# Pressure lesson Note- Science (Preparation for SEE)

# Pressure

Pressure is defined as the force acting per unit area of the surface.

From equation (1), we can say that pressure is inversely proportional to area provided the force that acts on the area is constant.

Pressure directly proportional to force provided the area on which the force acts is constant.

# Atmospheric Pressure

Pressure exerted by the atmospheric air is called atmospheric pressure. This is also termed as standard atmospheric pressure or normal pressure. The normal pressure is the pressure exerted by 760 mm long column of mercury at 0oC at sea level and at a altitude of 45o. It is equal to 1.01 105Nm-2.

# Some consequences of pressure

All cutting and piercing tools such as scissors, needles, knives, etc. have sharp edges. This is because they have very small areas of cutting or piercing edges, so a small force can create a very high pressure.

A sharp knife is more effective in cutting an object than a blunt knife. This is because the applied force is spread over a small area in the case of sharp knife but the force is spread over a larger area in the case of blunt knife. Hence, higher pressure is exerted on the object by the sharp knife than the blunt knife.

It is difficult for a man to walk on sand or muddy surface but a camel or an elephant walks easily on such surface even though camel or elephant is much heavier than man. This is because these animals have feet of larger area than the feet of the man. Due to large area, less pressure exerted is on the surface and walking is easier.

The foundation of a building is made wider than walls. This is because the walls of a building and its roof exert pressure on the wide area of foundation and this reduces pressure over earth’s surface and the building is saved from collapsing.

A girl with heeled pointed shoe would make deep impressions on the clay road than an elephant moving along he same road. This is because the elephant exerts a large amount of force over comparatively large area but the girl exerts force on the pointed heel that with less area, thus the pressure exerted by the girl is more than that of the elephant.

# Liquid Pressure

Consider a liquid of density ‘d’ is contained in a beaker of cross-sectional area ‘A’ Let ‘h’ be the depth of the liquid column from its free surface as shown in the figure.

Then, the volume of liquid = Area height

or , V = A h

The weight of the liquid column = Mass gravity

or, W = m g

= (v d)g [since d = , m= d]

W = A h d g V=A h]

This weight of the liquid column is the force acting on the area ‘A’, normally. Therefore,

W = F

or, P =

or,

Thus, pressure exerted by a liquid of density‘d’ on the base of the beaker is dgh. Therefore, the pressure exerted by liquid at rest depends on three factors:

Depth or height of the liquid column (h),

Density of the liquid (d) and

Acceleration due to gravity (g)

# Laws of liquid pressure

The pressure in liquid is directly proportional to the depth from the free surface of the liquid or the height of the liquid column.

The pressure at the same depth, in a liquid is the same in all directions.

The pressure of a liquid is independent of the shape of the container is which it is kept.

A liquid finds its own level.

The pressure at any point in a liquid depends on its density.

# Some consequences of liquid pressure

The pressure exerted by a liquid increases with depth:

This principle is used in the construction of dams. Dams in rivers and water canals are made with their base wide so that they can withstand greater pressure due to water flowing in the dams and water canals.

Similarly, a bucket can be filled faster at the bottom floor than at the top floor of a building, provided that the size of water taps are the same, due to this principle.

A liquid finds its own level:

This principle is used in the water supply system. Water supply tanks or water reservoir are constructed at a taller place so that water rises in buildings without using water pump.

The pressure at any point in a liquid acts in all the directions:

# Pascal’s law

This law is formulated by the French Scientist Blaise Pascal in 1647 AD who was born in 1623 AD.

“When pressure is applied on a liquid enclosed in a vessel, it is transmitted equally in all the directions.”

# Applications of Pascal’s law

Pascal’s law tells that if pressure at any point in a liquid is changed, there will be equal change in pressure at other points in the liquid. This fact is used in hydraulic machines such as hydraulic press, hydraulic brake, hydraulic garage lift, hydraulic cranes etc.

# Hydraulic garage lift

Hydraulic garage lift is a machine used to lift light vehicles in service stations during their services.

# Hydraulic Brake

A hydraulic brake is a braking system that is used in heavy automobiles.

# Density and relative density

The density of a substance is defined as the mass per unit volume.

When the mass of a particular volume of substance is divided with the mass of the same volume of water at 4oC, it is called relative density of that substance.

Relative density is a simple ratio to that why it does not have any unit.

# Upthrust

When a body is partially or wholly immersed in a liquid, the resultant force acted on it by the liquid is called upthrust or buoyant force.

# Density of liquid and upthrust

The density of liquid directly influences Upthrust. As density of liquid increases the pressure given by the liquid, the upthrust is also affected by the density of the liquid.

It is found that the egg sinks in water but floats in saturated salt solution. It is due to the more density of salt solution, which applies more upthrust on the egg.

Due to the same reason a cargo loaded ship floats more in seas than in rivers. In dead seas, which contains about 270 gram of salt in one litre of water, a person does not sink, due to the great upthrust.

# Archimedes’s Principle

When a body is partially or wholly immersed in a liquid, it experiences an upthrust which is equal to the liquid displaced.

Proof:

Let us consider a cylinder of height h and cross sectional area A be completely immersed in a liquid of density as shown in the figure. The horizontal thrust on the cylinder balances each other because they are equal in magnitude but opposite in direction.

Let the top face M of the cylinder be at depth h1, and the bottom face N at depth h2 below the free surface of the liquid.

Now, force on M is

Or,

Also, force on N is

Or,

Thus, total upward force on the cylinder is,

Or,

Or,

Or,

The cylinder is wholly immersed in the liquid, so its volume is equal to the volume of displaced liquid. Also, is upthrust on the cylinder.

So,

Or.

Thus, upthrust on the cylinder is

Directly proportional to volume of the body

Directly proportional to the density of liquid

Directly proportional to the acceleration due to gravity.

Equation (iii) is called the mathematical form of Archimedes’s principle. Vd is the mass of the cylinder, so we have upthrust given by

As mg is weight of the body, upthrust is equal to weight of the body. This is Archenemies’ principle.

# Principle of floatation:

According to principle of floatation- A floatation body displaces liquid equal to its own weight.

# Hydrometer:

Hydrometer is a device which is used to measure the specific gravity/density of liquids.It is based on principle of floatation. When a hydrometer is placed in a liquid, under test, it sinks until it diplaces the liquid which is equal to its own weight.

The cylindrical bulb of the hydrometer is weighted in its lower side to make the hydrometer float vertically in liquids. Similarly, the stem is made narrow for greater sensitivity.

A specially designed constant weight hydrometer to test the purity of milk is called Lactometer.

Some Solved Problems

Q1. Calculate the pressure exerted by a mercury column of 76cm high at its bottom. Given that the density of mercury is 13600kg/m³ and g = 9.8m/sec².

Solution:

Given,

Height of mercury column (h) = 76cm = 0.76m

Density of mercury (d) = 13600kg/m³

Gravity (g) = 9.8m/sec²

Now,

Pressure (p) = hdg

= 0.76×13600×9.8

= 1.01×105 pa

Therefore, the pressure exerted by mercury column at the bottom is 1.01×105 pa.

Q2. A cube of wood of volume of 0.2m³ and density 600kg/m³ is placed in a liquid of density 800kg/m³. What fraction of the volume of the wood be immersed in the liquid?

Solution:

Given, Volume of cube of wood (v) = 0.2m³

Density of wood (dı) = 600kg/m³

Density of liquid (d) = 800kg/m³

Mass of wood (m) =

= 600×0.2

= 120kg

By principle of floatation,

Mass of wood = mass of liquid displaced

Therefore, mass of liquid displaced = 120kg.

Now,

Volume of liquid displaced (v) =

=

= 0.15m³

Now,

Volume of wood inside water = =

Therefore, part of the volume of the wood is immersed inside water.

Q3. A block of wood of mass 24kg floated in water. The volume of the wood is 0.032 m³. Find the volume of the block below the surface of the water and the density of the wood. (Density of water = 1000kg/m³)

Solution:

Given, Mass of the block of wood (m) = 24 kg.

Volume of wood (d) = 0.032 m³

Density of water (d) = 1000kg/m³

Now,

Density of wood (d) = = 750kg/m³

Therefore, the density of wood is 750kg/m³

By principle of floatation,

Mass of wood = mass of liquid displaced

Therefore, mass of liquid displaced = 24kg.

Volume of liquid displaced (v) = = 0.024m³

Therefore, volume of wood inside water =

Q4. A cube made of oak and of side 15cm floats in water with 10.5cm of its depth below the surface and with its sides vertical. What is the density of oak? (Density of water = 1000kg/m³).

Solution:

Let cross sectional area of wood = A and density = d

Therefore, volume =15Am³

So, mass of wood = dv

= d×15A

= 15Adkg

→ Mass of displaced .liquid = 15Adkg

Therefore, volume of displaced liquid = m/d

= 15Ad/1000

But volume of displaced liquid = A×10.5

Therefore, 15Ad/1000 = A×10.5

D = 10.5×1000/15 = 700kg/m³

Q5. The depth of water in a rectangular tank is 6 m, find pressure exerted by water at the bottom of it. (g = 10m/s², density of water = 1000kg/m³)

Solution:

Given, Height of the liquid column (h) = 6m

Acceleration due to gravity (g) = 10m/s²

Density of water (d) = 1000kg/m³

Now,

Pressure (p) = hdg

= 6×1000×10

= 6×104 pa

Therefore, the pressure exerted by the water in the rectangular tank is 6×104 pa.

Q6. The weight of a piece of stone when immersed in water is 18N and it displaces 4N of water. What is the weight of a stone in air?

Solution:

Given,

Weight of stone in water (w) = 18N

Weight of displaced liquid (v) = 4N

Weight of stone in air (w) =?

We have,

W = w +v

= 18+ 4

= 22N

Therefore, the weight of a piece of stone in air is 22N.

Q7. A circular well has depth 5m. Water level is below 2m from the upper face of the wall. Find the pressure at its base? (g = 9.8m/s²)

Solution:

Given, Total height of well = 5m

Actual height of liquid column (h) = 5-2 = 3m

Acceleration due to gravity (g) =9.8m/s²

Density of water (d) = 1000kg/m³

Pressure (p) =?

We have,

P = hdg

= 3×1000×9.8

= 2.9×104 pa

Therefore, the pressure exerted at the bottom of the well is 2.9×104 pa.

Q8. Calculate the pressure exerted by a mercury column of height 75cm at its bottom. Give that the density of mercury is 13600kg/m³ and g = 10m/s².

Solution:

Given, Height of mercury column (h) = 75cm = 0.75m

Density of mercury (d) = 13600kg/m³

Pressure (p) =?

Acceleration due to gravity (g) = 10m/s²

We have,

P = hdg

= 0.75 13600×10

= 1.01 ×105

Therefore, the pressure exerted by a mercury column is 1.01×105 pa.

Q9. Calculate the mass of displaced water when30cm thick iceberg with surface area 1000cm² floats in water. (Density of ice = 0.9g/cm³ and density of water = 1gm/cm³)

Solution:

Given, Height of iceberg (h) = 30cm

Surface area (A) = 1000cm²

Volume of iceberg (V) = A × h =1000×30 =3 ×10000cm³

Density of iceberg (d) = 0.9gm/cc

Density of water (dı) = 1gm/cc

Now,

Mass of iceberg (m) = v×d

= 3×10000× 0.9

= 2.7 × 104 gm

By principle of floatation,

Mass of iceberg = mass of liquid displaced

Therefore, mass of liquid displaced = 2.7 × 104 gm= 27kg

So, the mass of liquid displaced is 27kg.

Q10. A rectangle body is dipped into water as shown in the figure. The upper or lower surface area of it is 2m²; find out the upthrust acted on it due to water. (The density of water is 1000kg/m³)

Solution:

Given,

Density of water (d) = 1000kg/m³

Surface area (A) = 2m²

Acceleration due to gravity (g) =9.8m/s²

Height (h0 =4m

Now,

U = hdgA = 4×1000×9.8×2 = 8000×9.8 = 78400N

Therefore, the upthrust is 78400N.

Some Reasonable Facts:

# An egg floats in salt water.

An egg floats in salt water because density of an egg is less than salt water. The salt solution exerts more upthrust on egg. So, when the weight of egg equals the upthrust, the egg floats on water.

# An iron nail sinks in water but a ship made of same material floats in water.

Bodies which are less dense than water float, those bodies which are denser than water sink. An iron nail sinks in water as weight of the piece of an iron nail is more than the weight of the water displaced by it or upthrust. On the other hand, a ship made of iron is hollow and contains air, so its average density is less than that of water. Because of this it displaces more water than its weight, causing the ship float in water.

# It is easier to pull a bucket of water from a well until it is inside the water but difficult when it is out of water.

It is easier to pull a bucket of water from a well until i t is inside the water but difficult when it is out of water because water exerts upthrust on the bucket and thus reduces the apparently weight of the bucket Thus force required to pull a bucket when immersed in water is smaller than the force for the same pull when it is out of water.

# Bucket of water is filled faster in the downstairs tap than the bucket in the upstairs tap.

Bucket of water is filled faster in the downstairs tap than the bucket in the upstairs tap because the pressure exerted by a liquid increases with depth.

# The blood pressure in human body is greater at the feet than at the brain.

The blood pressure in human body is greater at the feet than at brain because the pressure exerted by a liquid increase with depth .

# Deep – sea divers wear diving suits.

Deep sea divers wear diving suits because there is great pressure at the bottom of the sea, so to with stand the great pressure deep sea divers wear diving suits.

# Wooden sleepers are used below the rails.

Wooden sleepers are used below the rails because as the area of the rails is very small and the force applied on is more, there is the chance of sinking the rails due to pressure. So, in order to avoid the sinking of the rails, wooden sleepers are used below the rails.

# A steel belt is provided over the toothed wheels of a tank

A steel belt is provided over the toothed wheels of a tank in order to decrease the pressure exerted on the ground. If steel belt is not provided then the toothed wheel may sink on ground which may cause some difficulty in moving. Hence, if belt is provided pressure is less as it is distributed.

# Some Differences

# Archimedes’s principle and principle of floatation

Archimedes’s principle Principle of floatation

When a body is wholly or partially immersed in a liquid, it experiences an upthrust which is equal to the weight of liquid displaced. A body floats in a liquid only when the weight of the body is equal to the upthrust acting on it. This is called principle of floatation.

It can also be applied in gas. It can only be applied in liquid.

# Force and Pressure

Force Pressure

It is an external agency that changes or tends to change the state of motion or state of rest of a body in a straight line. Normal force per unit area acting on a surface is called pressure.

Its SI unit is Newton (N). Its SI unit is N/m² or Pascal (Pa).

It is a vector quantity. It is a scalar quantity.

It is measured by spring balance. It is measured by manometer, barometer and press guage,etc.

# Upthrust and Pressure

Upthrust Pressure

When a body is partially or wholly immersed in a liquid, the resultant force acted on it by the liquid is called up-thrust. Normal force per unit area acting on a surface is called pressure.

Its SI unit is Newton (N) Its SI unit Pascal (Pa).

# Exercise for Practice

Mention two other factors, besides acceleration due to gravity (g) on which pressure exerted by a liquid depends. If two water tanks, one (A) having a larger base area than the other (B), carry equal volume of water, explain whose bottom experiences more pressure.

According to the law of floatation, on what condition does a piece of wood float on water? If a piece of wood having a total volume of 1.6m3 and density of 800 kgm-3 floats on water of density 1000 kgm-3, calculate what part (fraction) of the wood is in water?

On what factors does pressure of liquid depend on? One of the two ladies with same weight is wearing pointed healed shoes and the other wearing flat shoes. Explain which one exerts more pressure on the ground.

State Archimedes' principle. The volume of an ice block is 24000 cm3and its density is 0.9 gm/cm3. How much part of it remains above the surface of water when it is kept in water? [Ans: 1/10 part]

As in the given figure, if two hydrometers are partially immersed in fresh water and salt water, which one is salt water in between 'A' and ‘B’? On which law does the working of hydrometer base? An iron nail sinks in the water but a huge ship made of iron floats on water, why?

State Pascal's law. Name equipment which works in the basis of Pascal's law and write its one use. Prove liquid pressure (P) = hdg.

What factors affect the pressure of liquid? What is change in pressure at the bottom of a drum filled with water if it is brought to Himalaya from Terai. Write with reason.

Name the instrument given in the diagram. What weight of water is displaced by the instrument? Give reason also. Name the law on which it is based. Give one use of this instrument.

State Pascal's law. What are the two factors that effect pressure of liquid? If a man has mass 60 kilogram and the area of his one foot is 150 cm2, what is pressure exerted by him on the ground when he stepped with one foot. [Ans: 39200 Pa]

Figure shows a hydrometer floating in a certain liquid. The hydrometer reads the density of the liquid 800 kgm-3 and the volume of the submerged part of the hydrometer is 5x 10 -5 m3. Calculate the weight of the hydrometer.

'Liquid transmits the pressure equally in all direction. 'Which law does this statement represent? Speed of flow of water from tap of upper floor is less than that from the tap of down floor, why? Prove that liquid pressure (P)=hdg

State Archemede's principle. An iron ball sinks in water but it floats on mercury, why?

Prove that P = dgh. A rectangular body is completely dipped in to water as shown in the figure. The upper or lower surface area of its 2m2. Find the upthrust acted upon it due to water. (density of water is 1000 kg/m 3) [Ans.78400N]

what is a hydrometer? Why does a hydrometer have a heavy bulb and a narrow stem? On which principle does it work?

Write down any two factors on which the upthrust depends? Two objects having same volume are shown in the figure. Which one experiences move upthrust? Describe with reason.

How does the hydrometer remain upright in the liquid? A 4 cm long glass rod weighing 10g weighs 7 g in water of density 1 g/cm 3. Calculate the density of glass and area of cross-section of the glass rod. [Ans: 3.33 gm/cm3, 0.75cm2]

In which law is the given experiment based? State this law. Prove that P hdg.

What is liquid pressure? Name a factor on which liquid pressure depends. Prove that upthrust = vdg (Where v= volume, d= density, and g=acceleration due to gravity).

Why does the hydrometer sink less in the liquid with high density? State the law of flotation. An object with dimensions 24cm x 18cm x12cm is dropped into a vessel containing water. If 3cm height of the object is seen above the water surface, calculate the density of the object and weight of the water displaced. [Ans: 0.75 gm/cm3, 3.88kg]

State the Pascal's law. An iceberg floats in the sea water. The density of water is 1030 kg/m3 and that of the iceberg is 920kg/m3. What is the percentage by volume of iceberg is lying outside the water? [Ans: 89.32%]

Write the relationship between upthrust and density of a liquid. Two rectangular water tanks, one (A) having a larger base area than the other (B) carry equal volume of water. Explain whose bottom experiences more pressure?

In which law does the equipment given in figure works? State the law. Give a name of equipment, which works on this law and its one use.

Prove that the pressure of a certain liquid at a certain place depends only on its depth. Name equipment which is based on law of floatation. An iron nail sinks in water but a huge ship made of iron floats on water, why?

Pressure is defined as the force acting per unit area of the surface.

From equation (1), we can say that pressure is inversely proportional to area provided the force that acts on the area is constant.

Pressure directly proportional to force provided the area on which the force acts is constant.

# Atmospheric Pressure

Pressure exerted by the atmospheric air is called atmospheric pressure. This is also termed as standard atmospheric pressure or normal pressure. The normal pressure is the pressure exerted by 760 mm long column of mercury at 0oC at sea level and at a altitude of 45o. It is equal to 1.01 105Nm-2.

# Some consequences of pressure

All cutting and piercing tools such as scissors, needles, knives, etc. have sharp edges. This is because they have very small areas of cutting or piercing edges, so a small force can create a very high pressure.

A sharp knife is more effective in cutting an object than a blunt knife. This is because the applied force is spread over a small area in the case of sharp knife but the force is spread over a larger area in the case of blunt knife. Hence, higher pressure is exerted on the object by the sharp knife than the blunt knife.

It is difficult for a man to walk on sand or muddy surface but a camel or an elephant walks easily on such surface even though camel or elephant is much heavier than man. This is because these animals have feet of larger area than the feet of the man. Due to large area, less pressure exerted is on the surface and walking is easier.

The foundation of a building is made wider than walls. This is because the walls of a building and its roof exert pressure on the wide area of foundation and this reduces pressure over earth’s surface and the building is saved from collapsing.

A girl with heeled pointed shoe would make deep impressions on the clay road than an elephant moving along he same road. This is because the elephant exerts a large amount of force over comparatively large area but the girl exerts force on the pointed heel that with less area, thus the pressure exerted by the girl is more than that of the elephant.

# Liquid Pressure

Consider a liquid of density ‘d’ is contained in a beaker of cross-sectional area ‘A’ Let ‘h’ be the depth of the liquid column from its free surface as shown in the figure.

Then, the volume of liquid = Area height

or , V = A h

The weight of the liquid column = Mass gravity

or, W = m g

= (v d)g [since d = , m= d]

W = A h d g V=A h]

This weight of the liquid column is the force acting on the area ‘A’, normally. Therefore,

W = F

or, P =

or,

Thus, pressure exerted by a liquid of density‘d’ on the base of the beaker is dgh. Therefore, the pressure exerted by liquid at rest depends on three factors:

Depth or height of the liquid column (h),

Density of the liquid (d) and

Acceleration due to gravity (g)

# Laws of liquid pressure

The pressure in liquid is directly proportional to the depth from the free surface of the liquid or the height of the liquid column.

The pressure at the same depth, in a liquid is the same in all directions.

The pressure of a liquid is independent of the shape of the container is which it is kept.

A liquid finds its own level.

The pressure at any point in a liquid depends on its density.

# Some consequences of liquid pressure

The pressure exerted by a liquid increases with depth:

This principle is used in the construction of dams. Dams in rivers and water canals are made with their base wide so that they can withstand greater pressure due to water flowing in the dams and water canals.

Similarly, a bucket can be filled faster at the bottom floor than at the top floor of a building, provided that the size of water taps are the same, due to this principle.

A liquid finds its own level:

This principle is used in the water supply system. Water supply tanks or water reservoir are constructed at a taller place so that water rises in buildings without using water pump.

The pressure at any point in a liquid acts in all the directions:

# Pascal’s law

This law is formulated by the French Scientist Blaise Pascal in 1647 AD who was born in 1623 AD.

“When pressure is applied on a liquid enclosed in a vessel, it is transmitted equally in all the directions.”

# Applications of Pascal’s law

Pascal’s law tells that if pressure at any point in a liquid is changed, there will be equal change in pressure at other points in the liquid. This fact is used in hydraulic machines such as hydraulic press, hydraulic brake, hydraulic garage lift, hydraulic cranes etc.

# Hydraulic garage lift

Hydraulic garage lift is a machine used to lift light vehicles in service stations during their services.

A hydraulic brake is a braking system that is used in heavy automobiles.

# Density and relative density

The density of a substance is defined as the mass per unit volume.

When the mass of a particular volume of substance is divided with the mass of the same volume of water at 4oC, it is called relative density of that substance.

Relative density is a simple ratio to that why it does not have any unit.

# Upthrust

When a body is partially or wholly immersed in a liquid, the resultant force acted on it by the liquid is called upthrust or buoyant force.

# Density of liquid and upthrust

The density of liquid directly influences Upthrust. As density of liquid increases the pressure given by the liquid, the upthrust is also affected by the density of the liquid.

It is found that the egg sinks in water but floats in saturated salt solution. It is due to the more density of salt solution, which applies more upthrust on the egg.

Due to the same reason a cargo loaded ship floats more in seas than in rivers. In dead seas, which contains about 270 gram of salt in one litre of water, a person does not sink, due to the great upthrust.

# Archimedes’s Principle

When a body is partially or wholly immersed in a liquid, it experiences an upthrust which is equal to the liquid displaced.

Proof:

Let us consider a cylinder of height h and cross sectional area A be completely immersed in a liquid of density as shown in the figure. The horizontal thrust on the cylinder balances each other because they are equal in magnitude but opposite in direction.

Let the top face M of the cylinder be at depth h1, and the bottom face N at depth h2 below the free surface of the liquid.

Now, force on M is

Or,

Also, force on N is

Or,

Thus, total upward force on the cylinder is,

Or,

Or,

Or,

The cylinder is wholly immersed in the liquid, so its volume is equal to the volume of displaced liquid. Also, is upthrust on the cylinder.

So,

Or.

Thus, upthrust on the cylinder is

Directly proportional to volume of the body

Directly proportional to the density of liquid

Directly proportional to the acceleration due to gravity.

Equation (iii) is called the mathematical form of Archimedes’s principle. Vd is the mass of the cylinder, so we have upthrust given by

As mg is weight of the body, upthrust is equal to weight of the body. This is Archenemies’ principle.

# Principle of floatation:

According to principle of floatation- A floatation body displaces liquid equal to its own weight.

# Hydrometer:

Hydrometer is a device which is used to measure the specific gravity/density of liquids.It is based on principle of floatation. When a hydrometer is placed in a liquid, under test, it sinks until it diplaces the liquid which is equal to its own weight.

The cylindrical bulb of the hydrometer is weighted in its lower side to make the hydrometer float vertically in liquids. Similarly, the stem is made narrow for greater sensitivity.

A specially designed constant weight hydrometer to test the purity of milk is called Lactometer.

Some Solved Problems

Q1. Calculate the pressure exerted by a mercury column of 76cm high at its bottom. Given that the density of mercury is 13600kg/m³ and g = 9.8m/sec².

Solution:

Given,

Height of mercury column (h) = 76cm = 0.76m

Density of mercury (d) = 13600kg/m³

Gravity (g) = 9.8m/sec²

Now,

Pressure (p) = hdg

= 0.76×13600×9.8

= 1.01×105 pa

Therefore, the pressure exerted by mercury column at the bottom is 1.01×105 pa.

Q2. A cube of wood of volume of 0.2m³ and density 600kg/m³ is placed in a liquid of density 800kg/m³. What fraction of the volume of the wood be immersed in the liquid?

Solution:

Given, Volume of cube of wood (v) = 0.2m³

Density of wood (dı) = 600kg/m³

Density of liquid (d) = 800kg/m³

Mass of wood (m) =

= 600×0.2

= 120kg

By principle of floatation,

Mass of wood = mass of liquid displaced

Therefore, mass of liquid displaced = 120kg.

Now,

Volume of liquid displaced (v) =

=

= 0.15m³

Now,

Volume of wood inside water = =

Therefore, part of the volume of the wood is immersed inside water.

Q3. A block of wood of mass 24kg floated in water. The volume of the wood is 0.032 m³. Find the volume of the block below the surface of the water and the density of the wood. (Density of water = 1000kg/m³)

Solution:

Given, Mass of the block of wood (m) = 24 kg.

Volume of wood (d) = 0.032 m³

Density of water (d) = 1000kg/m³

Now,

Density of wood (d) = = 750kg/m³

Therefore, the density of wood is 750kg/m³

By principle of floatation,

Mass of wood = mass of liquid displaced

Therefore, mass of liquid displaced = 24kg.

Volume of liquid displaced (v) = = 0.024m³

Therefore, volume of wood inside water =

Q4. A cube made of oak and of side 15cm floats in water with 10.5cm of its depth below the surface and with its sides vertical. What is the density of oak? (Density of water = 1000kg/m³).

Solution:

Let cross sectional area of wood = A and density = d

Therefore, volume =15Am³

So, mass of wood = dv

= d×15A

= 15Adkg

→ Mass of displaced .liquid = 15Adkg

Therefore, volume of displaced liquid = m/d

= 15Ad/1000

But volume of displaced liquid = A×10.5

Therefore, 15Ad/1000 = A×10.5

D = 10.5×1000/15 = 700kg/m³

Q5. The depth of water in a rectangular tank is 6 m, find pressure exerted by water at the bottom of it. (g = 10m/s², density of water = 1000kg/m³)

Solution:

Given, Height of the liquid column (h) = 6m

Acceleration due to gravity (g) = 10m/s²

Density of water (d) = 1000kg/m³

Now,

Pressure (p) = hdg

= 6×1000×10

= 6×104 pa

Therefore, the pressure exerted by the water in the rectangular tank is 6×104 pa.

Q6. The weight of a piece of stone when immersed in water is 18N and it displaces 4N of water. What is the weight of a stone in air?

Solution:

Given,

Weight of stone in water (w) = 18N

Weight of displaced liquid (v) = 4N

Weight of stone in air (w) =?

We have,

W = w +v

= 18+ 4

= 22N

Therefore, the weight of a piece of stone in air is 22N.

Q7. A circular well has depth 5m. Water level is below 2m from the upper face of the wall. Find the pressure at its base? (g = 9.8m/s²)

Solution:

Given, Total height of well = 5m

Actual height of liquid column (h) = 5-2 = 3m

Acceleration due to gravity (g) =9.8m/s²

Density of water (d) = 1000kg/m³

Pressure (p) =?

We have,

P = hdg

= 3×1000×9.8

= 2.9×104 pa

Therefore, the pressure exerted at the bottom of the well is 2.9×104 pa.

Q8. Calculate the pressure exerted by a mercury column of height 75cm at its bottom. Give that the density of mercury is 13600kg/m³ and g = 10m/s².

Solution:

Given, Height of mercury column (h) = 75cm = 0.75m

Density of mercury (d) = 13600kg/m³

Pressure (p) =?

Acceleration due to gravity (g) = 10m/s²

We have,

P = hdg

= 0.75 13600×10

= 1.01 ×105

Therefore, the pressure exerted by a mercury column is 1.01×105 pa.

Q9. Calculate the mass of displaced water when30cm thick iceberg with surface area 1000cm² floats in water. (Density of ice = 0.9g/cm³ and density of water = 1gm/cm³)

Solution:

Given, Height of iceberg (h) = 30cm

Surface area (A) = 1000cm²

Volume of iceberg (V) = A × h =1000×30 =3 ×10000cm³

Density of iceberg (d) = 0.9gm/cc

Density of water (dı) = 1gm/cc

Now,

Mass of iceberg (m) = v×d

= 3×10000× 0.9

= 2.7 × 104 gm

By principle of floatation,

Mass of iceberg = mass of liquid displaced

Therefore, mass of liquid displaced = 2.7 × 104 gm= 27kg

So, the mass of liquid displaced is 27kg.

Q10. A rectangle body is dipped into water as shown in the figure. The upper or lower surface area of it is 2m²; find out the upthrust acted on it due to water. (The density of water is 1000kg/m³)

Solution:

Given,

Density of water (d) = 1000kg/m³

Surface area (A) = 2m²

Acceleration due to gravity (g) =9.8m/s²

Height (h0 =4m

Now,

U = hdgA = 4×1000×9.8×2 = 8000×9.8 = 78400N

Therefore, the upthrust is 78400N.

Some Reasonable Facts:

# An egg floats in salt water.

An egg floats in salt water because density of an egg is less than salt water. The salt solution exerts more upthrust on egg. So, when the weight of egg equals the upthrust, the egg floats on water.

# An iron nail sinks in water but a ship made of same material floats in water.

Bodies which are less dense than water float, those bodies which are denser than water sink. An iron nail sinks in water as weight of the piece of an iron nail is more than the weight of the water displaced by it or upthrust. On the other hand, a ship made of iron is hollow and contains air, so its average density is less than that of water. Because of this it displaces more water than its weight, causing the ship float in water.

# It is easier to pull a bucket of water from a well until it is inside the water but difficult when it is out of water.

It is easier to pull a bucket of water from a well until i t is inside the water but difficult when it is out of water because water exerts upthrust on the bucket and thus reduces the apparently weight of the bucket Thus force required to pull a bucket when immersed in water is smaller than the force for the same pull when it is out of water.

# Bucket of water is filled faster in the downstairs tap than the bucket in the upstairs tap.

Bucket of water is filled faster in the downstairs tap than the bucket in the upstairs tap because the pressure exerted by a liquid increases with depth.

# The blood pressure in human body is greater at the feet than at the brain.

The blood pressure in human body is greater at the feet than at brain because the pressure exerted by a liquid increase with depth .

# Deep – sea divers wear diving suits.

Deep sea divers wear diving suits because there is great pressure at the bottom of the sea, so to with stand the great pressure deep sea divers wear diving suits.

# Wooden sleepers are used below the rails.

Wooden sleepers are used below the rails because as the area of the rails is very small and the force applied on is more, there is the chance of sinking the rails due to pressure. So, in order to avoid the sinking of the rails, wooden sleepers are used below the rails.

# A steel belt is provided over the toothed wheels of a tank

A steel belt is provided over the toothed wheels of a tank in order to decrease the pressure exerted on the ground. If steel belt is not provided then the toothed wheel may sink on ground which may cause some difficulty in moving. Hence, if belt is provided pressure is less as it is distributed.

# Some Differences

# Archimedes’s principle and principle of floatation

Archimedes’s principle Principle of floatation

When a body is wholly or partially immersed in a liquid, it experiences an upthrust which is equal to the weight of liquid displaced. A body floats in a liquid only when the weight of the body is equal to the upthrust acting on it. This is called principle of floatation.

It can also be applied in gas. It can only be applied in liquid.

# Force and Pressure

Force Pressure

It is an external agency that changes or tends to change the state of motion or state of rest of a body in a straight line. Normal force per unit area acting on a surface is called pressure.

Its SI unit is Newton (N). Its SI unit is N/m² or Pascal (Pa).

It is a vector quantity. It is a scalar quantity.

It is measured by spring balance. It is measured by manometer, barometer and press guage,etc.

# Upthrust and Pressure

Upthrust Pressure

When a body is partially or wholly immersed in a liquid, the resultant force acted on it by the liquid is called up-thrust. Normal force per unit area acting on a surface is called pressure.

Its SI unit is Newton (N) Its SI unit Pascal (Pa).

# Exercise for Practice

Mention two other factors, besides acceleration due to gravity (g) on which pressure exerted by a liquid depends. If two water tanks, one (A) having a larger base area than the other (B), carry equal volume of water, explain whose bottom experiences more pressure.

According to the law of floatation, on what condition does a piece of wood float on water? If a piece of wood having a total volume of 1.6m3 and density of 800 kgm-3 floats on water of density 1000 kgm-3, calculate what part (fraction) of the wood is in water?

On what factors does pressure of liquid depend on? One of the two ladies with same weight is wearing pointed healed shoes and the other wearing flat shoes. Explain which one exerts more pressure on the ground.

State Archimedes' principle. The volume of an ice block is 24000 cm3and its density is 0.9 gm/cm3. How much part of it remains above the surface of water when it is kept in water? [Ans: 1/10 part]

As in the given figure, if two hydrometers are partially immersed in fresh water and salt water, which one is salt water in between 'A' and ‘B’? On which law does the working of hydrometer base? An iron nail sinks in the water but a huge ship made of iron floats on water, why?

State Pascal's law. Name equipment which works in the basis of Pascal's law and write its one use. Prove liquid pressure (P) = hdg.

What factors affect the pressure of liquid? What is change in pressure at the bottom of a drum filled with water if it is brought to Himalaya from Terai. Write with reason.

Name the instrument given in the diagram. What weight of water is displaced by the instrument? Give reason also. Name the law on which it is based. Give one use of this instrument.

State Pascal's law. What are the two factors that effect pressure of liquid? If a man has mass 60 kilogram and the area of his one foot is 150 cm2, what is pressure exerted by him on the ground when he stepped with one foot. [Ans: 39200 Pa]

Figure shows a hydrometer floating in a certain liquid. The hydrometer reads the density of the liquid 800 kgm-3 and the volume of the submerged part of the hydrometer is 5x 10 -5 m3. Calculate the weight of the hydrometer.

'Liquid transmits the pressure equally in all direction. 'Which law does this statement represent? Speed of flow of water from tap of upper floor is less than that from the tap of down floor, why? Prove that liquid pressure (P)=hdg

State Archemede's principle. An iron ball sinks in water but it floats on mercury, why?

Prove that P = dgh. A rectangular body is completely dipped in to water as shown in the figure. The upper or lower surface area of its 2m2. Find the upthrust acted upon it due to water. (density of water is 1000 kg/m 3) [Ans.78400N]

what is a hydrometer? Why does a hydrometer have a heavy bulb and a narrow stem? On which principle does it work?

Write down any two factors on which the upthrust depends? Two objects having same volume are shown in the figure. Which one experiences move upthrust? Describe with reason.

How does the hydrometer remain upright in the liquid? A 4 cm long glass rod weighing 10g weighs 7 g in water of density 1 g/cm 3. Calculate the density of glass and area of cross-section of the glass rod. [Ans: 3.33 gm/cm3, 0.75cm2]

In which law is the given experiment based? State this law. Prove that P hdg.

What is liquid pressure? Name a factor on which liquid pressure depends. Prove that upthrust = vdg (Where v= volume, d= density, and g=acceleration due to gravity).

Why does the hydrometer sink less in the liquid with high density? State the law of flotation. An object with dimensions 24cm x 18cm x12cm is dropped into a vessel containing water. If 3cm height of the object is seen above the water surface, calculate the density of the object and weight of the water displaced. [Ans: 0.75 gm/cm3, 3.88kg]

State the Pascal's law. An iceberg floats in the sea water. The density of water is 1030 kg/m3 and that of the iceberg is 920kg/m3. What is the percentage by volume of iceberg is lying outside the water? [Ans: 89.32%]

Write the relationship between upthrust and density of a liquid. Two rectangular water tanks, one (A) having a larger base area than the other (B) carry equal volume of water. Explain whose bottom experiences more pressure?

In which law does the equipment given in figure works? State the law. Give a name of equipment, which works on this law and its one use.

Prove that the pressure of a certain liquid at a certain place depends only on its depth. Name equipment which is based on law of floatation. An iron nail sinks in water but a huge ship made of iron floats on water, why?

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