The derivative of $f(x) = 5^{x^2}$ with respect to $x$ is:

B) $2x \cdot 5^{x^2} \cdot \ln 5$

The sum of the first n terms of an arithmetic sequence (S_n) is formulated as:

S_n = n / 2 * [ 2a + (n - 1) * d ] = n / 2 * [ a + l ]

The sum to infinity (S_infinity) of a geometric series exists and converges if and only if the common ratio satisfies:

|r| < 1 (i.e. -1 < r < 1), and S_infinity = a / (1 - r)

كيف يحسب الرقم الهيدروجيني pH للمحاليل؟ How to calculate pH and pOH?

يحسب الرقم الهيدروجيني بالعلاقة: pH = -log[H⁺]، وللمحاليل المائية عند 25°C: pH + pOH = 14. إذا كان pH 7 فالمحلول قاعدي. [الصيغة الرمزية الموزونة: pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴]

Couples and the Resultant Couple

Q: كيف يحسب الرقم الهيدروجيني pH للمحاليل؟ How to calculate pH and pOH?

يحسب الرقم الهيدروجيني بالعلاقة: pH = -log[H⁺]، وللمحاليل المائية عند 25°C: pH + pOH = 14. إذا كان pH 7 فالمحلول قاعدي. [الصيغة الرمزية الموزونة: pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴]

This lesson covers lessons 3-1 and 3-2 of the statics unit "Couples". After it you should be able to:

define a couple and find its moment;
decide when two couples are equivalent or in equilibrium;
show that a system of forces is equivalent to a couple, and find the resultant of several couples.

🌳 Concept tree

Couples and the Resultant Couple
├── 3-1 (a) What is a couple?
│   ├── From the previous lesson
│   ├── Definition
│   └── Experiment
├── 3-1 (b) The moment of a couple
│   ├── Definition
│   ├── The moment is a constant vector
│   └── Observations
├── 3-1 (c) Equilibrium under coplanar couples
│   └── Condition
├── 3-1 (d) Equivalent couples
│   └── Definition
├── 3-2 (a) Systems equivalent to a couple
│   └── Two conditions
├── 3-2 (b) Forces represented by the sides of a triangle
│   ├── Rule
│   ├── Why
│   └── Generalisation to polygons
├── 3-2 (c) The three-point test
│   ├── Rule
│   └── Why
├── 3-2 (d) The resultant couple
│   └── Definition
└── 3-2 (e) Summary
    ├── Comparison
    └── Key points

01 3-1 (a) What is a couple?

From the previous lesson

Two unlike parallel forces of unequal size have a resultant $F_{1} - F_{2}$ .
If the two forces become equal, the resultant force is zero, but the body does not stay still: it turns.
Such a pair cannot be replaced by a single force, nor by two non-parallel forces; it has its own name.

Definition

A couple is a system of two forces of equal magnitude, opposite directions and different lines of action.
Examples: turning a tap, a screwdriver or the steering wheel with two hands, a key in a lock.
The perpendicular distance between the two lines of action is the arm of the couple.

Experiment

Two forces of 15 N act in opposite directions at the ends of a graduated ruler. Take the sum of their moments about different points of the ruler.
You find the same number at every point: the turning effect does not depend on the point chosen.

02 3-1 (b) The moment of a couple

Definition

The moment of a couple is the sum of the moments of its two forces about any point in space.
Its magnitude is the product of one force and the arm: $M = F \times L$ where $F$ is the magnitude of each force and $L$ is the arm.
Its direction is perpendicular to the plane of the couple, given by the right-hand rule. In a plane we use a sign: anticlockwise positive, clockwise negative.
Units: N·m, N·cm, kg·wt·cm, dyne·cm.

The moment is a constant vector

Let the forces $\vec{F}$ and $- \vec{F}$ act at $A$ and $B$ , and let $O$ be any point.
$\vec{M} = \vec{O A} \times \vec{F} + \vec{O B} \times (- \vec{F}) = (\vec{O A} - \vec{O B}) \times \vec{F} = \vec{B A} \times \vec{F}$ .
The point $O$ has vanished from the result: the moment is the same about every point.

Observations

If the arm becomes larger, the same moment needs a smaller force.
Only the moment (size and sense) matters; the couple can be moved anywhere in its plane or into a parallel plane without changing its effect.
A couple can be rotated in its own plane, and its forces and arm can be changed, provided the moment stays the same.

✍️ Worked examples

Example 1:

Forces of 40 N form a couple with arm 4 cm in one case, and forces of 20 N form a couple with arm 8 cm in another.
Compare the moments.

Solution:

First couple: $40 \times 4 = 160$ N·cm.
Second couple: $20 \times 8 = 160$ N·cm.
The moments are equal in magnitude: halving the force while doubling the arm leaves the turning effect unchanged.

03 3-1 (c) Equilibrium under coplanar couples

Condition

A body acted on only by coplanar couples is in equilibrium if the sum of their moments is zero: $M_{1} + M_{2} + \dots + M_{n} = 0$
The sum is algebraic: anticlockwise couples positive, clockwise negative.
A couple can never be balanced by a single force; only by another couple.

✍️ Worked examples

Example 5:

A light rod is acted on by a pair of 10 N forces (arm 7 m), a pair of $40 \sqrt{3}$ N forces (making $6 0^{\circ}$ with a line of 3 m between them), and a pair of 250 N forces (arm 1 m).
Show that the rod is in equilibrium (given the senses: the first two clockwise, the third anticlockwise).

Solution:

$M_{1} = - 10 \times 7 = - 70$ N·m.
$M_{2} = - 40 \sqrt{3} \times 3 \sin 6 0^{\circ} = - 40 \sqrt{3} \times 3 \times \frac{\sqrt{3}}{2} = - 180$ N·m.
$M_{3} = + 250 \times 1 = 250$ N·m.
Sum $= - 70 - 180 + 250 = 0$ , so the rod is in equilibrium.

04 3-1 (d) Equivalent couples

Definition

Two couples in the same plane are equivalent if their moments are equal (in size and in sense).
Equivalent couples have the same effect on a rigid body even if their forces and arms are different.

✍️ Worked examples

Example 8:

$A B C D$ is a rectangle with $A B = 12$ cm and $B C = 16$ cm.
Equal forces of 96 N act along $A B$ and $C D$ (opposite directions).
Find the equal forces acting at $A$ and $C$ parallel to $B D$ whose couple is equivalent to this one.

Solution:

First couple: arm $= B C = A D = 16$ cm, moment $96 \times 16 = 1536$ N·cm (clockwise).
$B D = \sqrt{1 2^{2} + 1 6^{2}} = 20$ cm.
The distance of $A$ from $B D$ is $\frac{12 \times 16}{20} = 9.6$ cm, and the same for $C$ on the other side, so the arm of the second couple is $2 \times 9.6 = 19.2$ cm.
Equal moments: $F \times 19.2 = 1536$ , so $F = 80$ N, acting clockwise.

05 3-2 (a) Systems equivalent to a couple

Two conditions

A system of coplanar forces is equivalent to a couple if both hold:
the resultant force is zero (the sum of components along any direction is zero);
the sum of the moments about a point is not zero.
If the first condition holds but the moments also sum to zero, the system is in equilibrium, not a couple.
If concurrent forces have a zero resultant, they are in equilibrium and cannot be a couple.

✍️ Worked examples

Example 10:

A light rod $A B$ carries a 15 N force at $A$ and opposite forces of 8 N at $C$ ( $A C = 5$ cm) and 7 N at $B$ ( $A B = 14$ cm).
Show that the system is equivalent to a couple and find its moment.

Solution:

Take the 15 N direction as positive: $R = 15 - 8 - 7 = 0$ .
Moments about $A$ : $- 8 \times 5 - 7 \times 14 = - 40 - 98 = - 138$ N·cm.
The resultant is zero but the moment is not, so it is equivalent to a clockwise couple of moment $- 138$ N·cm.
Check about $B$ : $- 15 \times 14 + 8 \times 9 = - 138$ , the same value, as it must be for a couple.

06 3-2 (b) Forces represented by the sides of a triangle

Rule

If three coplanar forces acting on a rigid body are completely represented (in magnitude, direction and line of action) by the sides $\vec{A B}$ , $\vec{B C}$ , $\vec{C A}$ of a triangle taken in one cyclic order, the system is equivalent to a couple.
The magnitude of its moment is $M = 2 \times (areaofthetriangle) \times m$ where $m$ is the force that represents one unit of length.

Why

The vector sum $\vec{A B} + \vec{B C} + \vec{C A} = \vec{0}$ , so the resultant force is zero.
Replace ${\vec{F}}_{1}$ at $A B$ and ${\vec{F}}_{2}$ at $B C$ by their resultant $- {\vec{F}}_{3}$ at $B$ ; together with ${\vec{F}}_{3}$ at $C A$ they form a couple.
Its arm is the altitude $B D$ from $B$ to $A C$ , so $M = F_{3} \times B D = (m \cdot A C) B D = m \times 2 area$ .

✍️ Worked examples

Example 12:

$A B C$ is isosceles with $A B = B C = 17$ cm and $A C = 16$ cm.
Forces of 340, 340 and 320 N act along $A B$ , $B C$ , $C A$ .
Show the system is a couple and find its moment.

Solution:

The ratios $\frac{340}{17} = \frac{340}{17} = \frac{320}{16} = 20$ are equal, so the forces are represented by the sides at $m = 20$ N per cm.
They act in one cyclic order, so the system is equivalent to a couple.
Altitude from $B$ : $\sqrt{1 7^{2} - 8^{2}} = 15$ cm; area $= \frac{1}{2} \times 16 \times 15 = 120$ cm $^{2}$ .
$M = 2 \times 120 \times 20 = 4800$ N·cm.

Generalisation to polygons

If a system of coplanar forces is completely represented by the sides of a closed polygon taken in one cyclic order, the system is equivalent to a couple of moment $2 \times (areaofthepolygon) \times m$ .

✍️ Worked examples

Example 14: A square $A B C D$ of side 10 cm has forces of 10 N along $A B$ , $B C$ , $C D$ and $D A$ (in that cyclic order). Find the moment.

Solution:

$m = \frac{10}{10} = 1$ N per cm and the area is $100$ cm $^{2}$ .
$M = 2 \times 100 \times 1 = 200$ N·cm.
Check by moments about $A$ : the forces along $A B$ and $D A$ pass through $A$ (zero moment); the force along $B C$ has arm 10 and the force along $C D$ has arm 10, both in the same sense: $100 + 100 = 200$ .

07 3-2 (c) The three-point test

Rule

If the sum of the moments of a system of coplanar forces about three non-collinear points in its plane has the same non-zero value $k$ , the system is equivalent to a couple of moment $k$ .

Why

A coplanar system reduces to a single force, a single couple, or equilibrium.
It is not in equilibrium (the moments are non-zero).
If it reduced to a single force $F$ at distances $L_{1}, L_{2}, L_{3}$ from the three points, then $F L_{1} = F L_{2} = F L_{3}$ , so the three points would lie on a line parallel to the force; they do not. So it must be a couple.

✍️ Worked examples

Example 16: A square $A B C D$ of side 10 cm is acted on by the forces in Example 14. Use the test to confirm that it is a couple.

Solution:

Moments about $A$ : $+ 200$ (computed above).
Moments about $B$ : the forces along $A B$ and $B C$ pass through $B$ ; $C D$ has arm 10 and $D A$ has arm 10, both in the same sense: $+ 200$ .
Moments about $C$ : similarly $+ 200$ .
Equal non-zero values at three non-collinear points, so the system is a couple of moment $200$ N·cm.

08 3-2 (d) The resultant couple

Definition

The sum of two or more coplanar couples is a single couple whose moment is the algebraic sum of the moments: $M = M_{1} + M_{2} + \dots + M_{n}$
It is called the resultant couple. If $M = 0$ the couples are in equilibrium.

✍️ Worked examples

Example 17:

Three couples act on a plate: 200 N forces with arm 0.8 m (anticlockwise), 500 N forces with arm 2.3 m (clockwise), and 100 N forces with arm 4 cm (clockwise).
Find the resultant couple.

Solution:

$M_{1} = + 200 \times 0.8 = 160$ N·m.
$M_{2} = - 500 \times 2.3 = - 1150$ N·m.
$M_{3} = - 100 \times 0.04 = - 4$ N·m.
$M = 160 - 1150 - 4 = - 994$ N·m: a clockwise couple of magnitude 994 N·m.

09 3-2 (e) Summary

Comparison

Point of comparison	Single force	Couple
Resultant force	the force itself	zero
Effect	translates (and may rotate)	pure rotation
Moment	depends on the point chosen	the same about every point
Can it be balanced?	by an equal opposite force on the same line	only by another couple of opposite moment

Key points

A couple is two equal, opposite, non-collinear parallel forces; its moment is $F \times L$ .
A couple's moment is a constant vector, so its position in the plane does not matter.
Couples are equivalent when their moments are equal, in equilibrium when their moments sum to zero, and combine by adding moments.
A system is equivalent to a couple if its resultant force is zero while its moment is not zero.

Couples and the Resultant Couple

🧭 Lesson points — pick a point and study it with its question bank

🌳 Concept tree

01 3-1 (a) What is a couple?

From the previous lesson

Definition

Experiment

02 3-1 (b) The moment of a couple

Definition

The moment is a constant vector

Observations

03 3-1 (c) Equilibrium under coplanar couples

Condition

04 3-1 (d) Equivalent couples

Definition

05 3-2 (a) Systems equivalent to a couple

Two conditions

06 3-2 (b) Forces represented by the sides of a triangle

Rule

Why

Generalisation to polygons

07 3-2 (c) The three-point test

Rule

Why

08 3-2 (d) The resultant couple

Definition

09 3-2 (e) Summary

Comparison

Key points

Comprehensive Curriculum Encyclopedia & Core Terminology — Couples and the Resultant Couple

📌 Comprehensive Scientific Terminology & Definitions:

⚖️ Governing Scientific Laws, Principles & Mathematical Formulas:

🔍 In-Depth Explanations, Deductions & Core Concepts:

📊 Core Criteria Distinguishing Single Resultant, Couple, and Complete Equilibrium:

💡 High-Yield Exam Traps & Key Takeaways: