Density and pressure

A giant squid with a length of $20.0\,\text{m}$ is positioned vertically in seawater, and the upper end of the squid is at a depth of $8.00\,\text{m}$. The density of the seawater is $1050\,\text{kg m}^{-3}$. What is the pressure difference between the top and the bottom of the squid?

Feb/March 2016

Which force is produced solely by a difference in pressure?

Feb/March 2017

The density of paper is $800\,\text{kg m}^{-3}$. A standard sheet of paper measures $210\,\text{mm}$ in width and $300\,\text{mm}$ in length. A stack of $500$ sheets of paper is $50\,\text{mm}$ thick. What is the mass of one sheet of paper?

Feb/March 2018

A hemisphere is placed so that its flat face is in contact with a table. It has radius $r$ and density $\rho$. The volume of a sphere is $\frac{4}{3}\pi r^3$. What is the mean pressure the object exerts on the table?

Feb/March 2019

What would be a sensible estimate of the density of sand?

Feb/March 2021

A balance is used to determine the mass $m$ of several cylindrical metal rods of length $l$. Each of the metal rods has the same radius $r$. The graph shows how $m$ varies with $l$. The gradient of the graph is $G$. Which expression gives the density of the metal?

Feb/March 2021

State one unit used for density.

Feb/March 2022

A number of small solid cubes each have a mass of $1.0\,\text{kg}$ and edge lengths of $5.0\,\text{cm}$. These small cubes are arranged to make one large solid cube with edge lengths of $2.0\,\text{m}$. What is the weight of the large cube?

Feb/March 2024

A borehole that is $2000\,\text{m}$ deep has oil and water in it, as illustrated. At the base of the borehole, the pressure caused by the liquids is $17.5\,\text{MPa}$. The density of the oil is $830\,\text{kg m}^{-3}$ and the density of the water is $1000\,\text{kg m}^{-3}$. What is the depth $x$ of the oil?

Feb/March 2024

A solid is entirely beneath the surface of a liquid. A student finds the ratio $\frac{\text{upthrust acting on the object}}{\text{weight of the object}}$. Which one change would make this ratio twice as large?

Feb/March 2025

An object submerged in a liquid in a tank experiences an upthrust. What physical cause gives rise to this upthrust?

May/June 2010

At sea level, atmospheric pressure is $100\,\text{kPa}$. Sea water has a density of $1020\,\text{kg m}^{-3}$. At what depth in the sea is the total pressure $110\,\text{kPa}$?

May/June 2010

As ice melts, it gets smaller in volume. Which row shows the correct result for ice changing into water?

May/June 2010

When an object is placed in a liquid inside a tank, it experiences an upthrust. What physical cause produces this upthrust?

May/June 2010

As ice melts, it shrinks. Which row gives the correct result for ice changing into water?

May/June 2010

The atmospheric pressure at sea level is $100\,\text{kPa}$. Sea water has a density of $1020\,\text{kg m}^{-3}$. At what depth in the sea would the total pressure be $110\,\text{kPa}$?

May/June 2010

A liquid in a tank exerts an upthrust on an object that is immersed in it. What physical cause produces this upthrust?

May/June 2010

The atmospheric pressure at sea level is $100\,\text{kPa}$. Sea water has a density of $1020\,\text{kg m}^{-3}$. At what depth in the sea will the total pressure be $110\,\text{kPa}$?

May/June 2010

As ice melts, it contracts. Which row is correct for ice changing into water?

May/June 2010

State the observation that supports the assumption that the intermolecular forces in a solid are significant.

May/June 2010

$1.5\,\text{m}^3$ of water is combined with $0.50\,\text{m}^3$ of alcohol. Water has a density of $1000\,\text{kg m}^{-3}$ and alcohol has a density of $800\,\text{kg m}^{-3}$. Determine the density of the mixture when its volume is $2.0\,\text{m}^3$?

May/June 2011

The figure depicts two vessels, P and Q, each having sides that are tilted at $45^{\circ}$. As illustrated, vessel P widens upwards while vessel Q narrows upwards. Each vessel contains a liquid, and the liquids are at the same depth. The liquid in vessel P has twice the density of the liquid in vessel Q. Find the ratio $\dfrac{\text{pressure due to the liquid on the base of P}}{\text{pressure due to the liquid on the base of Q}}$?

May/June 2011

Two solid substances P and Q consist of atoms with masses $M_P$ and $M_Q$ respectively. They contain $n_P$ and $n_Q$ atoms in each unit volume. The density of P is higher than the density of Q. Which statement must be true?

May/June 2011

$1.5\,\text{m}^3$ of water is combined with $0.50\,\text{m}^3$ of alcohol. The density of water is $1000\,\text{kg m}^{-3}$ and the density of alcohol is $800\,\text{kg m}^{-3}$. What is the density of the mixture with volume $2.0\,\text{m}^3$?

May/June 2011

Explain the differences in density in solids, liquids and gases by referring to how far apart the molecules are.

May/June 2011

A cylindrical wooden block has cross-sectional area $A$ and weight $W$. It is completely submerged in water with its axis vertical. The pressures on its top and bottom faces are $p_t$ and $p_b$ respectively. Which expression gives the upthrust on the block?

May/June 2012

In each box, the gas molecules are shown as circles, and arrows indicate both the direction of movement and the molecule’s speed. A longer arrow shows a greater speed. Which box contains a gas with the greatest density and the lowest temperature?

May/June 2012

When the temperature is room temperature, the density of liquid mercury is five times greater than the density of solid aluminium. What accounts for this?

May/June 2012

A quantity of gas trapped inside a cylinder by a piston is warmed gently. Meanwhile, the piston is adjusted so that the pressure stays constant. Because of this, which change will not happen?

May/June 2012

A pipe is sealed at one end and contains gas, which is confined by a column of water. The atmospheric pressure is $1.0 \times 10^{5}\ \text{Pa}$. The density of water is $1000\ \text{kg m}^{-3}$. Calculate the pressure of the gas. (Use $g = 10\ \text{m s}^{-2}$.)

May/June 2012

A wooden cylinder has cross-sectional area $A$ and weight $W$. It is completely submerged in water, with its axis aligned vertically. The pressures acting on its upper and lower faces are $p_t$ and $p_b$ respectively. Which expression is equal to the upthrust on the block?

May/June 2012

At room temperature, the density of liquid mercury is five times higher than the density of solid aluminium. What accounts for this?

May/June 2012

In each box, identical gas molecules are shown as circles, and arrows indicate the direction of movement and the speed of each molecule. A longer arrow shows a higher speed. Which box contains a gas with the greatest density and the lowest temperature?

May/June 2012

Show that the pressure $P$ produced by a liquid of density $\rho$ is proportional to the depth $h$ below the liquid surface.

May/June 2012

The diagram shows a sphere submerged in water. $P$, $Q$, $R$ and $S$ are forces on the sphere caused by the pressure of the water. Each force acts at right angles to the sphere’s surface. $P$ and $R$ act vertically in opposite directions. $Q$ and $S$ act horizontally in opposite directions. Which statement about the sizes of these forces is correct?

May/June 2013

A bore-hole with a depth of $2000\,\text{m}$ contains oil and water, as illustrated. The pressure caused by the liquids at the base of the bore-hole is $17.5\,\text{MPa}$. The density of the oil is $830\,\text{kg m}^{-3}$ and the density of the water is $1000\,\text{kg m}^{-3}$. What is the depth $x$ of the oil?

May/June 2013

A submarine is balanced while completely underwater. What is the reason for the upthrust acting on the submarine?

May/June 2013

The diagram depicts an ice cube floating in water. Both the ice cube and the water are at $0^\circ\text{C}$. Which statement gives the correct comparison between the molecular properties of the ice and the water?

May/June 2013

Two bulbs X and Y, each filled with air at different pressures, are linked by a tube P that has two mercury threads inside it. The density of mercury is $13600\,\text{kg m}^{-3}$. Which pair of values for $h_1$ and $h_2$ could be correct?

May/June 2013

Ice at a temperature of $0^{\circ}\text{C}$ is an unusual solid because it floats on its liquid form, water, when both are at the same temperature. What explains this?

May/June 2013

The hydrostatic pressure formula is $p = \rho g h$. Which equation, or principle of physics, is applied in deriving this formula?

May/June 2013

Define pressure in words.

May/June 2013

If the Universe were such that, as temperature rose, the speed of the molecules in a substance increased, but at any given temperature every molecule in that substance had the same speed, which process would not take place?

May/June 2014

The density of liquid Q is double that of liquid R. When the depth in liquid R is $x$, the pressure caused by the liquid is $4\,\text{kPa}$. At what depth in liquid Q does the pressure caused by the liquid become $7\,\text{kPa}$?

May/June 2014

The Mariana Trench in the Pacific Ocean is roughly $10\,\text{km}$ deep. If sea water is taken to be incompressible and to have a density of about $1020\,\text{kg m}^{-3}$, what is the approximate pressure at that depth?

May/June 2014

The coin has the form of a thin cylinder, as illustrated in Fig. 2.1. Fig. 2.2 presents the measurements taken to find the density \(\rho\) of the substance from which the coin is made.

May/June 2014

The diagram illustrates the arrangement of atoms in a specific crystal. Each atom is positioned at a corner of a cube. The mass of each atom is $3.5 \times 10^{-25}\,\text{kg}$. The density of the crystal is $9.2 \times 10^3\,\text{kg m}^{-3}$. What is the smallest separation between the centres of two neighbouring atoms?

May/June 2015

Which row correctly shows how the molecules are arranged and how they move in liquid water and in ice when each is at a temperature of $0^{\circ}\text{C}$?

May/June 2015

The greatest pressure granite rock can tolerate is $2.0 \times 10^{8}\,\text{N m}^{-2}$. If the pressure rises above this value, the rock starts to behave like a liquid. Granite has a density of $2.7 \times 10^{3}\,\text{kg m}^{-3}$. How tall would a mountain made only of granite be if its base were just on the point of flowing?

May/June 2015

Four cuboids, each having the same length, breadth and height, are placed in water. They are maintained at an equal depth and in the same orientation by vertical rods, as illustrated. Water has density $\rho$. Cuboid $W$ is made of a material with density $4\rho$. Cuboid $X$ is made of a material with density $2\rho$. Cuboid $Y$ is made of a material with density $\rho$. Cuboid $Z$ is made of a material with density $0.5\rho$. Which statement is correct?

May/June 2015

In an engine, the gas exerts work on a piston with cross-sectional area $80\,\text{cm}^2$. The pressure acting on the piston remains constant at $4.6 \times 10^5\,\text{Pa}$. What amount of work is carried out by the gas on the piston as it travels a distance of $25\,\text{cm}$?

May/June 2015

As ice melts, it contracts. Which row shows ice changing into water correctly?

May/June 2015

A tube shaped like a W contains two separate portions of mercury, and each is exposed to the atmosphere. A pocket of air at pressure $P$ is trapped between them. The diagram labels two vertical distances $x$ and $y$. Atmospheric pressure is the same as the pressure produced by a column of mercury $760\,\text{mm}$ high. The pressure $P$ is written in this form. Which combinations of $x$, $y$ and $P$ could be correct?

May/June 2015

A solid metal cylinder is placed on a horizontal surface, as illustrated. The cylinder has length $x$ and cross-sectional area $A$. The cylinder produces a pressure $p$ on the surface. The acceleration of free fall is $g$. Which expression gives the density of the metal in the cylinder?

May/June 2016

A micrometer screw gauge is used to measure the diameter of a solid metal sphere. The diagram gives a magnified view of the shaft of the micrometer screw gauge as the measurement is being taken. The sphere has a mass of $0.450\,\text{g}$. What is the density of the metal from which the sphere is made?

May/June 2016

Inside a large tank at an oil refinery, three oils with different densities are blended together. There is no chemical reaction. The mixture contains $1200\,\text{kg}$ of oil with density $1100\,\text{kg m}^{-3}$, $1500\,\text{kg}$ of oil with density $860\,\text{kg m}^{-3}$, $4000\,\text{kg}$ of oil with density $910\,\text{kg m}^{-3}$. Determine the density of the mixture.

May/June 2016

The density of air in the Earth’s atmosphere falls by almost a straight-line relationship as height increases, dropping from $1.22\,\text{kg m}^{-3}$ at sea level to $0.74\,\text{kg m}^{-3}$ at an altitude of $5000\,\text{m}$. On a certain day, the atmospheric pressure at the Earth’s surface is $100\,000\,\text{Pa}$. Over the region from the Earth’s surface up to an altitude of $5000\,\text{m}$, $g$ may be taken as constant at $9.7\,\text{m s}^{-2}$. What is the atmospheric pressure at an altitude of $5000\,\text{m}$?

May/June 2016

For a change in depth $\Delta h$ within a liquid of density $\rho$, the corresponding change in pressure $\Delta p$ is given by $\Delta p = \Delta h\rho g$, where $g$ represents the acceleration of free fall. Which equation, or principle of physics, is used to derive this formula?

May/June 2016

Fig. 2.1 shows a liquid in a cylindrical container. The container has cross-sectional area $A$. The liquid column has height $h$ and the liquid density is $\rho$. Show that the pressure $p$ exerted by the liquid on the cylinder base is $p = \rho g h$.

May/June 2016

A spring balance is used to find the weight of a cylinder while it is fully immersed in oil, as illustrated in Fig. 4.1. The spring balance reads $4.8\,\text{N}$. The cylinder has a length of $5.0\,\text{cm}$ and a cross-sectional area of $13\,\text{cm}^2$. The weight of the cylinder is $5.3\,\text{N}$.

May/June 2016

A metal block has a mass of 750 g. 60% of its mass is magnesium, and the rest is copper. The density of magnesium is $1.7\,\text{g cm}^{-3}$. The density of copper is $9.0\,\text{g cm}^{-3}$. What is the density of the block?

May/June 2017

The speed $v$ at which a liquid exits a tube is affected by the pressure change $\Delta P$ and the liquid density $\rho$. This speed is described by $v = k \left( \frac{\Delta P}{\rho} \right)^n$ where $k$ is a constant with no units. What is the value of $n$?

May/June 2017

The diagram shows a sphere submerged in water. $P$, $Q$, $R$ and $S$ are forces on the sphere caused by the water pressure. Each force acts at right angles to the sphere’s surface. $P$ and $R$ act vertically in opposite directions. $Q$ and $S$ act horizontally in opposite directions. Which statement about the sizes of the forces is correct?

May/June 2017

A cylinder is formed from a substance with density $2.7\,\text{g cm}^{-3}$. Its diameter is $2.4\,\text{cm}$ and its length is $5.0\,\text{cm}$. Show that the cylinder’s weight is $0.60\,\text{N}$.

May/June 2017

Blocks P, Q, R and S are made from materials with the same density. Block T is made from a material whose density is twice that of the material used for the other blocks. The cross-sectional area and the height of each block are shown. Which two blocks produce the same pressure on the ground?

May/June 2018

Liquid X and liquid Y are kept in large open tanks. Liquid X has a density of $800\ \text{kg m}^{-3}$ and liquid Y has a density of $1200\ \text{kg m}^{-3}$. At what depths will the pressures be the same?

May/June 2018

Water has a density of $1.0\,\text{g cm}^{-3}$, whereas glycerine has a density of $1.3\,\text{g cm}^{-3}$. Water is poured into a measuring cylinder that already contains $40\,\text{cm}^3$ of glycerine until the mixture has a density of $1.1\,\text{g cm}^{-3}$. Assume that the mixing process does not change the total volume of the liquid. What is the volume of water added?

May/June 2018

A block of wood with a cylindrical shape has cross-sectional area $A$ and weight $W$. It is completely submerged in water, with its axis in a vertical position. The pressures acting on its top and bottom faces are $p_t$ and $p_b$ respectively. Which expression gives the upthrust acting on the block?

May/June 2018

Four cuboids with the same lengths, breadths and heights are submerged in water. As shown, vertical rods keep the cuboids at the same depth and in matching orientations. Water has density $\rho$. Cuboid W is made from a material with density $4\rho$. Cuboid X is made from a material with density $2\rho$. Cuboid Y is made from a material with density $\rho$. Cuboid Z is made from a material with density $0.5\rho$. Which option is correct?

May/June 2018

A rectangular block of lead, with density $1.13 \times 10^{4}\,\text{kg m}^{-3}$, has edge lengths of $12.0\,\text{cm}$, $15.0\,\text{cm}$ and $10.0\,\text{cm}$. What is the greatest pressure the block can apply when it is placed on a table?

May/June 2018

Full-fat milk is formed by combining fat-free milk with fat. A volume of $1.000 \times 10^{-3}\,\text{m}^3$ of full-fat milk has a mass of $1.035\,\text{kg}$. In this milk, $4.00\%$ of the volume is fat. The density of fat-free milk is $1.040 \times 10^{3}\,\text{kg m}^{-3}$. Calculate the density of the fat.

May/June 2018

A piston with cross-sectional area $A$ confines gas inside a cylinder. The piston is not frictionless. When the gas is heated, it expands and drives the piston back by distance $r$, moving it from position P to position Q. The gas column then has length $s$. Which expression gives the work done by the gas against the atmosphere during this expansion?

May/June 2018

A solid cylinder is hoisted from oil by a wire connected to a motor. Figure 5.1 shows the cylinder at two stages, X and Y, during the lifting. The motor is mounted on an overhead beam. The cylinder has cross-sectional area $0.018\,\text{m}^2$, length $1.2\,\text{m}$ and weight $560\,\text{N}$. The oil density is $940\,\text{kg m}^{-3}$. During the whole lift, the cylinder travels straight upward at a steady $0.020\,\text{m s}^{-1}$. The viscous force from the oil on the cylinder is negligible.

May/June 2018

State the meaning of work done.

May/June 2018

A cylindrical iceberg with height $H$ is floating in sea water. Its upper end rises to a height $h$ above the water surface. The density of ice is $\rho_i$ and the density of sea water is $\rho_w$. What is the value of $h$ for the iceberg above the sea water?

May/June 2019

An empty glass beaker has a mass of $103\ \text{g}$. If it is filled with water, its overall mass becomes $361\ \text{g}$. If it is filled with cooking oil, its overall mass becomes $351\ \text{g}$. The density of water is $1.00\ \text{g cm}^{-3}$. What is the density of the cooking oil?

May/June 2019

A volume of $1.5\,\text{m}^3$ of water is blended with $0.50\,\text{m}^3$ of alcohol. The density of water is $1000\,\text{kg m}^{-3}$ and the density of alcohol is $800\,\text{kg m}^{-3}$. The volume of the mixture is $2.0\,\text{m}^3$. What is the density of the mixture?

May/June 2019

The diagram illustrates how the atoms are arranged in one crystal. Each atom is located at a corner of a cube. The mass of one atom is $3.5 \times 10^{-25}\,\text{kg}$. The density of the crystal is $9.2 \times 10^{3}\,\text{kg m}^{-3}$. What is the shortest distance between the centres of two neighbouring atoms?

May/June 2019

Water has a density of $1.0\,\text{g cm}^{-3}$. Glycerine has a density of $1.3\,\text{g cm}^{-3}$. A student measures $40\,\text{cm}^3$ of glycerine into a container. The student then adds water so that the container contains a mixture of water and glycerine. Assume that the combined volume of the water and glycerine remains unchanged when the two liquids are mixed. What volume of water must be added to obtain a mixture with density $1.1\,\text{g cm}^{-3}$?

May/June 2019

Osmium is a naturally occurring element, and its density is $23\,000\,\text{kg m}^{-3}$. What is another value for the density of osmium?

May/June 2019

A cylindrical disc with mass $0.24\,\text{kg}$ and a circular cross-sectional area $A$ is shown in Fig. 3.1. It rests on horizontal ground, as in Fig. 3.2. A force X, with magnitude $8.9\,\text{N}$, acts at an angle of $30^{\circ}$ to the horizontal. The disc moves across the ground at constant speed $0.60\,\text{m s}^{-1}$.

May/June 2019

A rectangular metal bar produces a pressure of $15\,200\,\text{Pa}$ on the level surface beneath it. The height of the metal bar is $80\,\text{cm}$. What is the density of the metal?

May/June 2020

What would be a sensible estimate for the mass of a raindrop?

May/June 2020

A pipe with one end open is floating in a liquid, as illustrated. The pipe has a cross-sectional area of $0.012\,\text{m}^2$. Its weight is $32\,\text{N}$. What is the density of the liquid?

May/June 2020

A submarine is in equilibrium while it is completely underwater. What produces the upthrust acting on the submarine?

May/June 2020

Liquid $Q$ has a density that is twice that of liquid $R$. In liquid $R$, the pressure from the liquid at depth $x$ is $4\,\text{kPa}$. At what depth in liquid $Q$ will the pressure from the liquid be $7\,\text{kPa}$?

May/June 2020

A cylinder is heated, so the air trapped inside expands while the pressure remains constant at $2.2 \times 10^5\,\text{Pa}$. As the air expands, the piston moves upward by $0.50\,\text{m}$ and does $11\,\text{kJ}$ of work. Frictional effects are negligible. What is the area of the piston’s cross-section?

May/June 2020

A gas sample has a mass of $4.8\,\mu\text{g}$ and takes up a volume of $1.2\,\text{dm}^3$. What is the density of the gas sample?

May/June 2020

What would be a sensible estimate for the volume of an adult person?

May/June 2021

A layer of water with depth $9.0\,\text{cm}$ has a layer of oil with depth $5.0\,\text{cm}$ above it in a measuring cylinder. The density of the water is $1000\,\text{kg m}^{-3}$ and the density of the oil is $800\,\text{kg m}^{-3}$. What total pressure is exerted on the base of the measuring cylinder by the oil and the water?

May/June 2021

The cross-section of each elephant foot is circular, and its circumference is $1.4\,\text{m}$. The elephant’s mass is $5400\,\text{kg}$. The elephant is motionless, with all four feet touching the ground. Take the pressure beneath each foot to be equal. What is the approximate pressure exerted on the ground by one of the elephant’s feet?

May/June 2021

To derive the pressure equation $\Delta p = \rho g \Delta h$, several links between quantities are used. Which relationship is not involved in deriving this equation?

May/June 2021

A sample of gas is enclosed in a cylinder by a piston. Because the piston is frictionless and can move freely, the gas pressure stays constant at $1.80 \times 10^5\,\text{Pa}$. At first, the gas has a volume of $2.40 \times 10^{-4}\,\text{m}^3$. The gas now does $14.4\,\text{J}$ of work. What is the gas volume after this work has been done?

May/June 2021

What is density?

May/June 2021

Complete Table 1.1 by indicating whether each quantity is a vector or a scalar. The quantities shown are acceleration, electrical resistance, and momentum.

May/June 2021

A solid block has dimensions $L$, $2L$ and $4L$. It is placed in water of uniform density with the faces of greatest area lying horizontally, as illustrated. The upthrust on the block is $U$. The block is then turned into a different orientation so that the faces of least area are horizontal. The block stays completely submerged in the water. What is the upthrust acting on the block in its new orientation?

May/June 2022

Inside a sizeable vessel at an oil refinery, three oils with different densities are blended together. No chemical reaction takes place. The mixture contains: 1200 kg of oil with density $1100\ \text{kg m}^{-3}$ 1500 kg of oil with density $860\ \text{kg m}^{-3}$ 4000 kg of oil with density $910\ \text{kg m}^{-3}$. What is the density of the mixture?

May/June 2022

A granite rock located at the Earth's surface has density $\rho$. It is then moved to the surface of a different planet. The acceleration of free fall at the surface of the other planet is 2 times the value at the Earth's surface. What is the density of the rock on the other planet?

May/June 2022

A sealed U-shaped tube holds a liquid of density $\rho$ that is at rest. On one side of the tube there is a gas at pressure $p_1$, and on the other side there is a gas at pressure $p_2$, as illustrated. The acceleration of free fall is $g$. Which equation is correct?

May/June 2022

A wooden block with density $\rho_w$ has side lengths of $a$. It is immersed in a liquid whose density is $\rho_L$. The upper face of the block lies at a depth $h$ beneath the liquid surface. The acceleration of free fall is $g$. What upthrust does the liquid exert on the block?

May/June 2022