Name: resultant force from tugboats
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 23 s
Channel: JoVE

2つ以上の力ベクトルのベクトル加算は、平行四辺形の法則を使用して、合力を求めることができます。
2隻の小さなタグボートに引っ張られている船を考えてみましょう。F1 と F2 の 2 つの力は、船に同時に作用します。
平行四辺形の法則を使用して正味の力を求めるには、平行四辺形が形成されるように、両方の力ベクトルに平行線を引きます。
これで、平行四辺形の対角線は、2 隻のタグボートによる船の正味の力 FN を表します。これは、力F1とF2のベクトル和として表されます。
ここで、別のタグボートからの3番目の力が船に作用すると仮定します。
FNと3番目の力F3を使用して、別の平行四辺形が形成されます。 この新しい平行四辺形の対角線は、3隻のタグボートすべてによる船の合力となります。
ここでは、船の合力は、3 つの力すべてのベクトル合計として表されます。

<ol>
	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス \u2012 静力学と力学。 ニュージャージー州ホーボーケン:ピアソンプレンティスホール。18-22ページ</li>
	<li>ビール、FP;ジョンストン、ER;マズレック、D.F;クロムウェル、PJおよびセルフ、BP(2019)。エンジニアのためのベクトル力学 \u2012 静力学と動力学。ニューヨーク:マグロウヒル。29ページ</li>
</ol>

力のベクトルの加法

複数の力が物体に作用する影響を理解するために、ベクトルの加法は重要な概念です。この数学的概念は、2つ以上の力が物体に作用する場合の合力を計算するために用いられます。

ベクトルの加法の概念を理解するために、2つの小さな曳船によって引かれる船のシナリオを考えてみましょう。2つの力、F1とF2は、異なる方向で船に同時に作用します。平行四辺形の法則を使用して船に作用する合力を計算することができます。

平行四辺形の法則は、2つのベクトルを加算するための幾何学的な方法を提供するため、ベクトルの加法において役立ちます。この法則は、平行四辺形の隣り合う2辺が2つのベクトルを表す場合、それらの交点を通る対角線が2つのベクトルのベクトル和を表すと述べています。2つの曳船によって引かれる船の文脈でこの法則を適用するために、両方の力ベクトルに平行な線を引いて平行四辺形を形成します。そして、2つのベクトルの交点を通る対角線を引きます。この対角線は、2つの曳船が生み出す船への合力を表します。この合力は、2つの力のベクトル和として表されます。

しかし、第3の力F3が導入された場合、別の平行四辺形が描かれます。今度は、以前の合力F1 + F2を新しい平行四辺形の1辺として使用し、第3の力F3をもう1つの辺とします。再び、この平行四辺形の対角線は、すべての3つの曳船による船に作用する結果または合力を表します。この船に対する合力は、F1 + F2 + F3として表されます。

ベクトルの加法は、物体に複数の力が作用する影響を理解するための強力なツールです。物理学、工学、数学など、さまざまな分野において重要です。平行四辺形の法則とベクトルの加法の概念を理解することで、物体に作用する合力の正確な計算と予測が可能になります。

物体に作用する複数の力の影響を理解する場合、ベクトル加算は把握すべき重要な概念です。この数学的概念は...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Force Vectors

力のベクトル

KEY TERMS AND CONCEPTS

主要用語と概念

Vector Addition of Forces

resultant force from tugboats

Resultant force can be found with vector addition when two or more forces act on the same object. This idea helps explain how multiple pulls combine to create one net force. It is a key tool for understanding motion in physics.

A clear example is a ship pulled by two small tugboats. The forces, F1 and F2, act at the same time but in different directions. The net force on the ship is the vector sum of those two forces.

The parallelogram law gives a geometric way to add vectors. In this method, the two force vectors form adjacent sides of a parallelogram. The diagonal through their intersection shows the resultant, or net force.

To use the law for the tugboat example, parallel lines are drawn to both force vectors to complete the parallelogram. Then a diagonal is drawn across the shape. That diagonal represents the net force acting on the ship from the two tugboats.

A third force, F3, can be added in the same way. A new parallelogram is drawn using the previous resultant, F1 + F2, as one side and F3 as the other. The diagonal of this second parallelogram gives the total resultant force, written as F1 + F2 + F3.

Vector addition is useful in physics, engineering, and mathematics. It helps scientists and engineers calculate and predict the net force on an object. The parallelogram law makes that prediction easier to picture and measure.

Resultant Force from Tugboats

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vector addition of forces

Resultant force can be found with vector addition when two or more forces act on the same object. This idea helps explain how multiple pulls combine to create one net force. It is a key tool for understanding motion in physics. A clear example is a ship pulled by two small tugboats. The forces, F1 and F2, act at the same time but in different directions. The net force on the ship is the vector sum of those two forces. The parallelogram law gives a geometric way to add vectors. In this method,...

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key terms and concepts

Learn how vector addition uses the parallelogram law to find the resultant force on a ship pulled by two or three tugboats in physics.

Resultant Force from Tugboats | JoVE K12

JoVE Core 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Resultant Force from Tugboats

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