Thrust and Pressure

Icebergs float in the cold water of the North Atlantic. Although the visible portion of an iceberg may tower over a passing ship, only about $11\%$ of the iceberg is above water. 'The magnitude of the normal force on unit area of a surface is called pressure.

1.0Introduction

What happens when you walk on soft snow or dry sand? Your feet sink into the snow or sand and walking can be difficult. If you ride a bicycle with narrow tires over these surfaces, the tires will sink even deeper than your feet. How deep you sink depends on your weight as well as the area over which you make contact with the sand or snow. For example, when you stand on snow with your normal shoes, you may sink deep into the soft snow [see figure(a)]. However, if you use special kinds of supports (or snow boards) having larger area than your shoes [see figure(b)], you can stand on the snow very easily.

In both cases, your weight exerted a downward force on the snow. What changed was the area of contact between you and the snow. By changing the area of contact, you changed the 'pressure' you exerted on the snow due to your weight. When you stood with supports (or snow shoes), the area of contact increased, so that the same force was applied over a larger area. As a result, the pressure that was exerted on the snow decreased and you didn't sink as deep.

2.0Pressure

Thrust : The force acting on an object perpendicular to the surface is called thrust. Pressure : The thrust on unit area is called pressure.

SI unit of pressure : $\mathrm{N}/{\mathrm{m}}^2$ or $\mathrm{N}{\mathrm{m}}^{-2}$. In honour of scientist Blaise Pascal, the SI unit of pressure is called "pascal", denoted as Pa .

Definition of 1 pascal :

If 1 newton force acts perpendicular to a surface of area $1{\mathrm{m}}^2$, then the pressure acting on the surface is 1 pascal.

Pressure is a scalar quantity. Always remember it is the component of the force normal (perpendicular) to the area under consideration for calculating pressure, not the force vector.

A common unit of pressure is the atmosphere (atm), i.e. the pressure exerted by the atmosphere at sea level, $1\mathbf{atm}=1.013\times {10}^5\mathbf{Pa}$

Another unit of pressure is bar, $1\mathrm{Bar}={10}^5\mathbf{Pa}$

• The weight of a hundred rupee note resting completely flat on a table exerts a pressure of about 1 Pa on the table. Since 1 Pa is a very small unit of pressure, pressure sometimes is expressed in units of kPa (kilo pascal), which is $1,000\mathrm{Pa}$.

For a given force, pressure is inversely proportional to the area. Thus, the same force acting on a smaller area exerts a larger pressure, and a smaller pressure on a larger area.

• Q. When you push a nail using a hammer into a wooden plank by its head, the nail cannot be inserted in the plank. When you push the nail by the pointed end, the nail can easily be inserted in the plank. Why? Explanation: When you apply a force on the nail using a hammer with its head touching the wooden plank, the pressure exerted on the plank is quite small. This is because pressure is inversely proportional to area i.e., the larger the area, smaller the pressure on a surface for the same force. When you apply the same force on the nail with its pointed end touching the plank, pressure exerted on the plank is sufficient to insert it into the plank (see figure). This is because the area of the pointed end of the nail is much smaller
  

Some more examples related to pressure

(1) You cannot cut vegetables with a blunt knife. But, if you use a sharp knife, it is quite easy to cut the vegetables. This is because the area of sharp edge of the knife is quite small and thus, pressure is large enough to cut the vegetables. (2) The straps of shoulder bags are made wide (broad). This is because the wide straps have larger area so that the pressure exerted on the shoulder becomes quite small. Thus, it becomes quite comfortable to carry such shoulder bags. (3) Porters (coolies) place on their heads a round piece of cloth, when they have to carry heavy load. By this they increase the area of contact of the load with their head. So, the pressure on their head is reduced and they find it easier to carry the load. (4) Buildings have wide foundations so that they exert less pressure on the earth.

Numerical Ability

• Q. A block of wood is kept on a table top. The mass of wooden block is 5 kg and its dimensions are $40\mathrm{cm}\times 20\mathrm{cm}\times 10\mathrm{cm}$. Find the pressure exerted by the wooden block on the table top if it is made to lie on the table top with its sides of dimensions (a) 20 cm × 10 cm and (b) 40 cm × 20 cm (see figure, take g = 9.8 m/s²).

Solution: Decode the problem To find the pressure, we should know the formula pressure $=\frac{\text{ force }}{\text{ area }}$ The force exerted by wooden block = weight of the wooden block. We have to take that area which is in contact with the surface. Given, mass of the wooden block, $\mathrm{m}=5\mathrm{kg}$ Thrust, $\mathrm{F}=\mathrm{m}\times \mathrm{g}=5\times 9.8=49\mathrm{N}$ (a) Area, $\mathrm{A}=20\mathrm{cm}\times 10\mathrm{cm}=200{\mathrm{cm}}^2$ $=200\times 10^{-4}{\mathrm{m}}^2=0.02{\mathrm{m}}^2$ Now, pressure, $\mathrm{P}=\frac{\mathrm{F}}{\mathrm{A}}=\frac{49}{0.02}=2450{\mathbf{N}\mathbf{m}}^{-2}$

3.0Pressure in a fluid

A fluid is any substance that has no definite shape and has the ability to flow. You might think of a fluid as being a liquid, such as water or motor oil. But gases are also fluids. When you are outside on a windy day, you can feel the air flowing past you. Because air can flow and has no definite shape, air is a fluid. Gases, liquids, and the state of matter called plasma, which is found in the Sun and other stars, are fluids and can flow.

Fluids can exert pressure on the base and walls of the container in which they are enclosed.

Pascal's law

The French scientist Blaise Pascal observed that 'the pressure in a fluid at rest is the same at all points if they are at the same height'. A change in pressure at any point in an enclosed fluid at rest is transmitted undiminished to all points in the fluid. This is called Pascal's principle. Pascal's principle was discovered in the seventeenth century by Blaise Pascal.

Some important points related to fluid pressure. (1) Fluid pressure acts in all directions, not just the direction of the applied force. When you inflate a car tire, you are increasing the pressure in the tire. This force acts up, down and sideways in all directions inside the tire. (2) The fluid pressure at any point on the object is perpendicular to the surface of the object at that point (see figure).

A hydraulic lift : The pressure is the same on both sides of the enclosed fluid, allowing a small force to lift a heavy object.

Pressure, $P=\frac{F_1}{A_1}=\frac{F_2}{A_2}$ or $F_2=\left(\frac{A_2}{A_1}\right)F_1$ Since, ${\mathrm{A}}_2\gg {\mathrm{A}}_1,{\mathrm{F}}_2\gg {\mathrm{F}}_1$. (4) The pressure depends only on the height of the column of fluid above the surface where you measure the pressure. It does not depend on the area of the surface in contact or the shape of the liquid column. The greater the height of the column of fluid above a surface, the greater the pressure exerted by the fluid on the surface.

4.0Density of a substance

The density $\rho$ of a substance is defined as its mass per unit volume, $\rho =\frac{m}{v}$ where $m$ is the mass of a sample and $v$ is its volume. Density is a characteristic property of a solid or a liquid. This means for a given solid (or liquid), its density remains constant whatever be its size or mass. Density of a gas is not a constant, it is variable. It depends on temperature, pressure and volume. S.I. unit of density: $\mathbf{kg}/{\mathbf{m}}^3$ or $\mathbf{kg}\mathbf{m}{\mathbf{m}}^{-3}$.

Its C.G.S. unit is $\mathbf{g}/{\mathbf{cm}}^3$ or $\mathrm{g}{\mathrm{cm}}^{-3}$.

The density of water is $1000\mathrm{kg}{\mathrm{m}}^{-3}$ or $1\mathrm{g}{\mathrm{cm}}^{-3}$ at $4^{\circ }\mathrm{C}$.

• Solids and liquids tend to be almost incompressible, meaning that their density changes very little with changes in pressure. Thus, the densities for solids and liquids are approximately independent of pressure.
• Gases are compressible and can have densities over a wide range of values. Thus, there is not a standard density for a gas, as there is for solids and liquids. The densities for gases are usually the values of the density at STP. For deviations of temperature and pressure from these values, the density of the gas will vary significantly.

(All values are at standard atmospheric temperature and pressure (STP), defined as $0^{\circ }\mathrm{C}$ (273 K ) and $1\mathrm{atm}\left(1.013\times 10^5\mathrm{Pa}\right)$. To convert kilograms per cubic meter $\left(\mathrm{kg}/{\mathrm{m}}^3\right)$ to grams per cubic centimeter ( $\mathrm{g}/{\mathrm{cm}}^3$ ), multiply it by $10^{-3}$ ). What does density depend on? The densities of some solids, liquids and gases are listed in table. The table shows that the density of gold, for example, is more than 19 times greater than the density of water. Also, the density of some solids and liquids, such as mercury, can be more than 10,000 times greater than the density of some gases, such as helium. Density of a material depends on following factors: (1) Mass of particles : The density of a material depends on the mass of the particles, such as atoms or molecules, that make up the material. The more mass these particles have, the greater the density of the material. For example, the mass of a gold atom is more than seven times the mass of an aluminium atom. As a result, the density of gold is much greater than the density of aluminium. (2) Distance between particles: The density of a material also depends on the distance between the particles in the material. The greater the distance between the atoms or molecules, the smaller the density. Table shows that in gases, particles are much farther apart than in solids or liquids. As a result, the density of a gas is usually much less than the density of a solid or a liquid.

• Q. After an exciting but exhausting lecture, a physics professor stretches out for a nap on a bed of nails, as seen in figure, suffering no injury and only moderate discomfort. How is this possible?
  
  Explanation: If you try to support your entire weight on a single nail, the pressure on your body is your weight divided by the very small area of the end of the nail. The resulting pressure is large enough to penetrate the skin. If you distribute your weight over several hundred nails, as demonstrated by the professor, the pressure is considerably reduced because the area that supports your weight is the total area of all nails in contact with your body. Thus, he suffers no injury but only a little discomfort.

Relative density (R.D.)

The relative density of a substance is the ratio of its density to the density of water. It is also called 'specific gravity'.

Where, ${\rho }_s=$ density of substance ; ${\rho }_w=$ density of water

Variation of pressure with depth

The fluid pressure depends only on the height of the column of fluid above the surface where you measure the pressure. It does not depends on the area of the surface in contact or the shape of the liquid column. The greater the height of the column of fluid above a surface, the greater is the pressure exerted by the fluid on the surface.

If ${\mathrm{P}}_1$ is the pressure on the surface of liquid (see figure) and ${\mathrm{P}}_2$ is the pressure at a point within the liquid at a depth $h$, then, their pressure difference ( ${\mathrm{P}}_2-{\mathrm{P}}_1$ ) is given by,

$\Delta P=P_2-P_1=\rho gh$

Pressure depends only on the height of the fluid above a surface, not on the shape of the vessel. Pressure at the bottom of each section of the vessel is same (called hydrostatic paradox).

5.0Buoyancy

The tendency for an immersed body to be lifted up in a fluid, due to an upward force that acts opposite to the action of gravity is called buoyancy.

Buoyant force

It is an upward force that is exerted by a fluid on any object immersed partly or wholly in the fluid.

Cause of the buoyant force

The buoyant force is caused by the pressure that is exerted by a fluid on an object in the fluid. Figure shows a cube shaped object submerged in a glass of water. The water exerts pressure everywhere over the surface of the object. The direction of the pressure on a surface is always perpendicular to the surface. Also, the pressure exerted by a fluid increases as you go deeper into the fluid. In figure, the bottom of the cube is deeper in the water. Therefore, the pressure that is exerted by the water at the bottom of the cube is greater than it is at the top of the cube.

The higher pressure near the bottom means that the water exerts a net upward force on the cube. This net upward force is the buoyant force.

• A buoyant force acts on all objects that are placed in a fluid, whether they are floating or sinking.

6.0Archimedes' principle

According to Archimedes' principle, any object completely or partially submerged in fluid experiences an upward buoyant force equal in magnitude to the weight of the fluid displaced by the liquid. Buoyant force $=$ Weight of displaced fluid $={\mathrm{m}}_{\ell }\mathrm{g}({\mathrm{m}}_{\ell }=$ mass of displaced fluid $)$ ${\mathrm{m}}_{\ell }={\rho }_{\ell }\times {\mathrm{v}}_{\ell }$ ( ${\mathrm{v}}_{\ell }=$ volume of displaced liquid) ${\mathbf{F}}_{\mathbf{B}}={\mathbf{\rho }}_{\ell }\times {\mathbf{v}}_{\ell }\times \mathbf{g}$

Apparent weight : Because of an upward force acting on a body immersed in a fluid, either wholly or partially, there occurs an apparent loss in weight of the body. The net weight of an object immersed in a fluid is called apparent weight.

Apparent weight, ${\mathbf{W}}^{\prime }=$ Weight of body in air - Buoyant force

Sinking and floating

This means, the body will just float or remain hanging at whatever height it is left inside the liquid [see figure (b)]. (b) If ${\rho }_{\mathrm{s}}<{\rho }_{{\mathrm{L}}^{\prime }}$, apparent weight $=0$ i.e., weight of the body in air = buoyant force

This means the body will float, but it is immersed partly in the liquid [see figure (c)].

Changing the buoyant force

An object sinks or floats depends on whether the buoyant force is smaller than or equal to its weight. The fluid exerts upward pressure on the entire lower surface of the object that is in contact with the fluid. If this surface is made larger, then upward pressure is exerted on larger surface of the object and the buoyant force becomes large enough to float the object. E.g. If an aluminium sheet is crumpled, the buoyant force on it is less than the weight, so the aluminium sheet sinks. When the aluminium is flattened into a thin curved sheet, the buoyant force is large enough so that the sheet floats.

• Metal ships are able to float because the metal is formed into a curved sheet that has a large surface area in contact with the water, which is called the hull of the ship. The contact area of the hull with the water is much greater than if the metal were a solid block. As a result, the buoyant force on the hull is greater than it would be on a metal block and thus, the ship floats.

• Q. Relative density of silver is 10.8 . The density of water is $10^3{\mathbf{kg}\mathbf{m}}^{-3}$. What is the density of silver in SI unit? Solution: Decode the problem To find the relative density, we should know the formula Now, R.D. = $\frac{\text{ Density of silver }}{\text{ Density of water }}=\frac{\rho }{{\rho }_w}$ Given, Relative density of silver, R.D. $=10.8$; density of water, ${\rho }_{\mathrm{w}}=10^3\mathrm{kg}{\mathrm{m}}^{-3}$; density of silver, $\rho =$ ? Now, R.D. $=\frac{\text{ Density of silver }}{\text{ Density of water }}=\frac{\rho }{{\rho }_w}$ or $\rho =$ R.D. $\times {\rho }_{\mathrm{w}}$ or $\rho =10.8\times {10}^3{\mathbf{kg}\mathbf{m}}^{-3}$

Measuring relative density of an object using Archimedes' principle

Let us take an object having weight ${\mathrm{W}}_1$ in air. The weight ${\mathrm{W}}_1$ is measured by using a spring balance [see figure(a)].

Now take a vessel filled with water and immerse the object completely in water [see figure(b)] and again note the spring balance reading. This reading gives the weight $\left({\mathrm{W}}_2\right)$ of the object in water. Thus, the buoyant force on the object due to water is given by, ${\mathrm{F}}_{\mathrm{B}}={\mathrm{W}}_1-{\mathrm{W}}_2$ Now, relative density is given by, R.D. $=\frac{\text{ Density of object }}{\text{ Density of water }}=\frac{\rho }{{\rho }_w}=\frac{\rho Vg}{{\rho }_{\mathrm{w}}\mathrm{Vg}}$ (Multiplying and dividing by Vg ) Where, ${\mathrm{V}}_{\mathrm{s}}={\mathrm{V}}_{\mathrm{L}}=\mathrm{V}=$ volume of object = volume of water displaced, as the object is immersed completely, and ' g ' is the acceleration due to gravity. Now, weight of object in air, $W_1=\rho Vg$ Buoyant force, ${\mathrm{F}}_{\mathrm{B}}={\rho }_{\mathrm{w}}\mathrm{Vg}$ From (2), (3) and (4), we get, R.D. $=\frac{W_1}{F_B}$ From (1) and (4), we get, R.D. $=\frac{W_1}{W_1-W_2}$

• Q. A lady purchases a cheap "gold" crown at a flea (street) market. After she gets home, she hangs the crown from a scale and finds its weight to be 7.84 N [see figure(a)]. She then weighs the crown while it is immersed in water, and the scale reads 6.86 N [see figure (b)]. Is the crown made of pure gold? Explain. Take, Density of gold= $19,300\mathrm{kg}/{\mathrm{m}}^3$; density of water $=1000\mathrm{kg}/{\mathrm{m}}^3;\mathrm{g}=9.8\mathrm{m}/{\mathrm{s}}^2$.
  
  Solution: Decode the problem To find the relative density, we should know the formula,

Given, weight of crown in air, ${\mathrm{W}}_1=7.84\mathrm{N}$; density of water, ${\rho }_{\ell }=1000\mathrm{kg}/{\mathrm{m}}^3;$ weight of crown in water, ${\mathrm{W}}_2=6.86\mathrm{N}$; density of crown, ${\rho }_{\mathrm{s}}=$ ? R.D. $=\frac{{\mathrm{W}}_1}{{\mathrm{W}}_1-{\mathrm{W}}_2}=\frac{7.84}{7.84-6.86}=\frac{7.84}{0.98}=8$ R.D. $=\frac{{\rho }_s}{{\rho }_{\ell }}$ $\Rightarrow 8=\frac{{\rho }_{\mathrm{s}}}{1000}$ $\Rightarrow {\rho }_{\mathrm{s}}=8000\mathbf{kg}/{\mathrm{m}}^3$

• Since, the density of gold is $19,300\mathrm{kg}/{\mathrm{m}}^3$, the crown is certainly not pure. It is either hollow, made of an alloy, or both.

• Q. Three cubes A, B, C of equal size are made of ice, aluminium and iron respectively. All these cubes are placed in water (see figure). Which one of the three cubes experiences greatest buoyant force on it? Given, ${\rho }_{\text{ice }}=920\mathrm{kg}/{\mathrm{m}}^3$; ${\rho }_{\mathrm{Al}}=2700\mathrm{kg}/{\mathrm{m}}^3$;
  
  ${\rho }_{\mathrm{Fe}}=7800\mathrm{kg}/{\mathrm{m}}^3$. Explanation: The buoyant force depends only on the density of the fluid and the volume of the fluid displaced. The buoyant force on an object does not depend on the object's density or its weight. Thus, the buoyant force on each cube is the same, because each cube has the same volume and displaces the same amount of water.
• Q. How does buoyant force on an object that is completely submerged in water or any liquid change as it is moved deeper and deeper in that liquid? Explanation: The buoyant force doesn't change with depth. Suppose you drop a steel cube into the ocean (see figure).
  
  Since the cube on the right is deeper, the pressure on its upper surface is increased. But, the pressure on the bottom surface also increases by the same amount. You might think that the cube would sink only to a depth where the buoyant force on the cube balances its weight. However, the steel sinks to the bottom, no matter how deep the ocean is. The buoyant force on the cube is the difference between the downward force due to the water pressure on the top of the cube, and the upward force due to water pressure on the bottom of the cube. When the cube is deeper, the pressure on the top surface increases, but the pressure on the bottom surface also increases by the same amount. As a result, the difference between the forces on the top and bottom surfaces is the same, no matter how deep the cube is submerged. The buoyant force on the submerged object is the same at any depth.

7.0Mindmap

Important units, formulae, and quantity

8.0Some Basic Terms

• Sink :- To go down or make something go down under the surface of liquid or a soft substance.
• Perpendicular :- perpendicular is defined as a straight line that makes the right angle $\left(90^{\circ }\right)$ with the other line. $\mathrm{PQ}\perp \mathrm{RS}$
  
• Blunt :- Without a sharp edge or point.
• Dimensions :- A measurement of the length, width or height of something
• Fluid :- The substances which can flow easily are called fluids.
• Enclosed :- To surround something with a wall, fence, etc.; to put one thing inside another
• Incompressible :- Incapable of being compressed; resisting compression.
• STP. :- In physics STP stands for standard temperature and pressure. It is commonly used as a reference point for the properties of gases. At STP, the standard temperature is defined as $0^{\circ }\mathrm{C}$. At STP standard pressure is 1 atmosphere (atm).
• Atoms :- Atoms are defined as "the basic building blocks of matter".
• Molecules :- Molecules are made up of one or more atoms.
• Piercing :- To poke a hole in something with a sharp object.
• Immersed :- The term immersed is used to describe an object that it submerged in a fluid (liquid or gas).
• Hull :- The bottom of a boat or ship is commonly referred to as the "hull."
• Elongation :- Elongation refers to the process or state of becoming longer or stretched out.