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

נניח שמטוטלת פשוטה של מסה m מחוברת למחרוזת באורך L, מתנודדת בהשפעת כוח הכבידה g. מהי צורת המשוואה לתקופת הזמן של המטוטלת?
בתחילה, לזהות ולרשום את המשתנים המעורבים בבעיה. ניתן לבטא את פרק הזמן T כמכפלה של משתנים אלה, שכל אחד מהם מועלה למעריך לא ידוע. כאן, k הוא קבוע חסר ממד.
למעט הקבוע חסר הממדים מתקבלת משוואה המתייחסת לממדי המשתנים עם פרק הזמן.
כעת, על ידי השוואת המעריכים של הממדים משני הצדדים ופתרון המשוואות, נקבעים הערכים של המעריכים הלא ידועים.
בהחלפת המעריכים, מתקבל הביטוי הסופי לפרק הזמן, שהוא מכפלה של הקבוע k והשורש הריבועי של האורך על תאוצת הכבידה.
אחת המגבלות של ניתוח ממדי היא שהוא אינו מאפשר לנו למצוא את הערך של הקבוע חסר הממדים k.

<ol>
	<li>OpenStax. (2019). פיזיקה אוניברסיטאית כרך 1. [גרסת אינטרנט]. מקור: <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>; סעיף 1.4; עמודים 22-25.</li>
</ol>

פתרון בעיות: ניתוח ממדים

<div style="direction:rtl">כל משוואה מתמטית המחברת בין גדלים פיזיקליים נפרדים חייבת להיות עקבית מבחינה מימדית, מה שמרמז שהיא חייבת לעמוד בשני כללים. מסיבה זו, מושג הממד הוא קריטי. הכלל הראשון הוא שהביטויים של משוואה משני צדי השוויון חייבים להיות בעלי אותו ממד בדיוק, כלומר, ניתן להוסיף או להסיר כמויות מאותו ממד. הכלל השני קובע שכל הפונקציות המתמטיות הפופולריות, כגון פונקציות אקספוננציאליות, לוגריתמיות וטריגונומטריות, חייבות להיות עם ארגומנטים חסרי ממדים במשוואה. משוואה אינה עקבית מבחינה מימדית כדי לשבור אחד משני הכללים הללו, כך שמשוואה אינה יכולה להיות ייצוג של קביעה מדויקת של חוק פיזיקלי כלשהו. ניתוח ממדים יכול לעזור לזכור את חוקי הפיזיקה השונים, לבדוק אם יש שגיאות אלגבריות או שגיאות הקלדה, ואפילו להעלות השערות לגבי הצורה שחוקי הפיזיקה העתידיים עשויים לקבל. ניתן להשתמש בכמויות הבסיס ליצירת כל כמויות פיזיות רצויות. כמות מצוינת כמכפלה של עצמות שונות של כמויות הבסיס כאשר היא מבוטאת במונחים של כמויות הבסיס. מימד הכמות באותו בסיס הוא המעריך של כמות בסיס המופיעה במשוואה. חשבו על הכוח הגודל הפיזיקלי, המוגדר כמסה כפול תאוצה. התאוצה מחושבת כשינוי המהירות חלקי מרווח זמן, בעוד שהאורך חלקי מרווח הזמן שווה למהירות. כתוצאה מכך, לכוח יש את הממדים הבאים: אחד במסה, אחד באורך ומינוס שניים בזמן.</div>

כל משוואה מתמטית המחברת כמויות פיזיקליות נפרדות חייבת להיות עקבית מבחינה ממדית, מה שמרמז שהיא חייבת לעמוד בשני ...

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Education

JoVE Core

Physics

פיזיקה

Unit 1

יחידה 1

Units, Dimensions, and Measurements

יחידות, ממדים ומדידות

KEY TERMS AND CONCEPTS

מונחים ומושגים מרכזיים

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...

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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...

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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...

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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

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