Forces, density and pressure

Welcome to the World of Forces, Density, and Pressure!

In this chapter, we are going to explore why things balance, why some objects spin while others stay still, and why heavy ships can float on water while a tiny pebble sinks. These concepts are the "rules of the game" for engineering and everyday life. Don't worry if some of the math looks new—we’ll break it down step-by-step!

4.1 Turning Effects of Forces

Up until now, you’ve mostly learned about forces pushing or pulling objects in a straight line. But what happens when a force makes something rotate? That is what we call a turning effect.

Centre of Gravity

Every object behaves as if all its weight is concentrated at a single point. This point is called the centre of gravity.
Analogy: If you try to balance a ruler on your finger, the spot where it stays perfectly level is directly under its centre of gravity.

Moment of a Force

The moment is just a fancy Physics word for the "turning effect" of a force. It depends on two things: how hard you push and how far you are from the pivot.

The formula is: \( \text{Moment} = \text{Force} \times \text{perpendicular distance from the pivot} \)

Important: The distance must be perpendicular (at a 90-degree angle) to the line of action of the force. If you push a door right next to the hinges, it’s hard to open. If you push at the handle (far from the hinges), it’s easy! That is the power of a moment.

Couples and Torque

Sometimes, we use two forces to make something spin without moving it from its spot. This is called a couple.

A couple is a pair of forces that are:
1. Equal in magnitude.
2. Parallel to each other but acting in opposite directions.
3. Separated by a distance.

The turning effect of a couple is called torque.
\( \text{Torque of a couple} = \text{One of the forces} \times \text{perpendicular distance between the forces} \)

Example: Turning a steering wheel with both hands. One hand pulls up, the other pulls down. They work together to produce rotation only.

Quick Review:
- Moment: Turning effect of a single force.
- Torque: Turning effect of a couple (two forces).
- Centre of Gravity: The point where weight acts.

4.2 Equilibrium of Forces

When an object is "in equilibrium," it means it is perfectly balanced. It isn't moving in any direction, and it isn't spinning faster or slower.

The Two Rules for Equilibrium

For an object to be in total equilibrium, it must satisfy two conditions:
1. No Resultant Force: The sum of all forces acting on it must be zero (e.g., all the "up" forces must equal the "down" forces).
2. No Resultant Torque: The sum of all the moments must be zero.

The Principle of Moments

For an object in equilibrium, the sum of the clockwise moments about any point is equal to the sum of the anticlockwise moments about that same point.

Step-by-Step for Problems:
1. Identify the pivot point.
2. List all forces that want to turn the object clockwise.
3. List all forces that want to turn the object anticlockwise.
4. Set them equal: \( \text{Sum of Clockwise Moments} = \text{Sum of Anticlockwise Moments} \)

Vector Triangles

If three forces are acting on an object and it is in equilibrium, you can draw these forces tip-to-tail to form a closed triangle. If the triangle closes perfectly, the resultant force is zero!

Common Mistake: Forgetting that weight acts from the centre of gravity. When doing moment calculations for a beam, always draw the weight of the beam acting right in the middle (unless told otherwise).

4.3 Density and Pressure

Now we look at how mass is spread out through a volume, and how force is spread out over an area.

Density

Density is the mass per unit volume. It tells us how "tightly packed" the particles in a material are.
\( \rho = \frac{m}{V} \)
Where \( \rho \) (rho) is density in \( kg \, m^{-3} \), \( m \) is mass, and \( V \) is volume.

Pressure

Pressure is the normal force acting per unit area.
\( p = \frac{F}{A} \)
The unit is the Pascal (\( Pa \)), which is the same as \( N \, m^{-2} \).
Analogy: Why do snowshoes help you walk on deep snow? They increase your area, which decreases the pressure you put on the snow, so you don't sink!

Pressure in a Fluid (Hydrostatic Pressure)

The deeper you go in a liquid (or gas), the higher the pressure becomes because there is more "stuff" weighing down on you from above.

The Derivation (You should know this!):
1. Consider a column of liquid with height \( h \) and cross-sectional area \( A \).
2. The Volume \( V = A \times h \).
3. The Mass \( m = \text{density} \times V = \rho A h \).
4. The Weight (Force) \( F = m \times g = \rho A h g \).
5. Pressure \( p = \frac{F}{A} = \frac{\rho A h g}{A} \).
6. So, the change in pressure is: \( \Delta p = \rho g \Delta h \)

Upthrust and Archimedes’ Principle

Have you ever noticed that you feel lighter when you are in a swimming pool? This is because of upthrust.

Why does upthrust happen?
Pressure increases with depth. This means the water pressure pushing up on the bottom of an object is greater than the water pressure pushing down on the top of it. This difference in pressure creates an upward force.

Archimedes’ Principle:
The upthrust acting on an object in a fluid is equal to the weight of the fluid that the object has displaced.
The formula is: \( F = \rho g V \)
(Where \( \rho \) is the density of the fluid, and \( V \) is the volume of the submerged part of the object).

Did you know?
A huge steel ship floats because it is hollow. It displaces a massive volume of water, and the weight of that displaced water (the upthrust) is equal to the heavy weight of the ship!

Key Takeaways:
- Density: \( \text{Mass} / \text{Volume} \).
- Pressure: \( \text{Force} / \text{Area} \).
- Liquid Pressure: Increases with depth (\( \rho g \Delta h \)).
- Upthrust: Caused by pressure differences; equals the weight of the displaced fluid.