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Rigid Body Statics

Fundamentals: Are you ready to explore the world of rigid bodies?

In this chapter, we will dive into the fascinating world of rigid body mechanics and explore:

  • The invisible forces that act on objects and set them in motion or slow them down.
  • The secret of cutting free, which allows us to isolate the most important forces acting on an object.
  • The magical free body diagrams, which show us how forces and moments act on an object.
  • The rigid body and its six degrees of freedom, which give it its mobility.
  • The 6 axioms of rigid body statics, which are the foundation for everything we know about objects at rest.

Are you ready to uncover these secrets?

Then buckle up and let's go on an exciting journey into the world of rigid body mechanics!

It's going to be exciting!

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Table of Contents

4.5 The Parallelogram of Forces

An Indispensable Tool in Rigid Body Statics

The axiom of the parallelogram of forces is an important principle in rigid body statics, which states that the resultant force of a system of two forces can be represented as an equivalent parallelogram of forces. This means that the resultant force has the same effect as if two arbitrary, non-parallel forces were acting on the body.

To understand the axiom of the parallelogram of forces, let's consider a simple example: A body is pulled by two forces F1 and F2 in different directions:

This figure 2.4.9 shows an object being pulled by two different forces F1 and F2. The resultant R is determined using the parallelogram of forces.
Fig. 2.4.9: Example of Parallelogram of Forces

The resultant force Fres or R can be represented as the diagonal of a parallelogram, whose sides are formed by the forces F1 and F2. The magnitude and direction of the resultant force can be determined by the side lengths and angles of the parallelogram.

In practice, the axiom of the parallelogram of forces is often used to calculate the resultant forces in complex static systems. By representing the individual forces as vectors and adding them according to the parallelogram law, one can determine the resultant force and its effect on the body.

Whether you're an engineer, a physicist, or just curious: The parallelogram of forces is an important principle in mechanics that helps you solve complex static problems and accurately analyze the distribution of forces in a system.

Math alert!

Okay, okay, I know math can be really boring sometimes. But don't worry! The parallelogram of forces is actually quite easy. You just need a little bit of vector calculus and the parallelogram law. In vector calculus, forces are represented as vectors, which have both a magnitude and a direction. The parallelogram law states that the resultant force of two non-parallel forces can be represented as the diagonal of a parallelogram, whose sides are formed by the two forces.

Mathematically, the axiom of the parallelogram of forces can be represented as follows: Let F1 and F2 be two forces acting on a body. To calculate the resultant force R, the forces are represented as vectors. The resultant force R then corresponds to the vector sum of F1 and F2:

$$ \begin{aligned} \vec{R} = \vec{F}_1 + \vec{F}_2 = \vec{F}_2 + \vec{F}_1 \end{aligned} $$

(2.2)

To construct the parallelogram, the vectors F1 and F2 are placed with their origin at the same point. The resultant force R then corresponds to the diagonal of the parallelogram, which runs from the origin of the vectors to their intersection point.

This figure 2.4.10 shows two non-parallel forces and their lines of action. They are each shifted along their lines of action to the intersection point, and a parallelogram of forces with resultant is constructed.
Fig. 2.4.10: Construction of Parallelogram of Forces

The magnitude of the resultant force R can be calculated using the law of cosines. The law of cosines states that in a triangle with side lengths a, b, and c

This figure 2.4.11 shows a triangle with sides a, b, c, and the angles opposite to them, alpha, beta, gamma.
Fig. 2.4.11: Cosine Law

the angles \(\alpha\), \(\beta\) and \(\gamma\) opposite the sides a, b, and c can be calculated as follows:

The Law of Cosines:

$$ \begin{aligned} a^2 &= b^2 + c^2 - 2bc\cdot \cos(\alpha) \\[7pt] b^2 &= a^2 + c^2 - 2ac\cdot \cos(\beta) \\[7pt] c^2 &= a^2 + b^2 - 2ab\cdot \cos(\gamma) \end{aligned} $$

By applying the law of cosines to the parallelogram, where a and b are the side lengths of the forces F2 and F1 and c is the length of the resultant force R,

This figure 2.4.11 shows a triangle with sides F2, F1, R, and the angles alpha and 180°-alpha.
Fig. 2.4.12: Cosine Law for the Triangle of Forces

the magnitude of the resultant force can be calculated exactly:

$$ \begin{align} \tag{1} R &= \sqrt{{F_2}^2 + {F_1}^2 - 2F_2F_1\cdot \cos(180°-\alpha)} \end{align} $$

With \(\cos(\pi-\alpha)= -\cos(\alpha)\) or \(\cos(180°-\alpha) = -\cos(\alpha)\), it follows:

$$ \begin{aligned} R &= \sqrt{{F_1}^2 + {F_2}^2 + 2F_1F_2\cdot \cos(\alpha)} \end{aligned} $$

(2.3)

where \(\alpha\) is the angle between \(F_1\) and \(F_2\)

Thanks to the mathematical foundations of vector calculus and the parallelogram law, the axiom of the parallelogram of forces can be applied precisely and serves engineers and physicists as an important basis for the calculation and analysis of forces in static systems.

Pure practicality!

The parallelogram of forces is important in many areas, for example:

  • Civil engineering: Analyzing structures and constructions
  • Mechanical engineering: Calculating the distribution of forces in machines
  • Vehicle technology: Optimizing the handling of cars
Conclusion:

The parallelogram of forces is a brilliant tool for understanding forces in static systems. It's easy to learn and has many practical applications.

So, what are you waiting for? Try it out!