Pressure from a solid object and Pascal's law

Why does someone in boots sink into snow much more than someone on skis? Their mass is almost unchanged. The result must depend not only on the total force on the snow but also on the area over which it is spread.

Imagine many equal small patches of snow beneath your feet. A wide ski spreads the load over more patches, so each receives less on average. The pressing force F is perpendicular to the surface, and the contact area S is measured in m². The distribution of this perpendicular force determines average pressure. A force pulling the ski along the surface has a different effect and cannot be substituted for F here.

To compare supports of different sizes, find perpendicular force per unit area: p = F/S. This is average pressure; if the force is uniformly distributed, pressure is the same across the surface. The unit is the pascal: 1 Па = 1 Н/м². Doubling force at fixed area doubles pressure; doubling area at fixed force halves it. This is why a narrow blade cuts more easily and a wide support makes a smaller indentation. The formula itself does not describe the material's deformation.

Area conversion can easily cause a hundredfold error. A square of side 1 cm has a side of 0.01 m, so 1 см² = 0,01·0,01 м² = 0,0001 м². Square the conversion factor for length. Imagine a square metre divided into centimetre squares: 100 rows of 100 make 10 000. Dividing an area in cm² by only 100 makes its value in m² a hundred times too large and the calculated pressure a hundred times too small.

Gas presses even on the top wall of a container, although its molecules are not resting there as a load. They move continuously and strike the walls from many directions; countless impacts produce measurable pressure. Compress the same amount of gas at constant temperature: less space is available, so impacts per unit wall area become more frequent and pressure rises. Heating at fixed volume makes the impacts more intense on average, also raising pressure.

Push a piston closing a liquid-filled container. If the added pressure stayed only beneath it, the liquid would redistribute. Once equilibrium is reached, the additional pressure is transmitted unchanged in every direction. This is Pascal's law for a liquid or gas in a closed container. The same added pressure produces a larger extra force over a larger area: F = pS. Equal pressure does not mean equal forces on pistons of different areas.

Does Pascal's law conflict with pressure increasing with depth? It concerns the increase due to an external action. The liquid's own weight already produces different pressures at different depths. Add the same amount to two different numbers: their difference remains. Likewise, pushing the piston raises pressure equally throughout a resting liquid without removing the original dependence on depth. We will derive that dependence next lesson.

A perpendicular force of 200 N is distributed uniformly over 0.02 m². What is the pressure, and what will it be if the area doubles?

Use p = F/S: 200/0,02 = 10 000 Па = 10 кПа.

With the same force over 0.04 m², we obtain 5000 Па = 5 кПа. Each square metre carries half as much force.

10 кПа; after doubling the area, 5 кПа.

Pressure compares force with area. The same force can have very different effects on each patch of a surface.

In p = F/S, use perpendicular force and area in compatible units; 1 Па = 1 Н/м².

Pascal's law describes transmission of added pressure. It produces different forces on areas of different sizes.

Next we will include the liquid's own weight and find why pressure is greater at greater depth.