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

質量mの単純な振り子が長さLのひもに取り付けられ、重力gの影響下で振動するとします。振り子の期間の方程式の形式は何ですか?
最初に、問題に関連する変数を特定してリストします。期間 T は、これらの変数の積として表すことができ、それぞれが未知の指数に上げられます。ここで、k は無次元定数です。
無次元定数を除けば、変数の次元と期間を関連付ける方程式が得られます。
ここで、両側の次元の指数を等価にし、方程式を解くことにより、未知の指数の値が決定されます。
指数を代入すると、定数kと重力加速度に対する長さの平方根の積である期間の最終的な式が得られます。
次元解析の限界の1つは、無次元定数kの値を見つけることができないことです。

<ol>
	<li>OpenStaxです。(2019). 大学物理学 Vol.1.【Web版】。<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 から取得</a>。セクション1.4;22-25ページ。</li>
</ol>

問題解決: 次元分析

個別の物理量を結び付けるすべての数学方程式は次元的に一貫している必要があり、これは 2 つの規則に従う必要があることを意味します。 このため、寸法の概念は非常に重要です。 最初のルールは、等式の両側にある方程式の式はまったく同じ次元を持たなければならないということです。つまり、同じ次元の数量は加算または削除できます。 2 番目のルールは、指数関数、対数関数、三角関数などの一般的な数学関数はすべて、方程式内に無次元の引数を持たなければならないことを規定しています。

方程式がこれら 2 つのルールのいずれかを破ると次元的に矛盾するため、方程式は物理法則の正確な主張を表すことはできません。 次元解析は、さまざまな物理法則を覚えたり、代数エラーやタイプミスをチェックしたり、将来の物理法則がどのような形になるかを推測したりするのに役立ちます。

基本量を使用して、任意の物理量を作成できます。 数量は、基準数量で表現される場合、基準数量のさまざまな乗の積として表されます。 その底の量の次元は、方程式に現れる基本量の指数です。

質量と加速度の積として定義される力の物理量を考えてみましょう。 加速度は速度の変化を時間間隔で割ったものとして計算され、長さを時間間隔で割った値が速度に等しくなります。 その結果、力は次のような次元になります。1 は質量、1 は長さ、マイナス 2 は時間です。

別々の異なる物理量をつなぐすべての数式は、次元的に一貫していなければならず、これは2つの...

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Education

JoVE Core

Physics

物理学

Unit 1

ユニット1

Units, Dimensions, and Measurements

単位、次元、および測定

KEY TERMS AND CONCEPTS

主要用語と概念

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

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problem solving: dimensional analysis

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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 物理学は、運動学、力学、流体力学、静力学、波動、音響などの基礎的な物理学の理論と応用を、簡潔でわかりやすいアニメーション ビデオで紹介します。さらに、科学者の実際のビデオでは、世界中の今日の研究所で行われている古典的および独自の研究実験における関連概念の適用を示しています。

Dimensional Checks for Physics Equations

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