Mathematics (9709) | Forces and equilibrium

📝 Forces and Equilibrium: Study Notes for Paper 4 (Mechanics)

Hello! Welcome to the exciting world of Mechanics. This chapter, Forces and Equilibrium, is fundamental to Paper 4. It teaches us how to analyze objects that are either standing perfectly still or moving smoothly at a constant speed. This might sound simple, but mastering how to break forces down and balance them is the most important skill you'll need for this entire paper!

1. Understanding Forces (The Basics)

A force is simply a push or a pull. In Mechanics, we treat all objects as particles, meaning we assume the entire mass is concentrated at a single point, and all forces act through that point.

Key Concept: Force is a Vector

Forces are vector quantities. This means they have both magnitude (size) and direction. You cannot add forces arithmetically; you must use vector addition (which usually involves trigonometry).

The Golden Rule: Start with a Force Diagram
Always, always, always draw a clear diagram showing all the forces acting on the particle. This is often called a Free Body Diagram.

• Weight (\(W\) or \(mg\)): Always acts vertically downwards. \(W = mg\), where \(g \approx 10 \, \text{m s}^{-2}\) (in Cambridge exams).
• Tension (\(T\)): The force exerted by a stretched string or cable, acting away from the particle along the line of the string.
• Normal Reaction (\(R\)): The contact force exerted by a surface, acting perpendicular (normal) to the surface.
• Friction (\(F\)): The component of the contact force acting parallel to the surface, opposing motion or the tendency to move.
• Thrust/Compression: Force exerted by a rigid rod, acting towards the particle along the rod.

Did you know? The term ‘Normal’ in Normal Reaction doesn't mean ordinary; it’s a mathematical term meaning perpendicular.

2. Components and Resultants

To deal with multiple forces pointing in different directions, we must resolve them. This means breaking each force down into two perpendicular components. We usually choose the axes (directions) that make the calculation easiest, typically horizontal and vertical, or parallel and perpendicular to an inclined plane.

How to Resolve a Force (F)

If a force \(F\) acts at an angle \(\theta\) to the horizontal (or your chosen axis):

Component parallel to the axis: \(F \cos \theta\)

Component perpendicular to the axis: \(F \sin \theta\)

Memory Aid: Think of 'cos' as being 'next to' the angle you are using, and 'sin' as being 'away' from it.

Finding the Resultant Force

The resultant force (\(R\)) is the single force that represents the combined effect of all the forces acting on the particle.

Step-by-Step Process:

1. Choose your two perpendicular directions (e.g., Horizontal and Vertical).
2. Resolve every force into components along these two axes.
3. Find the total resultant in the first direction (\(\sum F_x\)). (Sum of all 'right' forces minus sum of all 'left' forces).
4. Find the total resultant in the second direction (\(\sum F_y\)). (Sum of all 'up' forces minus sum of all 'down' forces).

The magnitude of the resultant force \(R\) is found using Pythagoras:
$$R = \sqrt{(\sum F_x)^2 + (\sum F_y)^2}$$

The direction angle \(\alpha\) (relative to the x-axis) is found using trigonometry:
$$\tan \alpha = \frac{|\sum F_y|}{|\sum F_x|}$$

3. The Principle of Equilibrium

Equilibrium is the core idea of this chapter. A particle is in equilibrium if it has zero acceleration.

This can happen in two scenarios:

1. The particle is at rest (static equilibrium).
2. The particle is moving at a constant velocity (dynamic equilibrium).

The Equilibrium Rule

If a particle is in equilibrium, the vector sum of all forces acting on it must be zero.

In component form, this means:

• Sum of forces in the x-direction is zero: $$\sum F_x = 0$$
• Sum of forces in the y-direction is zero: $$\sum F_y = 0$$

Don't worry if this seems tricky at first! This principle is simply a balance act. Imagine a tug-of-war where nobody moves—the pull to the left equals the pull to the right, and the upward lift equals the downward pull.

Alternative Methods (When Suitable)

While resolving forces is the standard method, if only three forces are acting on a particle in equilibrium, you can use geometrical methods:

• Triangle of Forces: Since the vector sum is zero, the three forces form a closed triangle when drawn head-to-tail.
• Lami's Theorem: If forces \(P, Q, R\) are in equilibrium, and \(\alpha, \beta, \gamma\) are the angles opposite to \(P, Q, R\) respectively, then: $$\frac{P}{\sin \alpha} = \frac{Q}{\sin \beta} = \frac{R}{\sin \gamma}$$

Note: You are expected to be able to solve problems by resolving. You do not need to memorise Lami's theorem or be referred to it in questions, but knowing the Triangle of Forces concept can confirm your understanding.

4. Contact Forces and Friction

When surfaces touch, the force they exert on each other (the Contact Force) is generally split into two parts: the Normal Reaction (\(R\)) and the Frictional Force (\(F\)).

Smooth vs. Rough Contacts

1. Smooth Contact

This is a simplification (a model). A smooth surface means there is no friction (\(F = 0\)). The only force exerted by the surface is the Normal Reaction (\(R\)), acting strictly perpendicular to the surface.

2. Rough Contact (Friction is present)

Friction always acts parallel to the surface and opposes the direction of motion (or potential motion).

Limiting Equilibrium

When a particle rests on a rough surface, the friction force \(F\) only acts if there is another force trying to move the particle.

• If the particle is stationary, \(F\) is just large enough to balance the driving force.
• The maximum value friction can reach is called Limiting Friction (\(F_{MAX}\)).

When the particle is 'about to slip', it is in limiting equilibrium. At this precise moment, the magnitude of the frictional force is related to the Normal Reaction by the Coefficient of Friction, \(\mu\).

The Friction Law:
$$F_{MAX} = \mu R$$

Since friction can never exceed this maximum value, the general rule is:
$$F \le \mu R$$

• If the particle is stationary/in equilibrium (not about to slip): $F$ is calculated using \(\sum F_{parallel} = 0\), and \(F < \mu R\).
• If the particle is in limiting equilibrium (just about to slip): \(F = \mu R\).

5. Newton's Third Law (N3L)

While N2L (\(F=ma\)) is covered in a later section, N3L is required here, often in the context of connected particles or interactions with surfaces.

Newton's Third Law (N3L) states that every action has an equal and opposite reaction.

If body A exerts a force on body B, then body B exerts a force of equal magnitude but opposite direction on body A.

Example:
When a particle rests on the ground:

1. The Earth pulls the particle down with force \(W\) (Weight).
2. The particle pulls the Earth up with force \(W\). (This is an N3L pair).

3. The ground pushes the particle up with force \(R\) (Normal Reaction).
4. The particle pushes the ground down with force \(R\). (This is a separate N3L pair).

It is crucial to remember that N3L pairs always act on different bodies. When solving equilibrium problems, we only consider the forces acting on the specific particle in question.