Name: dimensional checks for physics equations
Uploaded: 2023-05-16T08:18:40.000Z
Duration: 1 min 8 s
Channel: JoVE

Предположим, что простой маятник с массой m прикреплен к струне длины L, колеблющейся под действием силы тяжести g. Какова форма уравнения для временного периода маятника?
Сначала определите и перечислите переменные, участвующие в проблеме. Период времени T может быть выражен как произведение этих переменных, каждая из которых возведена в неизвестную экспоненту. Здесь k — безразмерная константа.
Исключая безразмерную константу, получается уравнение, связывающее размерность переменных с периодом времени.
Теперь, уравнивая показатели размерностей с обеих сторон и решая уравнения, определяются значения неизвестных показателей.
При подстановке экспонент получается итоговое выражение для временного периода, которое является произведением постоянной k и квадратного корня из длины при гравитационном ускорении.
Одним из ограничений размерного анализа является то, что он не позволяет нам найти значение безразмерной постоянной k.

<ol>
	<li>OpenStax. (2019). Университетская физика Том 1. [Веб-версия]. Получено из <a href="https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis">https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis</a>; раздел 1.4; Страницы 22-25.</li>
</ol>

Решение задачи: Размерностный анализ

Каждое математическое уравнение, которое связывает отдельные физические величины, должно быть размерно согласованным, что подразумевает соблюдение двух правил. По этой причине понятие размерности имеет важное значение. Первое правило состоит в том, что выражения уравнения по обе стороны от равенства должны иметь одинаковую размерность, то есть величины одной размерности могут быть сложены или убраны. Второе правило предписывает, что все популярные математические функции, такие как экспоненциальные, логарифмические и тригонометрические функции, должны иметь безразмерные аргументы в уравнении.

Соблюдение этих двух правил является несовместимым с размерностью уравнения, поэтому оно не может быть представлением точного утверждения какого-либо физического закона. Размерный анализ может помочь запомнить различные законы физики, проверить наличие алгебраических ошибок или опечаток и даже предположить форму, которую могут принимать будущие законы физики.

Базовые величины могут использоваться для создания любых желаемых физических величин. Величина выражается как произведение различных степеней базовых величин, когда она выражается через базовые величины. Размерность величины в данной системе является показателем степени, в которой базовая величина появляется в уравнении.

Рассмотрим физическую величину силу, которая определяется как произведение массы на ускорение. Ускорение рассчитывается как изменение скорости, делённое на интервал времени, в то время как длина, делённая на интервал времени, равна скорости. В результате, сила имеет следующие размерности: одна в массе, одна в длине и минус две во времени.

Каждое математическое уравнение, соединяющее отдельные различные физические величины, должно быть размерно согласованным, что подразумевает, что оно должно подчиняться двум ...

/files/media/science-education/science-education-thumbs/14687.jpg

Education

JoVE Core

Physics

Физика

Unit 1

Раздел 1

Units, Dimensions, and Measurements

Единицы, размерности и измерения

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

Problem Solving: Dimensional Analysis

dimensional checks for physics equations

Dimensional analysis helps check whether physics equations make sense. It compares the dimensions on each side of an equation and shows whether the units are consistent. This method is useful for spotting algebra errors, typos, and for remembering physical laws.

A valid equation must follow two rules. First, both sides of an equality must have the same dimension, so only quantities with matching dimensions can be added or subtracted. Second, common math functions such as exponential, logarithmic, and trigonometric functions must use dimensionless arguments.

If an equation breaks either rule, it is dimensionally inconsistent. That means the equation cannot correctly represent a physical law. Dimensional analysis also helps scientists think about the form future laws of physics might take.

Base quantities are the building blocks for other physical quantities. A quantity can be written as a product of powers of the base quantities. In that form, the dimension of a quantity is given by the exponent of each base quantity.

Force is a useful example. Force is defined as mass multiplied by acceleration. Acceleration is the change in velocity divided by time, and velocity is length divided by time. From these relationships, force has dimensions of mass to the first power, length to the first power, and time to the minus second power.

Dimensional Checks for Physics Equations

the scope of physics

How Physics Shapes Science and Technology

Physics studies the interactions of energy, matter, space, and time. It looks for the basic mechanisms that explain natural phenomena. The word physics comes from the Greek word physis, which means nature. Physics helps us understand the natural world at its most fundamental level. It relies on quantitative laws, or rules written in numbers and measurements, to describe how things behave. That approach can also help other fields improve the limits of current technology. Physics matters in...

orders of magnitude

Estimating Scale with Powers of 10

Orders of magnitude help estimate the scale of a number using powers of 10. It is a simple way to round values to the nearest power of 10. This makes it easier to compare very large and very small numbers. To find the order of magnitude of a number, take its base-10 logarithm and round to the nearest integer. The result gives the power of 10 that most closely matches the original value. This method gives a consistent estimate of size without using exact digits. Orders of magnitude are useful...

/files/media/science-education/science-education-thumbs/12596.jpg

models, theories, and laws

Scientific Models, Theories, and Laws

Scientific models, theories, and laws help physicists describe natural phenomena. A model is a simplified version of a physical system that is too complex to study in full. It acts as a rough mental or visual picture of what may be happening. Good models keep the important parts and simplify the rest so the problem becomes workable. The light wave model is one example. Light waves are usually invisible, so they cannot be seen in the same way as water waves. Even so, it is useful to think of...

/files/media/science-education/science-education-thumbs/14686.jpg

units and standards of measurement

SI Units for Physical Quantities

SI units give a standard way to measure physical quantities. A physical quantity can be defined by how it is measured or by how it is calculated from other measurements. For example, the mass and dimensions of a metallic cube can be found with a weighing machine and a meter scale. Some quantities are built from others. The volume of a cube is the cube of its side length. The density of the cube is then found by dividing mass by volume. Measurements are written in units, which are standardized...

estimation of the physical quantities

Fermi Estimation for Physical Quantities

Fermi estimation helps physicists, scientists, and engineers make rough calculations for physical quantities. These estimates are also called guesstimates, order-of-magnitude approximations, back-of-the-envelope calculations, or Fermi calculations. Enrico Fermi was known for making these kinds of estimates with surprising precision. Estimation is not random guessing. It uses prior experience and sound physical reasoning to get a rough value for a quantity. This kind of reasoning helps build...

/files/media/science-education/science-education-thumbs/14734.jpg

base quantities and derived quantities

SI Base Units Behind Measured Quantities

SI base units are the starting point for measuring physical quantities. In any system of units, some quantities must be defined through direct measurement. These are called base quantities, and their units are base units. Other physical quantities are built from these base values by algebraic combinations. These are called derived quantities, and their units are derived units. For example, area comes from two lengths multiplied together. It is a derived quantity and is written in SI base units...

conversion of units

Unit Conversion Steps and Real-World Errors

Unit conversion lets you change a measurement from one unit to another. It is useful in everyday life, such as when a cookbook gives amounts in liters or ounces and you need cups instead. It also helps when directions use feet or meters, but you want to know the distance in miles. The first step is to list the given unit and the unit you need. The second step is to choose a conversion factor, which is a ratio that shows how many of one unit equal another unit. For example, 1,000 meters equal 1...

accuracy and precision

Repeated Measurements in Science

Repeated measurements help scientists check the quality of their results. They are used to judge both precision and accuracy. These two ideas are related, but they do not mean the same thing. Measurements are precise when repeated trials give very similar results. Precision shows how closely the values agree with one another. Measurements are accurate when the result is very close to the true value or the accepted value. Accuracy shows how closely the values agree with the real target. High...

random and systematic errors

How Measurement Errors Affect Results

Measurement errors can change how accurate and precise a result appears. Scientists try to record measurements as carefully as possible, but errors can still happen. These errors are usually random or systematic. Random errors come from inconsistency or fluctuation in the measuring process. They can also come from changes in the quantity being measured. In repeated measurements, random error may make a value appear greater than the true value one time and less than it the next. A simple...

rules for significant figures

Measurement Precision and Significant Figures

Measurement precision depends on the tool used, and significant figures show that precision. A ruler can measure length to the closest millimeter. A caliper can measure length to the nearest 0.01 mm, so it gives a more precise measurement because it can detect very small changes in length. When a value is written from a measurement, the last digit is not exact. The person making the measurement must estimate that digit. For example, a stick that seems to fall between 13.2 cm and 13.3 cm must...

significant figures in calculations

Rounding Rules for Significant Figures

Rounding rules for significant figures help show the right level of certainty in measured results. When a calculation is reported with the correct number of significant figures, it better matches the uncertainty in the measurements. The rounding rule depends on the type of math being done. For addition and subtraction, the result should have the same number of decimal places as the number with the fewest decimal places. For multiplication and division, the result should have the same number of...

dimensional analysis

Dimensional Checks in Physics Equations

Dimensional consistency is a key check in physics equations. It means the dimensions on both sides of an equation must match. The idea helps show whether an equation can represent a real physical law. The first rule is simple: expressions on each side of the equal sign must have the same dimensions. That makes sense because you can only add or subtract quantities of the same type. The second rule says that the input to a standard mathematical function, such as a trigonometric function, a...

problem solving: dimensional analysis

Dimensional analysis helps check whether physics equations make sense. It compares the dimensions on each side of an equation and shows whether the units are consistent. This method is useful for spotting algebra errors, typos, and for remembering physical laws. A valid equation must follow two rules. First, both sides of an equality must have the same dimension, so only quantities with matching dimensions can be added or subtracted. Second, common math functions such as exponential,...

solving problems in physics

Physics Problem Solving: Strategy to Check

Physics problem solving uses general physical principles to work out specific situations. It is a key skill in physics, and it can also help with everyday life. The same analytical skills can be used in new situations, while a list of facts can never cover every possible case. Solving physics problems also takes creativity and insight. These skills grow with experience and practice. There is no simple step-by-step method that works for every problem, but a three-stage process can guide the...

key terms and concepts

Learn how dimensional analysis checks physics equations, keeps units consistent, and reveals the dimensions of force from mass, length, and time.

Dimensional Checks for Physics Equations | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

Dimensional Checks for Physics Equations

Description

Transcript