Governments should be channeled to Microfinance banks for on-lending to the poor and low to medium scale entrepreneurs. So far, very few states have taken steps to get this started. Lagos state is a shinning example that has taken the lead to effectively and efficiently put microfinance banking into focus. Their microfinance institutionalization, the dept of involvement of the Ministry of Women Affairs and Poverty Alleviation in developmental activities at the grassroots is equaled by non I know in Nigeria of today. This Ministry is not just a moribund ministry; it is a ministry that is empowered and achieving evidential results. This ministry needs to be copied by every state in Nigeria. Today, some 800 microfinance banks are established in Nigeria, but some states rather than use the services of these banks licensed by the CBN, choose to find a round about way of dealing with mega banks or establishing their â€Å"owned† micro credit institutions. It is my candid opinion that government involvements in microfinance banking would be counterproductive as they should concern themselves with policy formulation and control rather than implementation. How can someone formulate policies, implement it and effectively control it? Nigerian Microfinance banks have what it takes to manage micro funds for states and international bodies and should be given the chance to do so under strict supervision and controls.

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