Solid Pressure and Force-Area Relationships

Pressure is the force acting perpendicular to a surface per unit area, p = F ÷ A, measured in pascals (Pa), where 1 Pa = 1 N/m². In brief: for the same force, a smaller area gives a larger pressure. This is why knives are sharp, nails are pointed and snowshoes are wide. To reduce pressure, spread the force over a larger area; to increase it, concentrate the force on a smaller area.

A person can lie on a bed of nails without injury, but stepping on a single nail is painful, although the force is the same. This guide defines pressure and explains how force and area affect it, gives applications, compares examples in a table, works through calculations including a person standing and a brick and corrects common mistakes.

What is pressure?

Pressure p = F ÷ A, where F is the force normal (perpendicular) to the surface and A is the area over which it acts. The SI unit is the pascal (Pa): 1 Pa = 1 N/m². This is a small unit (a banknote resting on a table exerts about 1 Pa), so kilopascals (kPa) and megapascals (MPa) are often used. Other units are the bar (100,000 Pa), the atmosphere (101,325 Pa) and psi. Pressure is a scalar, although the force that causes it is a vector. For a solid resting on a surface, the force is usually its weight, F = mg.

How does area change pressure?

Since p = F ÷ A, pressure is proportional to force and inversely proportional to area. If the force is constant, halving the area doubles the pressure. A large pressure can break a surface (sharp knife, needle, nail), and a small pressure avoids sinking in (skis, snowshoes, tracks of a tractor, wide tires). The pressure under a solid is calculated by dividing the weight by the area of contact.

Where do we use these ideas?

• High pressure by small area: knives, axes, nails, drawing pins, needles, high heels, the studs on football boots.
• Low pressure by large area: skis, snowshoes, camel feet, caterpillar tracks, foundations of buildings, wide straps on bags.
• Engineering: the foundations of a building spread its weight over the ground so that the pressure is less than what the soil can support.

Pressure in everyday cases

Worked example: pressure calculations

1. Person standing. A person of mass 70 kg has a weight of 70 × 9.8 = 686 N. The area of the soles is 0.040 m². p = 686 ÷ 0.040 = 17,150 Pa = 17 kPa.
2. High heel. If all the weight is carried by a heel of area 4.0 cm² = 4.0 × 10⁻⁴ m²: p = 686 ÷ 4.0 × 10⁻⁴ = 1.7 × 10⁶ Pa = 1.7 MPa, which is 100 times larger than when standing on both feet.
3. Brick on different faces. A brick of mass 2.0 kg has dimensions 20 cm × 10 cm × 5.0 cm. Its weight is 19.6 N. On the largest face (0.020 m²), p = 19.6 ÷ 0.020 = 980 Pa. On the middle face (0.010 m²), p = 1,960 Pa. On the smallest face (0.0050 m²), p = 3,920 Pa.
4. Car on its tires. A car of mass 1,080 kg has a weight of 1,080 × 9.8 = 10,584 N. If each of the four tires touches the road over a patch of 0.012 m², the total contact area is 0.048 m² and p = 10,584 ÷ 0.048 = 220,500 Pa, about 220 kPa. This is close to the typical pressure of air in a car tire, because the tire pressure is what supports the load.

How was pressure named?

The unit pascal was named after Blaise Pascal, who studied pressure in fluids in the 1640s and 1650s, and it was adopted as the SI unit in 1971. The idea of pressure in solids is used in engineering, since the 1800s, for designing foundations, railways and tools.

Which mistakes are common?

Mistake 1: Confusing pressure with force

Pressure is force per area, in Pa. Force is in N.

Mistake 2: Using the total surface area

Use only the area in contact, perpendicular to the force.

Mistake 3: Forgetting to convert cm² to m²

1 cm² = 10⁻⁴ m².

Mistake 4: Thinking that a heavier object always exerts more pressure

The area matters. A wide heavy object can exert less pressure than a light narrow one.