Physics (9702) | Equilibrium of forces

Welcome to Equilibrium: How Things Stay Still!

Hello future Physicists! This chapter is all about balance. We're going to learn the rules that govern why a suspension bridge doesn't fall down, why a crane doesn't tip over, and why you can balance a ruler on your finger.

Understanding Equilibrium is fundamental because it moves beyond just linear motion (kinematics) and helps us analyze real-world structures that are designed to remain perfectly still. Don't worry if moments and torque seem confusing—we will break them down using familiar, everyday examples!

4.1 Turning Effects of Forces: Setting the Stage

4.1.1 Centre of Gravity (CG)

Before anything can balance, we need to know where its weight acts.

• The Centre of Gravity (CG) is the single point through which the entire weight of an object appears to act, regardless of the object's orientation.
• Analogy: When you balance a broomstick on your finger, you aren't fighting all the tiny forces across the stick; you are just counteracting the single, effective force of weight acting downwards at the CG.

4.1.2 Defining the Moment of a Force

A moment is the measure of the turning effect of a force. If you want something to rotate, you need a moment!

Definition: The moment of a force about a pivot (or point) is the product of the force and the perpendicular distance from the pivot to the line of action of the force.

Mathematically:
\[ \text{Moment} = \text{Force} \times \text{Perpendicular distance} \] \[ M = F \times d_{\perp} \]

The SI unit for the moment is the Newton metre (Nm).

Real-World Tip: Opening a Door
Why are door handles placed far away from the hinges? Because the hinges are the pivot! By maximizing the perpendicular distance \(d_{\perp}\), you need to apply a much smaller force (\(F\)) to create the necessary turning effect (\(M\)).

Key Takeaway: The longer the lever arm, the easier it is to turn.

4.1.3 Couples and Torque

Sometimes, rotation is caused not by one force, but by two equal and opposite forces acting on an object, separated by a distance. This is called a Couple.

• A couple is a pair of forces that are equal in magnitude, opposite in direction, and whose lines of action are parallel but do not coincide.
• Crucially, a couple produces rotation only; it does not cause any overall translational motion (it has zero resultant force).

The turning effect caused by a couple is called the Torque (\(\tau\)).

Definition: The torque of a couple is the product of one of the forces and the perpendicular distance between the lines of action of the two forces.

If \(F\) is the magnitude of one force and \(d\) is the perpendicular distance between them:
\[ \tau = F \times d \]

Analogy: Steering a Car
When you turn a steering wheel, your hands apply an equal and opposite force on opposite sides of the wheel. This is a classic example of a couple producing torque.

4.2 Equilibrium of Forces: The Conditions for Stillness

For a system to be in complete Equilibrium, it must satisfy two conditions simultaneously. This means the object is neither speeding up nor slowing down (no change in linear velocity) AND it is not changing its rate of rotation (no change in angular velocity). For most problems, this means the object is stationary.

4.2.1 Condition 1: Translational Equilibrium (No Resultant Force)

For an object to have no tendency to move linearly (side-to-side or up-and-down), the resultant force acting on it must be zero.

In simple terms: All forces must balance out.

• The sum of forces acting in one direction must equal the sum of forces acting in the opposite direction.
• If we consider horizontal (\(x\)) and vertical (\(y\)) directions:
  \[ \sum F_x = 0 \] \[ \sum F_y = 0 \]

4.2.2 Condition 2: Rotational Equilibrium (Principle of Moments)

For an object to have no tendency to rotate, the resultant turning effect (moment or torque) must be zero.

The Principle of Moments:

For a body to be in rotational equilibrium, the sum of the clockwise moments about any point must be equal to the sum of the anti-clockwise moments about that same point.
\[ \sum \text{Clockwise Moments} = \sum \text{Anti-clockwise Moments} \]

Applying the Principle of Moments (Solving Problems)

When solving problems involving beams, see-saws, or levers in equilibrium, follow these simple steps:

1. Draw a clear diagram: Mark all forces (including weight/CG) and distances.
2. Choose a Pivot: You can choose any point on the object as the pivot, but it's usually easiest to choose a point where an unknown force acts.
   Why? If you choose the pivot where an unknown force \(F_{\text{unknown}}\) acts, the perpendicular distance to \(F_{\text{unknown}}\) is zero, so the moment caused by that force is zero. This simplifies the calculation greatly!
3. Calculate Moments: Identify which forces cause clockwise rotation and which cause anti-clockwise rotation about your chosen pivot.
4. Apply the Principle: Set the sum of CW moments equal to the sum of ACW moments.
5. Solve: Find your unknown force or distance.

4.2.3 Representing Forces in Equilibrium using Vector Triangles

When three coplanar forces (forces acting in the same plane) keep an object in translational equilibrium (\(\sum F = 0\)), we can represent this condition visually using a closed vector shape.

Forces in Translational Equilibrium

Remember Condition 1: the vector sum of all forces must be zero.

Analogy: A Trip Around the Block
Imagine forces as displacement vectors. If you start at point A and follow the direction and magnitude of all forces (Force 1, then Force 2, then Force 3), and you end up exactly back at point A, the resultant (net) displacement is zero. The same is true for forces!

Using the Vector Triangle

If an object is held in equilibrium by exactly three forces, and these forces are coplanar, then these three vectors, when drawn head-to-tail, must form a closed triangle.

• Step 1 (Scale Drawing): Choose a suitable scale (e.g., 1 cm = 1 N).
• Step 2 (Draw): Draw the first known force vector.
• Step 3 (Connect): From the head of the first vector, draw the tail of the second known force vector, ensuring the correct direction relative to the first.
• Step 4 (Close the Loop): The third force vector must start at the head of the second vector and finish exactly at the tail of the first vector (closing the triangle).

This technique is vital for determining the magnitude and direction of an unknown force graphically. If the vectors form a closed loop, you have confirmed that the resultant force is zero, and Condition 1 for equilibrium is satisfied.

Key Takeaway

Complete equilibrium requires no linear acceleration and no angular acceleration. When dealing with three forces, the simplest way to check linear equilibrium is to see if the forces form a closed vector triangle.