Forces, Density and Pressure

This topic introduces fundamental principles governing the mechanical behaviour of objects, both static and in fluids. We will explore how forces can cause rotation, the conditions required for an object to be perfectly still, and the concepts of density and pressure.

### The Turning Effect of a Force: Moments

A force can cause an object to accelerate linearly, but it can also cause it to rotate. The turning effect of a force is called a moment.

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

Formula:

Moment = Force × Perpendicular distance from the pivot

M = F × d

The SI unit for a moment is the newton-metre (N m). Moments can be clockwise or anticlockwise. A related concept is a couple, which consists of two parallel forces that are equal in magnitude but opposite in direction, acting on different lines. A couple produces a pure turning effect without causing any linear acceleration. The moment of a couple is called torque (τ) and is calculated by multiplying one of the forces by the perpendicular distance between the forces.

### The Principle of Moments and Equilibrium

For an object to be in rotational equilibrium (i.e., not rotating or rotating at a constant angular velocity), the total turning effect must be zero. This leads to the Principle of Moments.

Principle of Moments: For an object to be in rotational equilibrium, the sum of the clockwise moments about any point must be equal to the sum of the anticlockwise moments about the same point.

Σ Clockwise Moments = Σ Anticlockwise Moments

This principle is fundamental to understanding structures like bridges, levers, and see-saws. For an object to be in total equilibrium, two conditions must be met:

### Centre of Gravity

The centre of gravity (CG) of an object is defined as the single point through which the entire weight of the object can be considered to act. For a uniform object, like a ruler, the CG is at its geometric centre. The position of the CG is crucial for determining an object's stability. An object is stable as long as a vertical line drawn downwards from its centre of gravity falls within its base of support.

### Density

Density (ρ) is an intrinsic property of a substance that describes the concentration of mass. It is defined as the mass per unit volume.

Formula:

Density (ρ) = Mass (m) / Volume (V)

The SI unit for density is kilograms per cubic metre (kg m⁻³). To determine the density of a substance, you must measure its mass (using a balance) and its volume. For a regularly shaped solid, volume can be calculated using geometric formulas. For an irregularly shaped solid, volume can be found by the displacement method using a measuring cylinder or eureka can.

### Pressure

Pressure (p) is defined as the force acting normally (perpendicularly) per unit area.

Formula:

Pressure (p) = Normal Force (F) / Area (A)

The SI unit for pressure is the Pascal (Pa), which is equivalent to one newton per square metre (N m⁻²). Pressure is a scalar quantity. The concept explains why a sharp knife cuts better than a blunt one – the same force is applied over a much smaller area, resulting in a very high pressure.

### Pressure in Fluids

Fluids (liquids and gases) exert pressure on the surfaces of any object immersed in them. The pressure within a fluid increases with depth. Consider a column of fluid of height h, cross-sectional area A, and density ρ.

The weight of this column of fluid is W = mg. Since m = ρV and V = Ah, the weight is W = (ρAh)g.

This weight is the force exerted on the area A at the bottom of the column. Therefore, the pressure is:

p = F/A = (ρAhg)/A

This simplifies to the formula for pressure in a fluid:

p = hρg

This equation shows that pressure at a certain depth in a fluid depends only on the depth (h), the density of the fluid (ρ), and the acceleration of free fall (g). This is why dams are built much thicker at the base. An important consequence is upthrust. When an object is submerged in a fluid, the pressure on its bottom surface is greater than the pressure on its top surface. This pressure difference results in a net upward force called upthrust or buoyant force, as described by Archimedes' Principle.