Pressure and hydrostatic pressure

A crate of books pushes against the floor with a force of $200\ \text{N}$ distributed over a base of $0.5\ \text{m}^2$, as shown in the figure.

Calculate the pressure it exerts on the floor, in pascals, using $p = \frac{F}{S}$.

In what unit is pressure measured in the International System of Units? Write its equivalence in units of force and surface area, and state after which scientist it is named.

A workshop punch pushes against a metal sheet with a force of $20\ \text{N}$ and its tip has a surface area of $4\ \text{mm}^2$.

1. a) Express the surface area of the tip in $\text{m}^2$.
2. b) Calculate the pressure exerted by the punch on the metal sheet.

A hiker walks through freshly fallen snow in boots and sinks up to her ankle; the next day she goes through the same place on skis and leaves hardly a footprint, as shown in the figure.

If her weight is the same in both cases, explain using the concept of pressure why she does not sink when wearing skis.

A skier of $60\ \text{kg}$ is standing on the snow. Take $g = 10\ \text{m/s}^2$.

1. a) Calculate her weight.
2. b) Wearing boots, she rests on two soles of $25\ \text{cm} \times 10\ \text{cm}$ each. Calculate the pressure she exerts on the snow (take care with unit conversions).
3. c) Wearing skis, she rests on two skis of $1.5\ \text{m} \times 0.2\ \text{m}$ each. Calculate the pressure now.
4. d) Compare the two results and explain why she does not sink when wearing skis.

Compare the following two situations, shown in the figure. Take $g = 10\ \text{m/s}^2$.

1. a) A $55\ \text{kg}$ person rests all their weight on two stiletto heels of $1\ \text{cm}^2$ each: calculate the pressure exerted on the ground (take care with unit conversions).
2. b) A $5\,000\ \text{kg}$ elephant rests on its four feet, of $0.1\ \text{m}^2$ each: calculate the pressure it exerts.
3. c) Who exerts more pressure on the ground? Compare the results and explain the role played by the contact area.

A swimming pool is $2\ \text{m}$ deep, as shown in the figure. Given: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 10\ \text{m/s}^2$.

Calculate the pressure exerted by the water on the bottom using $p = d\times g\times h$.

When swimming down to the bottom of a pool, you feel pressure in your ears, and the deeper you go, the more uncomfortable it becomes. Explain which quantity is changing with depth and why.

The water tank of a housing estate is at the top of a tower, $12\ \text{m}$ above the tap of a house, as shown in the figure. Given: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 9.8\ \text{m/s}^2$.

1. a) Calculate the pressure due to the column of water reaching the tap.
2. b) What will that pressure be at the tap on the ground floor, located $3\ \text{m}$ lower down?

A dam holds back water in a reservoir that reaches a depth of $30\ \text{m}$ next to the wall, as shown in the figure. Data: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 10\ \text{m/s}^2$.

1. a) What pressure does the water exert on the base of the dam?
2. b) A small sluice gate at the bottom has a surface area of $2\ \text{m}^2$; assume that the whole gate is subject to the pressure found in part a). With what force does the water push against that sluice gate?

To what depth must you descend in the water of a swimming pool for the pressure due to the water to be $49\,000\ \text{Pa}$? First rearrange for depth in the expression $p = d\times g\times h$. Data: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 9.8\ \text{m/s}^2$.

A dam holds back water in a reservoir with a maximum depth of $40\ \text{m}$ next to the wall, as shown in the figure. Data: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 10\ \text{m/s}^2$.

1. a) What pressure does the wall withstand at its base?
2. b) What about at a depth of $10\ \text{m}$?
3. c) With those two results, explain why dams are built much thicker at their base than at the top.

A building's water tank is on the roof, $15\ \text{m}$ above a tap on the second floor. Data: $d_{agua} = 1\,000\ \text{kg/m}^3$; take $g = 9.8\ \text{m/s}^2$.

1. a) Calculate the water pressure at that tap.
2. b) When closed, the tap has a cross-sectional area of $1\ \text{cm}^2$: with what force does the water push against it?
3. c) A neighbor on the ground floor has their tap $9\ \text{m}$ lower down: calculate the pressure at their tap and explain the difference compared to the second floor.

The snow accumulated on a flat roof exerts a pressure of $600\ \text{Pa}$ over an area of $30\ \text{m}^2$. Calculate the total force supported by the roof.

What is atmospheric pressure? Name the instrument used to measure it and write down its approximate value at sea level, in pascals.

A scuba diver dives at a depth of $10\ \text{m}$ in a lake, as shown in the figure. Data: $d_{agua} = 1\,000\ \text{kg/m}^3$, $p_{atm} = 101\,300\ \text{Pa}$; take $g = 10\ \text{m/s}^2$.

1. a) Calculate the pressure due solely to the water above him.
2. b) Calculate the total pressure he experiences, adding atmospheric pressure.
3. c) How many times atmospheric pressure is that total pressure? Comment on the result.

On the weather map, a low-pressure system has a pressure at its center of $990\ \text{hPa}$ and a high-pressure system, of $1\,030\ \text{hPa}$.

1. a) Express both pressures in pascals (remember: $1\ \text{hPa} = 100\ \text{Pa}$).
2. b) The normal value of atmospheric pressure at sea level is $1\,013\ \text{hPa}$: which of the two systems is closer to that value?
3. c) Given that depressions usually bring clouds and rain and anticyclones bring settled weather, indicate in which of the two you would expect bad weather and give reasons for your answer.

The atmosphere pushes down on your tabletop with a pressure of about $101\,300\ \text{Pa}$. The tabletop measures $1.2\ \text{m} \times 0.6\ \text{m}$.

1. a) Calculate the surface area of the tabletop.
2. b) Calculate the force with which the atmosphere crushes it.
3. c) If the force is so huge, why does the table not break or collapse? Think about the air underneath the tabletop.

In Torricelli's experiment, a glass tube is filled with mercury and inverted over a trough containing mercury: the column falls until it reaches a height of $76\ \text{cm}$ above the surface of the trough, as shown in the figure. Data: $d_{Hg} = 13\,600\ \text{kg/m}^3$; take $g = 9.8\ \text{m/s}^2$.

1. a) Calculate the pressure exerted by this mercury column on its base.
2. b) Compare the result with the value of atmospheric pressure at sea level and explain what the experiment demonstrates and what is at the top of the tube.
3. c) What height should the column be if the experiment were carried out with water instead of mercury?

A scuba diver descends to a depth of $20\ \text{m}$ in the sea. Data: $d_{mar} = 1\,025\ \text{kg/m}^3$, $p_{atm} = 101\,300\ \text{Pa}$; take $g = 10\ \text{m/s}^2$.

1. a) Calculate the pressure exerted by the water at that depth.
2. b) Calculate the total pressure experienced by the diver, adding atmospheric pressure.
3. c) How many times the atmospheric pressure is this total pressure?
4. d) Using the previous result, explain why divers must plan their dives and control their ascent.

A mountaineer climbs Mulhacén ($3\,479\ \text{m}$) and their barometer reads $670\ \text{hPa}$; before setting off, at sea level, it read $1\,013\ \text{hPa}$.

1. a) Express both pressures in pascals.
2. b) What percentage of the sea-level pressure remains at the summit?
3. c) Explain why atmospheric pressure decreases with altitude (think about how much air remains above you) and explain why water boils below $100\ \celsius$ at the summit.

In a classic laboratory experiment, a metal can containing a little water is heated until it boils and is quickly sealed; on cooling, the can crushes itself without anyone touching it. The figure outlines the interior and exterior pressures on the walls of the can.

1. a) Explain what happens to the water vapour inside on cooling, and what then happens to the pressure inside the can.
2. b) If at a given moment the interior pressure is $40\,000\ \text{Pa}$ and the exterior pressure is $100\,000\ \text{Pa}$, calculate the pressure difference between both sides of the walls.
3. c) Calculate the net force crushing the can if the exposed surface area is $0.04\ \text{m}^2$, and explain why it buckles inwards rather than outwards.